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Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation

  • Ting Luo EMAIL logo
Published/Copyright: February 1, 2024
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H 1 ( R ) energy space according to the modulation argument.

Keywords: orbital stability; modified Camassa-Holm-Novikov equation; shallow water equation; multi-peakons
MSC 2010: 35B35; 35G25

1 Introduction

In this article, we consider the modified Camassa-Holm-Novikov (mCH-Novikov) equation arising from the water wave theory with constant vorticity [5], namely,

(1.1) m t + k 1 ( ( u 2 − u x 2 ) m ) x + k 2 ( u 2 m x + 3 u u x m ) = 0 , t > 0 , x ∈ R ,

where m = u − u x x , u ( t , x ) is related to the horizontal velocity of the fluid: k 1 and k 2 are two arbitrary constants. This model was derived by Chen et al. [5] from the Euler equation for an incompressible fluid with simple bottom and surface conditions. Indeed, it is obtained with the application of suitable scaling, the truncation of the asymptotic expansions of the unknowns to an appropriate order, and the introduction of a special Kodama transformation. With the property of transport equation, the local well-posedness of the mCH-Novikov equation is given in in the study by Mi et al. [20]. It is shown in the study by Chen et al. [4] that this model possesses a peaked traveling wave solution, which is orbital stable in both H 1 ( R ) and H 1 ( R ) ∩ W 1 , 4 ( R ) . Moreover, a blow-up criterion is given in in the study by Chen et al. [5] by using the sign preservation of the momentum density and a Riccati-type inequality.

It is observed that the mCH-Novikov equation (1.1) becomes the modified Camassa-Holm (mCH) equation

(1.2) m t + ( ( u 2 − u x 2 ) m ) x = 0 , t > 0 , x ∈ R ,

if k 1 = 0 and k 2 = 1 , which can be regarded as the dual integrable nonlinear model of the Camassa-Holm (CH) equation [1,2]. It was originally proposed by Fuchssteiner and Fokas [10] from symplectic structures and admits Lax pair [24] and bi-Hamiltonian structure [22], i.e.,

(1.3) m t = J ˜ 1 δ ℱ ˜ δ m = J ˜ 2 δ ℰ ˜ δ m ,

where

(1.4) J ˜ 1 = ∂ x 3 − ∂ x , J ˜ 2 = − ∂ x m ∂ x − 1 m ∂ x

are two compatible Hamiltonian operators and

(1.5) ℰ ˜ = ∫ m u d x , ℱ ˜ = ∫ u 4 + 2 u 2 u x 2 − 1 3 u x 4 d x .

are two conserved quantities. Similar to the CH equation, some new features are captured by the mCH equation, which has attracted the attention of numerous mathematicians and physicians in the past decades, including the wave breaking phenomena, blow-up criteria, and peaked solitary wave [7,8,27,28] that differed from the traditional smooth wave pattern. Indeed, the mCH equation possesses the single peakon [12]

(1.6) φ ˜ c ( x − c t ) = 3 2 c e − ∣ x − c t ∣ , c > 0 .

and multipeakons

(1.7) u ( t , x ) = ∑ i = 1 N p i ( t ) e − ∣ x − q i ( t ) ∣ ,

where p i ( t ) and q i ( t ) satisfy the corresponding Hamiltonian system [14]. It also possesses a train of infinite many peaked solitary waves [3]. In the study by Qu et al. [25], with the construction of certain Lyapunov functionals, the single peakon soliton of the mCH equation is orbital stable under small perturbations in H 1 energy space. Li and Liu also establish the orbital stability of peakons in the Sobolev space H 1 ∩ W 1 , 4 [17] using the variational approach without the assumption on the positive momentum density initially. Inspired by the work of Qu et al. [25], the stability of ordered trains of peakons is proved by constructing two suitable functionals related to the two conserved quantities [18].

On the other hand, taking k 1 = 0 and k 2 = 1 , the mCH-Novikov equation reduces to the Novikov equation

(1.8) m t + u 2 m x + 3 u u x m = 0 , t > 0 , x ∈ R ,

which is another well-known CH-type equation with cubic nonlinearity. The Novikov equation was originally proposed from the classification of generalized CH-type equations, which possess infinite hierarchies of higher symmetries in [21]. It can also be derived from the shallow water wave theory [5]. Note that the Novikov equation also carries a bi-Hamiltonian structure, which can be written as follows:

(1.9) m t = J ˆ 1 δ ℱ ˆ δ m = J ˆ 2 δ ℰ ˆ δ m ,

where

(1.10) J ˆ 1 = − ∂ x m ∂ x − 1 m ∂ x , J ˆ 2 = ( 1 − ∂ x 2 ) 1 m ∂ x 1 m ( 1 − ∂ x 2 )

are two compatible Hamiltonian operators and

(1.11) ℰ ˆ = ∫ m u d x , ℱ ˆ = 1 6 ∫ u m ∂ − 1 m ( ∂ x 2 − 1 ) − 1 ( u 2 m x + 3 u u x m ) d x

are two conserved quantities. The integrability, well-posedness, blow-up criteria, and existence of peakons to this model have been investigated in [13,26,29–31]. More specifically, the single peaked solution to the Novikov equation was derived in [11,15,16], i.e.,

(1.12) φ ˆ c ( x − c t ) = c e − ∣ x − c t ∣ , c > 0 ,

due to the interaction between the nonlinear and dispersion terms. The multipeakon solution is given by Hone et al. with the inverse scattering approach in [15]. Using the conservation laws, the orbital stability of peakons for the Novikov equation in the sense of the energy space H 1 norm is given in the study by Liu et al. [19]. Moreover, by using the modulation argument, the exponential decay property of H 1 -localized solutions, and the rigidity property, the asymptotic stability for the Novikov peakon is given in the studies by Chen et al. [6] and Palacios [23].

As a successor to the mCH equation and the Novikov equation, equation (1.1) also enjoys the peaked traveling wave solution

(1.13) u ( t , x ) = φ c ( x − c t ) ≔ a e − ∣ x − c t ∣ ,

where a is related to c , k 1 , and k 2 . A direct computation shows that the mCH-Novikov equation retains a linear superposition of N -peaked solitary wave solutions with restrictions on the position functions and the amplitude functions [4]. Thus, inspired by the aforementioned work, we are wondering the dynamical properties of the mCH-Novikov solitary wave solution has how much in common with the mCH equation and the Novikov equation. More precisely, we are interested in the orbital stability issue on the wave profiles under small perturbation for the mCH-Novikov (equation (1.1), which has significant physical implication. In the present study, the idea for investigating the mCH-Novikov-peaked solitary wave solution developed from [9] and [18], and hence we will recall the approach briefly and reveal the main challenges.

The general framework for the proof of the stability of the N -peaked solitary waves has three principal ingredients: the first one is the modulation argument, which provides the orbits of each traveling wave following the implicit function theorem; the second one is the almost monotonicity of the functionals relating to the conservation laws that describe the energy at the right of i th bump, for i = 1 , 2 , … , N ; and the third one is the local and global estimates that connect the local maximum value of u with localized energy functionals ℰ i and ℱ i .

The case for the mCH-Novikov equation is considerably similar to the mCH equation and the Novikov equation. However, note that the expression of the mCH-Novikov equation contains interaction between those cubic nonlinearities. This presents a series of obstacles when applying Dika and Molinet’s method. The conservation laws of the mCH-Novikov equation include

ℰ ( u ) = ∫ R ( u 2 + u x 2 ) d x , ℱ ( u ) = ∫ R ( u 4 + 2 u 2 u x 2 − 1 3 u x 4 ) d x ,

which are exactly the same as two conserved quantities in mCH equation. Using these two conservation laws may still be able to give the local and global estimates. However, it is not clear whether the appearance of the interaction between those cubic nonlinearities ( ( u 2 − u x 2 ) m ) x and ( u 2 m x + 3 u u x m ) with coefficients k 1 and k 2 would fit the framework of the modulation argument that generates the orbits, and carry the same almost monotonicity of the localized energy functionals or not. Moreover, the existence of a peaked traveling wave solution depends on the parameters k 1 and k 2 , so prior to the stability of the N -peaked solitary waves, one shall place restrictions on these two parameters.

We now explain the idea and the strategy adopted here to overcome the above difficulties. To prove the orbital stability of the trains of ordered N -peaked solitary waves to the mCH-Novikov equation has the flavor of the Lyapunov sense. In order to apply the modulation argument, we have to estimate the ∫ R ( r ˜ ˙ j − c j ) ∂ x R j ⋅ ∂ x R j d x . Combining all the like terms, using the properties of the kernel function and a subtle analysis, we fulfill the first gap mentioned above. On the other hand, with the help of weighted functions and the tricks adopted in both the mCH equation, and Novikov equation, we achieve the almost monotonicity for the mCH-Novikov equation. This combining with local and global estimates implies the stability result according to the idea originated in [9] and [18].

Now, we are ready to state the orbital stability result on the ordered N -peaked solitary waves ∑ i = 1 N φ c i ( x − c i t ) = ∑ i = 1 N a i e − ∣ x − c i t ∣ (where c i > 0 is the wave speed and a i > 0 as defined in Proposition 2.3) of equation (1.1) in energy space H 1 ( R ) with small perturbation.

Theorem 1.1

Consider k 1 > 0 and k 2 > 0 . Let u 0 ∈ H s ( R ) , where s > 5 ∕ 2 be the initial data of the mCH-Novikov equation and u ∈ C ( [ 0 , T ) ; H s ( R ) ) ∩ C 1 ( [ 0 , T ) ; H s − 1 ( R ) ) be the corresponding solution of the Cauchy problem for the mCH-Novikov equation (1.1). Suppose that c 1 , … , c N are N wave velocities such that 0 < c 1 < ⋯ < c N . Given κ 0 > 0 , R 0 > 0 , and A > 0 such that for some 0 < κ < κ 0 and y i 0 − y i − 1 0 ≥ R > R 0 , if u 0 satisfies

(1.14) u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ( R ) ≤ κ 2 ,

then there exist r 1 ∗ ( t ) , … , r N ∗ ( t ) ∈ R , such that

(1.15) sup 0 < t < T u ( t , ⋅ ) − ∑ i = 1 N φ c i ( ⋅ − r i ∗ ( t ) ) H 1 ( R ) ≤ A κ 1 ∕ 2 ,

where r i ∗ ( t ) − r i − 1 ∗ ( t ) > R ∕ 2 , i = 2 , … , N and T is the maximum existence time depending only on initial data u 0 .

The remainder of the article is organized as follows: in Section 2, the local well-posedness, the existence of peaked solitary waves, and the orbital stability of single peaked solitary wave solution for the mCH-Novikov equation are recalled; and then, efforts are devoted to the modulation arguments, the almost monotonicity, and the local and global estimates in Section 3, which prove the orbital stability of ordered trains of N -peaked solitary wave solution to the mCH-Novikov equation.

2 Preliminary

This section gives a brief review on the existence and stability of peaked solitary waves solution for the mCH-Novikov equation (1.1). Prior to a discussion of the stability issue should be the well-posedness result.

2.1 Local well-posedness and conservation laws

Consider the following Cauchy problem, namely,

(2.1) m t + k 1 ( ( u 2 − u x 2 ) m ) x + k 2 ( u 2 m x + 3 u u x m ) = 0 , m = u − u x x , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R .

Note that

(2.2) u = ( 1 − ∂ x 2 ) − 1 m = p ∗ m ,

where p ( x ) = 1 2 e − ∣ x ∣ is the kernel function for the operator ( 1 − ∂ x 2 ) − 1 and “ ∗ ” represents the convolution. With the application of the operator ( 1 − ∂ x 2 ) − 1 , the first equation in equation (2.1) can be rewritten as follows:

(2.3) u t + k 1 u 2 − 1 3 u x 2 u x + k 1 p x ∗ 2 3 u 3 + u u x 2 + 1 3 k 1 p ∗ u x 3 + k 2 u 2 u x + k 2 p x ∗ u 3 + 3 2 u u x 2 + 1 2 k 2 p ∗ u x 3 = 0 .

Now, we recall the following local well-posedness result without proof, which is rudiment of the stability theory established here. For details, refer to the study by Mi et al. [20].

Proposition 2.1

[20] If s > 5 2 and u 0 ∈ H s ( R ) , then there exists a time T > 0 such that the Cauchy problem (2.1) has a unique strong solution u ∈ C ( [ 0 , T ) ; H s ( R ) ) ∩ C 1 ( [ 0 , T ) ; H s − 1 ( R ) ) . Furthermore, the mapping u 0 ↦ u is continuous from a neighborhood of u 0 in H s ( R ) into u ∈ C ( [ 0 , T ) ; H s ( R ) ) ∩ C 1 ( [ 0 , T ) ; H s − 1 ( R ) ) .

As aforementioned, the conservation laws play an essential role while proving the stability result. Hence, we also recall the following result.

Proposition 2.2

[4] For the strong solutions u obtained in Proposition 2.1, the following functionals

(2.4) ℰ ( u ) = ∫ R ( u 2 + u x 2 ) d x , ℱ ( u ) = ∫ R ( u 4 + 2 u 2 u x 2 − 1 3 u x 4 ) d x

are conserved with respect to time t.

Remark 2.1

Even though the mCH-Novikov equation does not admit conserved total mass, with the observation of the corresponding characteristic equation, it still enjoys the sign-persistence property. That is, if the initial data m 0 = u 0 − u 0 , x x ≥ 0 (or ≤ 0 ), then the corresponding solution satisfies m ( t , x ) = u ( t , x ) − u x x ( t , x ) ≥ 0 (or ≤ 0 ), which implies that ∣ u x ∣ ≤ ∣ u ∣ .

Moreover, we recall the existence of the peaked solution to equation (1.1), which discovers the relationship between wave amplitude and wave speed.

Proposition 2.3

[4] The function

(2.5) u ( t , x ) = φ c ( x − c t ) ≔ a e − ∣ x − c t ∣

is a peaked solution to equation (2.3), provided that

  1. 2 k 1 + 3 k 2 ≠ 0 , 3 c 2 k 1 + 3 k 2 > 0 , and a = ± 3 c 2 k 1 + 3 k 2 ≠ 0 ; or

  2. 2 k 1 + 3 k 2 = c = 0 and a ≠ 0 .

A direct computation reveals that

(2.6) ℰ ( φ c ) = 2 a 2 , ℱ ( φ c ) = 4 3 a 4 .

Note that, due to the emergence of singularity, the peaked solitary wave solution should be considered as a weak solution in the distribution sense.

Moreover, the H 1 -orbital stability of the single peaked wave is also given in the study by Chen et al. [4].

Proposition 2.4

[4] Let φ c ( x − c t ) = a e − ∣ x − c t ∣ be the peaked solutions given in Proposition 2.3. Assume that the initial data u 0 ∈ H s ( R ) , s > 5 2 . Then, φ c is H 1 -orbital stable in the following sense: there exists 0 < δ 0 ≪ 1 such that if

(2.7) ‖ u 0 − φ c ‖ H 1 < a δ , 0 ≤ δ ≤ δ 0 ,

then the corresponding solution u ( t , x ) to equation (1.1) satisfies

(2.8) sup t ∈ [ 0 , T ∗ ) ‖ u ( t , ⋅ ) − φ c ( ⋅ − r ∗ ( t ) ) ‖ H 1 < 2 ( 3 a + C ( u 0 ) 1 ∕ 4 ) δ 1 ∕ 4 ,

where r ∗ ( t ) is the point at which the solution u ( t , x ) attains its maximum and the constant

(2.9) C ( u 0 ) ≔ 2 2 a 3 ‖ u 0 x ‖ L ∞ 2 ‖ u 0 x ‖ L 2 .

Note that with a refined continuity argument, the H 1 -orbital stability of the single peaked solitary wave solution of the mCH-Novikov equation removes the sign preservation condition on the initial data. Moreover, the proof of this proposition is essentially based on the following four lemmas.

Lemma 2.1

[4] For any u ∈ H 1 ( R ) and z ∈ R , we have

(2.10) ℰ ( u ) − ℰ ( φ c ) = ‖ u − φ c ( ⋅ − z ) ‖ H 1 2 + 4 a ( u ( z ) − a ) .

Lemma 2.2

[4] Assume that u ∈ H s ( R ) , s > 5 2 .

(2.11) M ˜ = max x ∈ R { u ( t , x ) } , m ˜ = min x ∈ R { u ( t , x ) } .

  1. If M ˜ + m ˜ ≥ 0 , then ℱ ( u ) ≤ 4 3 M ˜ 2 ℰ ( u ) − 4 3 M ˜ 4 .

  2. If M ˜ + m ˜ ≤ 0 , then ℱ ( u ) ≤ 4 3 m ˜ 2 ℰ ( u ) − 4 3 m ˜ 4 .

Lemma 2.3

[4] Let u ∈ H s ( R ) , s > 5 2 , and assume that ‖ u − φ c ‖ H 1 < a δ , with 0 ≤ δ ≤ 1 . Then,

(2.12) ∣ ℰ ( u ) − ℰ ( φ c ) ∣ ≤ 4 a 2 δ ,

(2.13) ∣ ℱ ( u ) − ℱ ( φ c ) ∣ ≤ ( C ( u ) + 17 a 4 ) δ ,

where C ( u ) ≔ 2 2 a 3 ‖ u x ‖ L ∞ 2 ‖ u x ‖ L 2 .

Lemma 2.4

[4] Assume that u ( x ) ∈ H s ( R ) , s > 5 2 , which satisfies equations (2.12) and (2.13) with 0 ≤ δ ≪ 1 . Then, we have the following:

  1. If M ˜ ( t ) + m ˜ ( t ) ≥ 0 , then

    (2.14) ∣ M ˜ − a ∣ < 21 a 2 + 3 4 A 2 C ( u ) δ .

  2. If M ˜ ( t ) + m ˜ ( t ) < 0 , then

    (2.15) ∣ m ˜ + a ∣ < 21 a 2 + 3 4 A 2 C ( u ) δ .

These lemmas give a clue for proving the orbital stability of trains of N -ordered peaked solitary wave solutions.

3 Orbital stability of the sum of ordered trains of peaked solitary waves

On the one hand, solitons traveling with their own speed will finally be arranged in order. On the other hand, the soliton resolution conjecture says solutions to the dispersive equation with certain initial data should eventually resolve into a finite number of soliton plus a radiative term that goes to zero. Therefore, we mainly concern the orbital stability of the trains of N -ordered peaked solitary wave solution in the present article. Inspired by the strategy developed in the study by El Dika and Molinet [9], we carry out the framework of proof to Theorem 1.1 in the following.

For κ > 0 and R > 0 , the neighborhood of ∑ i = 1 N φ c i ( ⋅ ) with radius κ and spatial shifts r i satisfying r i − r i − 1 ≥ R , where i = 2 , … , N , is denoted as follows:

U ( κ , R ) = u ∈ H 1 ( R ) , inf r i − r i − 1 > R u − ∑ i = 1 N φ c i ( ⋅ − r i ) H 1 ( R ) < κ .

To prove the orbital stability of the ordered trains is equivalent to showing that there exist A > 0 , R 0 > 0 , and κ 0 > 0 such that for any R > R 0 , 0 < κ < κ 0 , if the initial data u 0 ∈ H s ( R ) ( s > 5 ∕ 2 ) belongs to U ( κ 2 , R ) , then for all t ∈ [ 0 , T ) , where T is the maximal existence time, the corresponding solution u ( t ) belongs to U ( A ( κ + R − 1 ∕ 8 ) , R ∕ 2 ) . We would like to point out that A is independent of time t .

Due to the fact that the mapping t ↦ u ( t ) from [ 0 , T ) to H s ( R ) is continuous, to prove Theorem 1.1 is sufficient to prove that if the initial data u 0 ∈ U ( κ 2 , R ) and if for some 0 < t * < T ,

(3.1) u ( t ) ∈ U ( A ( κ + R − 1 ∕ 8 ) , R ∕ 2 ) ∀ t ∈ [ 0 , t * ] ,

then

(3.2) u ( t * ) ∈ U A 2 κ + R − 1 8 , 2 R 3 .

Therefore, assume that equation (3.1) holds for some 0 < κ < κ 0 and R > R 0 , where A , κ 0 , and R 0 will be determined later, we are going to demonstrate equation (3.2) in the sequel.

3.1 Modulation argument

In the following lemma, with the application of orthogonal condition and conservation laws, we will show that the solution u of the mCH-Novikov equation satisfying equation (3.1) has the following properties: u ( t ) remains close to the sum of N -modulated peaked waves in H 1 ( R ) , where each peaked wave has no overlap for t ∈ [ 0 , t * ] . More precise, solution u can be decomposed into the following form:

u ( t , x ) = ∑ i = 1 N φ c i ( x − r ˜ i ( t ) ) + v ( t , x ) ,

where v ( t , x ) is an infinitesimal in H 1 -norm and orthogonal to ∂ x φ c j ( x − r ˜ j ( t ) ) in L 2 -norm for any t ∈ [ 0 , t * ] and j = 1 , 2 , … , N .

Lemma 3.1

Consider k 1 > 0 and k 2 > 0 . Let u 0 be the initial value to the mCH-Novikov equation.There exist κ 0 > 0 and R 0 > 0 depending on k 1 , k 2 , and { c i } i = 1 N . For arbitrary t ∈ [ 0 , t * ] , if 0 < κ < κ 0 , R > R 0 , u 0 ∈ U ( κ 2 , R ) and the corresponding solution u ( t ) satisfies

(3.3) u ( t ) ∈ U κ , R 2 ,

where 0 < t * < T , then there exist unique C 1 spatial functions r ˜ i ( t ) which maps from [ 0 , t * ] to R , ( i = 1 , … , N ) , such that defining v ( t , x ) by

v ( t ) = u ( t ) − ∑ i = 1 N φ ˜ c i ( t ) , w h e r e φ ˜ c i ( t , ⋅ ) = φ c i ( ⋅ − r ˜ i ( t ) ) ,

one has the orthogonal condition

(3.4) ∫ R v ( t ) ∂ x φ ˜ c i ( t ) d x = 0 ,

and the following estimates

(3.5) ‖ v ( t ) ‖ H 1 ( R ) ≤ O ( κ ) ,

(3.6) ∣ r ˜ ˙ i ( t ) − c i ∣ ≤ O ( κ ) + O ( R − 1 ) ,

(3.7) r ˜ i ( t ) − r ˜ i − 1 ( t ) ≥ 3 R 4 + ( c i − c i − 1 ) 2 t , i ≥ 2 .

Moreover, denote ℐ i ( t ) = [ x i ( t ) , x i + 1 ( t ) ] , with

(3.8) x 1 = − ∞ , x N + 1 = + ∞ a n d x i ( t ) = r ˜ i − 1 ( t ) + r ˜ i ( t ) 2 , i = 2 , … , N .

Let r i ∗ ( t ) be any point in the spatial line such that

(3.9) u ( t , r i ∗ ( t ) ) = max x ∈ ℐ i ( t ) u ( t , x ) , t ∈ [ 0 , t * ] , i = 1 , … , N ,

then there holds

(3.10) ∣ r i ∗ ( t ) − r ˜ i ( t ) ∣ = O ( 1 ) .

Proof

The standard modulation argument enable us to discover the translations of N -peaked solitary waves. Fix y = ( y 1 , … , y N ) ∈ R N with y i − y i − 1 > R ∕ 2 and set Φ y ( ⋅ ) = ∑ i = 1 N φ c i ( ⋅ − y i ) . For 0 < α 0 < 1 , we define the function

ℱ : ∏ i = 1 N ( − α 0 , α 0 ) × B H 1 ( Φ y , α 0 ) → R N ,

( f 1 , … , f N , u ) ⟼ ( ℱ 1 ( f 1 , … , f N , u ) , … , ℱ N ( f 1 , … , f N , u ) ) ,

where B H 1 ( Φ y , α 0 ) is the ball in H 1 ( R ) centered at Φ y with radius α 0 and

ℱ j ( f 1 , … , f N , u ) = ∫ R u − ∑ i = 1 N φ c i ( ⋅ − y i − f i ) ∂ x φ c j ( ⋅ − y j − f j ) d x .

To determine the spatial shift functions f i , one shall apply the implicit function theorem, which requires two preconditions satisfied. The first one is that function ℱ should be C 1 mapping. This can be proved by the dominated convergence theorem. The second condition is that the determinant of the matrix of the first partial derivatives at ( 0 , … , 0 , Φ y ) should be nonzero. For j = 1 , … , N ,

∂ ℱ j ∂ f j ( f 1 , … , f N , u ) = ∫ R u x − ∑ i = 1 , i ≠ j N ∂ x φ c i ( ⋅ − y i − f i ) ∂ x φ c j ( ⋅ − y j − f j ) d x ,

∂ ℱ j ∂ f i ( f 1 , … , f N , u ) = ∫ R ∂ x φ c i ( ⋅ − y i − f i ) ∂ x φ c j ( ⋅ − y j − f j ) d x , where i ≠ j .

A direct computation gives

∂ ℱ j ∂ f j ( 0 , … , 0 , Φ y ) = ‖ ∂ x φ c j ‖ L 2 ( R ) 2 ≥ a 1 2

and

∂ ℱ j ∂ f i ( 0 , … , 0 , Φ y ) = ∫ R ∂ x φ c i ( ⋅ − y i ) ⋅ ∂ x φ c j ( ⋅ − y j ) d x ≤ O e − R 4 .

Thus, for sufficiently large R 0 > 0 , given R > R 0 , we have

D ( f 1 , … , f N ) ℱ ( 0 , … , 0 , Φ y ) ≠ 0 .

Hence, with the application of the implicit function theorem we know, there exist 0 < β 0 < κ 0 and uniquely determined C 1 functions ( f 1 ( u ) , … , f N ( u ) ) mapping from B H 1 ( Φ y , β 0 ) to a neighborhood of ( 0 , … , 0 ) , such that

ℱ ( f 1 ( u ) , … , f N ( u ) , u ) = 0 ∀ u ∈ B H 1 ( Φ y , β 0 ) .

Furthermore, for 0 < β < β 0 and u ∈ B H 1 ( Φ y , β ) , then there exists K 0 > 0 such that

(3.11) ∑ i = 1 N ∣ f i ( u ) ∣ ≤ K 0 β ,

where K 0 and β 0 depend on k 1 , k 2 , c 1 , and R 0 . If u ∈ B H 1 ( Φ y , β 0 ) , take β 0 ≤ min { R 0 ∕ ( 8 K 0 ) , κ 0 } and set r ˜ i ( t ) = y i + f i , and one has

(3.12) r ˜ i ( u ) − r ˜ i − 1 ( u ) = y i − y i − 1 + f i ( u ) − f i − 1 ( u ) ≥ R 2 − 2 K 0 β 0 ≥ R 4 .

Therefore, at a fixed time t , for R > R 0 and 0 < κ < κ 0 , the modulation of u is in U ( κ , R ∕ 2 ) , which can be covered by

⋃ y ∈ R N , y i − y i − 1 > R 2 B H 1 ( Φ y , 2 ε ) .

In addition, the uniqueness in the implicit function theorem carries out the uniqueness of the modulation of u . Therefore, one shall be able to define the modulation of the solution u ( t ) to the mCH-Novikov equation if u ∈ U ( κ , R ∕ 2 ) , i.e.,

r ˜ i ( t ) = r ˜ i ( u ( t ) ) , v ( t ) = u ( t ) − ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) ,

where i = 1 , … , N , t ∈ [ 0 , t * ] and v satisfies the orthogonal condition ⟨ v , ∂ x φ ˜ c i ⟩ H − 1 , H 1 = 0 , which gives rise to equation (3.4).

Inspired by the translation r ˜ i ( t ) defined above and the estimates of f i ( u ( t ) ) in equation (3.11), it is obtained that

‖ v ( t ) ‖ H 1 ( R ) ≤ u ( t ) − ∑ i = 1 N φ c i ( ⋅ − y i ) H 1 ( R ) + ∑ i = 1 N ∥ φ c i ( ⋅ − y i ) − φ c i ( ⋅ − y i − f i ) ∥ H 1 ( R ) ≤ κ + ∑ i = 1 N ‖ φ c i ( ⋅ − y i ) ‖ H 1 ( R ) 2 + ‖ φ c i ( ⋅ − y i − f i ) ‖ H 1 ( R ) 2 − 2 ∫ R φ c i ( ⋅ − y i ) φ c i ( ⋅ − y i − f i ) d x − 2 ∫ R ∂ x φ c i ( ⋅ − y i ) ∂ x φ c i ( ⋅ − y i − f i ) d x 1 2 ≤ κ + 2 ∑ i = 1 N ∣ a i ∣ ⋅ ∣ f i ( u ) ∣ 1 2 ≤ O ( κ ) .

Attention is now turn to the rate of change for r ˜ i ( t ) . Recall that

(3.13) ∂ x 2 φ ˜ c j ( t ) = φ ˜ c j ( t ) − 2 a j δ ( r ˜ j ( t ) ) .

For the orthogonality condition, taking derivative with respect to time t , we derive that

∫ R v t ∂ x φ ˜ c i − r ˜ ˙ i ⟨ ∂ x 2 φ ˜ c i , v ⟩ H − 1 , H 1 = 0 ,

which, when combined with equation (3.13), implies that

(3.14) ∫ R v t ∂ x φ ˜ c i ≤ ( r ˜ ˙ i − c i ) O ( ‖ v ‖ H 1 ( R ) ) + O ( ‖ v ‖ H 1 ( R ) ) .

On the other hand, substituting u by v + ∑ i = 1 N φ ˜ c i into the mCH-Novikov equation (2.3) leads to

v t + ∑ i = 1 N φ ˜ c i , t + k 1 v + ∑ i = 1 N φ ˜ c i 2 − 1 3 v x + ∑ i = 1 N φ ˜ c i , x 2 v x + ∑ i = 1 N φ ˜ c i , x + k 1 ( 1 − ∂ x 2 ) − 1 ∂ x 2 3 v + ∑ i = 1 N φ ˜ c i 3 + v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 + 1 3 k 1 ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 + k 2 v + ∑ i = 1 N φ ˜ c i 2 v x + ∑ i = 1 N φ ˜ c i , x + k 2 ( 1 − ∂ x 2 ) − 1 ∂ x v + ∑ i = 1 N R i 3 + 3 2 v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 + 1 2 k 2 ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 = 0 .

For each c i , φ ˜ c i is also a solution to the mCH-Novikov equation (2.3), which implies that

∂ t φ ˜ c i + ( r ˜ ˙ i − c i ) φ ˜ c i , x + k 1 φ ˜ c i 2 − 1 3 φ ˜ c i , x 2 φ ˜ c i , x + k 1 ( 1 − ∂ x 2 ) − 1 ∂ x 2 3 φ ˜ c i 3 + φ ˜ c i φ ˜ c i , x 2 + 1 3 k 1 ( 1 − ∂ x 2 ) − 1 φ ˜ c i , x 3 + k 2 φ ˜ c i 2 φ ˜ c i , x + k 2 ( 1 − ∂ x 2 ) − 1 ∂ x φ ˜ c i 3 + 3 2 φ ˜ c i φ ˜ c i , x 2 + 1 2 k 2 ( 1 − ∂ x 2 ) − 1 φ ˜ c i , x 3 = 0 .

Then, the aforementioned two identities infer that v satisfies the following condition on [ 0 , t * ] ,

v t − ∑ i = 1 N ( r ˜ ˙ i − c i ) ∂ x φ ˜ c i = − k 1 v + ∑ i = 1 N φ ˜ c i 2 − 1 3 v x + ∑ i = 1 N φ ˜ c i , x 2 v x + ∑ i = 1 N φ ˜ c i , x + k 1 ∑ i = 1 N φ ˜ c i 2 − 1 3 φ ˜ c i , x 2 φ ˜ c i , x − k 1 ( 1 − ∂ x 2 ) − 1 ∂ x 2 3 v + ∑ i = 1 N φ ˜ c i 3 + v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 + k 1 ( 1 − ∂ x 2 ) − 1 ∂ x 2 3 ∑ i = 1 N φ ˜ c i 3 + ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 − 1 3 k 1 ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 + 1 3 k 1 ( 1 − ∂ x 2 ) − 1 ∑ i = 1 N φ ˜ c i , x 3 − k 2 v + ∑ i = 1 N φ ˜ c i 2 v x + ∑ i = 1 N φ ˜ c i , x + k 2 ∑ i = 1 N φ ˜ c i 2 φ ˜ c i , x − k 2 ( 1 − ∂ x 2 ) − 1 ∂ x v + ∑ i = 1 N φ ˜ c i 3 + 3 2 v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 + k 2 ( 1 − ∂ x 2 ) − 1 ∂ x ∑ i = 1 N φ ˜ c i 3 + 3 2 φ ˜ c i φ ˜ c i , x 2 − 1 2 k 2 ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 + 1 2 k 2 ( 1 − ∂ x 2 ) − 1 ∑ i = 1 N φ ˜ c i , x 3 .

Then, taking L 2 -inner product with ∂ x φ ˜ c j , the above identity deduces

∫ R ( r ˜ ˙ j − c j ) ∂ x φ ˜ c j ⋅ ∂ x φ ˜ c j d x = ∫ R v t ⋅ ∂ x φ ˜ c j d x − ∫ R ∑ i = 1 , i ≠ j i = N ( r ˜ ˙ i − c i ) ∂ x φ ˜ c i ⋅ ∂ x φ ˜ c j d x + ( k 1 + k 2 ) ∫ R v + ∑ i = 1 N φ ˜ c i 2 v x + ∑ i = 1 N φ ˜ c i , x − ∑ i = 1 N φ ˜ c i 2 φ ˜ c i , x ∂ x φ ˜ c j d x + 1 3 k 1 ∫ R v x + ∑ i = 1 N φ ˜ c i , x 3 − ∑ i = 1 N φ ˜ c i , x 3 ∂ x φ ˜ c j d x + 2 3 k 1 + k 2 ∫ R ( 1 − ∂ x 2 ) − 1 ∂ x v + ∑ i = 1 N φ ˜ c i 3 − ∑ i = 1 N φ ˜ c i 3 ∂ x φ ˜ c j d x + k 1 + 3 2 k 2 ∫ R ( 1 − ∂ x 2 ) − 1 ∂ x v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 ∂ x φ ˜ c j d x + 1 3 k 1 + 1 2 k 2 ∫ R ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 − ∑ i = 1 N φ ˜ c i , x 3 ∂ x φ ˜ c j d x ≔ ∫ R v t ⋅ ∂ x φ ˜ c j d x − ∫ R ∑ i = 1 , i ≠ j N ( r ˜ ˙ i − c i ) ∂ x φ ˜ c i ⋅ ∂ x φ ˜ c j d x + V 1 + V 2 + V 3 + V 4 + V 5 ,

where the technique of integration by parts and the properties of exponential decay for both φ ˜ c i and its first order derivative are applied.

Now, the estimate of ∣ r ˜ j − c j ∣ is converted to the estimates of V 1 , V 2 , V 3 , V 4 , and V 5 . First, for V 1 , we know

(3.15) V 1 = ( k 1 + k 2 ) ∫ R v + ∑ i = 1 N φ ˜ c i 2 v x + ∑ i = 1 N φ ˜ c i , x − ∑ i = 1 N φ ˜ c i 2 φ ˜ c i , x ∂ x φ ˜ c j d x .

Let

(3.16) Q 1 = v + ∑ i = 1 N φ ˜ c i 3 − ∑ i = 1 N φ ˜ c i 3 .

Then, V 1 has the following form:

(3.17) V 1 = 1 3 ( k 1 + k 2 ) ∫ R Q 1 , x ∂ x φ ˜ c j d x = − 1 3 ( k 1 + k 2 ) ∫ R Q 1 ∂ x 2 φ ˜ c j d x .

With the identity (3.13), V 1 can be rewritten as follows:

V 1 = 1 3 ( k 1 + k 2 ) ∫ R 2 a j δ ( r ˜ j ( t ) ) Q 1 d x − ∫ R Q 1 φ ˜ c j d x = 1 3 ( k 1 + k 2 ) 2 a j Q 1 ( t , r ˜ j ( t ) ) − ∫ R Q 1 φ ˜ c j d x ,

which, when combined with the embedding formula

(3.18) ‖ v ‖ L ∞ ( R ) ≤ 2 2 ‖ v ‖ H 1 ( R ) ≤ O ( κ )

and the exponential decay of φ ˜ c j , gives rise to

(3.19) ∣ Q 1 ( t , x ) ∣ ≤ ( O ( κ ) + O ( 1 ) ) O ( κ ) + O e − R 8 ,

(3.20) ∫ R Q 1 ( t , x ) φ ˜ c j ( t ) d x ≤ ( O ( κ ) + O ( 1 ) ) O ( κ ) + O e − R 8 .

Hence,

V 1 ≤ ( O ( κ ) + O ( 1 ) ) O ( κ ) + O e − R 8 .

Second, for the estimate of V 2 , a direct expansion infers

V 2 = − 1 3 k 1 ∫ R v x 3 ∂ x φ ˜ c j d x + 3 ∫ R v x 2 ∂ x φ ˜ c j ∑ i = 1 N ∂ x φ ˜ c i d x + 3 ∫ R v x ∂ x φ ˜ c j ∑ i = 1 N ∂ x φ ˜ c i 2 d x + ∫ R ∑ i = 1 N ∂ x φ ˜ c i 3 − ∑ i = 1 N φ ˜ c i , x 3 ∂ x φ ˜ c j d x .

With the sign-persistence property of solution u and equation (3.18), we know

(3.21) ‖ v x ‖ L ∞ ≤ 2 2 ‖ v ‖ H 1 + O ( 1 ) ,

which deduces that

(3.22) 1 3 k 1 ∫ R v x 3 ∂ x φ ˜ c j d x ≤ a j ‖ v x ‖ L ∞ ∫ R v x 2 d x ≤ ( O ( κ ) + O ( 1 ) ) ‖ v ‖ H 1 2 .

In the similar fashion, with Holder’s inequality, we obtain

3 ∫ R v x 2 ∂ x φ ˜ c j ∑ i = 1 N ∂ x φ ˜ c i d x + 3 ∫ R v x ∂ x φ ˜ c j ∑ i = 1 N ∂ x φ ˜ c i 2 d x ≤ C ∫ R v x 2 d x + C ‖ v ‖ H 1 ≤ C ( ‖ v ‖ H 1 + 1 ) ‖ v ‖ H 1 ≤ ( O ( κ ) + O ( 1 ) ) O ( κ ) .

Moreover, using equation (3.12) and the exponential decay of ∂ x φ ˜ c i , there holds

∫ R ∑ i = 1 N ∂ x φ ˜ c i 3 − ∑ i = 1 N φ ˜ c i , x 3 ∂ x φ ˜ c j d x ≤ O e − R 8 + O ( κ ) .

Hence, we obtain

(3.23) V 2 ≤ O ( κ ) + O e − R 8 .

Third, with the expression of Q 1 and integration by parts, V 3 can be written in the following form:

V 3 = − 2 3 k 1 + k 2 ∫ R ( 1 − ∂ x 2 ) − 1 Q 1 ∂ x 2 φ ˜ c j d x = − 2 3 k 1 + k 2 ∫ R p ∗ Q 1 ( − 2 a j δ ( r ˜ j ( t ) ) + φ ˜ c j ( t ) ) d x = 2 a j 2 3 k 1 + k 2 p ∗ Q 1 ( t , r ˜ j ( t ) ) − 2 3 k 1 + k 2 ∫ R p ∗ Q 1 φ ˜ c j ( t ) d x .

The application of equations (3.19) and (3.20) infers that

(3.24) V 3 ≤ O ( κ ) + O e − R 8 .

For the estimation V 4 ,

V 4 = k 1 + 3 2 k 2 ∫ R ( 1 − ∂ x 2 ) − 1 ∂ x v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 ∂ x φ ˜ c j d x = − k 1 + 3 2 k 2 ∫ R ( 1 − ∂ x 2 ) − 1 v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 ∂ x 2 φ ˜ c j d x .

With the fact that ∂ x 2 φ ˜ c j ( t ) = φ ˜ c j ( t ) − 2 a j δ ( r ˜ j ( t ) ) , we have

V 4 = − k 1 + 3 2 k 2 ∫ R p ∗ v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 ( φ ˜ c j ( t ) − 2 a j δ ( r ˜ j ( t ) ) ) d x .

Denote that

Q 4 = v + ∑ i = 1 N φ ˜ c i v x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i ( ∂ x φ ˜ c i ) 2 = v + ∑ i = 1 N φ ˜ c i v x 2 + 2 v x ∑ i = 1 N φ ˜ c i , x + ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i ( ∂ x φ ˜ c i ) 2 = v v x 2 + 2 v v x ⋅ ∑ i = 1 N φ ˜ c i , x + v ⋅ ∑ i = 1 N φ ˜ c i , x 2 + v x 2 ⋅ ∑ i = 1 N φ ˜ c i + 2 v x ⋅ ∑ i = 1 N φ ˜ c i ∑ i = 1 N φ ˜ c i , x + ∑ i = 1 N φ ˜ c i ∑ i = 1 N φ ˜ c i , x 2 − ∑ i = 1 N φ ˜ c i φ ˜ c i , x 2 .

Then, with equation (3.12) and the exponential decay of φ ˜ c i , we deduce

∣ Q 4 ( t , x ) ∣ ≤ O ( κ ) ( ‖ v ‖ H 1 2 + 1 ) + ( O ( κ ) + 1 ) O ( κ ) + O e − R 8 ≤ O ( κ ) + O e − R 8 ,

which then gives rise to

(3.25) V 4 ≤ O ( κ ) + O e − R 8 .

For the estimate of V 5 , we know

V 5 = 1 3 k 1 + 1 2 k 2 ∫ R ( 1 − ∂ x 2 ) − 1 v x + ∑ i = 1 N φ ˜ c i , x 3 − ∑ i = 1 N φ ˜ c i , x 3 ∂ x φ ˜ c j d x .

Let Q 5 = v x + ∑ i = 1 N φ ˜ c i , x 3 − ∑ i = 1 N φ ˜ c i , x 3 . Similar to the process in the estimation of V 2 , we know

V 5 = 1 3 k 1 + 1 2 k 2 ∫ R p ∗ Q 2 ⋅ ∂ x φ ˜ c j d x ≤ O ( κ ) + O e − R 8 .

Hence, this deduces that

(3.26) V 5 ≤ O ( κ ) + O e − R 8 .

Note that the constants involving in V 1 , V 2 , V 3 , V 4 , and V 5 depend on k 1 , k 2 , and ( c i ) i = 1 N . On the other hand, note that by the inequality (3.12), the construction of the modulation guarantees ‖ r ˜ i ( t ) ‖ C 1 = O ( κ ) . With the application of equations (3.12) and (3.14) and the decay of ∂ x φ ˜ c i , we obtain that

a j 2 ∣ r ˜ ˙ j − c j ∣ ≤ ∣ v t ∂ x φ ˜ c j d x ∣ + ∑ i = 1 , i ≠ j N ( ∣ r ˜ ˙ i ∣ + c i ) ∫ R ∂ x φ ˜ c i ⋅ ∂ x φ ˜ c j d x + O ( κ ) + O e − R 8 ≤ O ( κ ) ∣ r ˜ ˙ j − c j ∣ + O ( κ ) + O e − R 8 .

Taking κ 0 small enough and R 0 large enough depending on k 1 , k 2 , and ( c i ) i = 1 N , we establish the estimate (3.6) for 0 < κ < κ 0 and R > R 0 .

Furthermore, for all t ∈ [ 0 , t * ] , together with the initial condition y i 0 − y i − 1 0 ≥ R and equation (3.11), there exists τ ∈ [ 0 , t ] such that

r ˜ j ( t ) − r ˜ j − 1 ( t ) = r ˜ j ( 0 ) − r ˜ j − 1 ( 0 ) + ( r ˜ ˙ j ( τ ) − r ˜ ˙ j − 1 ( τ ) ) t = r ˜ j ( 0 ) − r ˜ j − 1 ( 0 ) + ( ( r ˜ ˙ j ( τ ) − c j ) + ( c j − 1 − r ˜ ˙ j − 1 ( τ ) ) ) t + ( c j − c j − 1 ) t ≥ 3 R 4 + c j − c j − 1 2 t ,

which yields the estimate (3.7).

Inspired by equations (3.5) and (3.7), taking x = r ˜ i ( t ) implies that

∣ u ( t , r ˜ i ( t ) ) ∣ = ∣ a i ∣ + O ( κ ) + O e − R 4 ≥ 3 ∣ a i ∣ 4 .

If x ∈ [ r ˜ i ( t ) − R 4 , r ˜ i ( t ) + R 4 ] \ ( r ˜ i ( t ) − 2 , r ˜ i ( t ) + 2 ) , then

∣ u ( t , x ) ∣ ≤ ∣ a i ∣ e − 2 + O ( ε ) + O e − R 4 ≤ ∣ a i ∣ 2 ,

which infers that r i ∗ belongs to [ r ˜ i ( t ) − 2 , r ˜ i ( t ) + 2 ] , where u ( t , r i ∗ ( t ) ) = max x ∈ I i ( t ) u ( t , x ) . This proves equation (3.9), which then completes the proof of Lemma 3.1.□

3.2 Monotonicity property

In this subsection, with the application of two conservation laws and the weighted function, we will show the almost monotonicity of functionals that describe the energy at the right side of i th peaked wave of u , where i = 1 , … , N .

Let ϒ ∈ C ∞ ( R ) , which satisfies

(3.27) 0 < ϒ ( x ) < 1 , ϒ ′ ( x ) > 0 , x ∈ R , ∣ ϒ ′ ′ ′ ( x ) ∣ ≤ 10 ϒ ′ ( x ) , x ∈ [ − 1 , 1 ] ,

and

(3.28) ϒ ( x ) = e − ∣ x ∣ , x < − 1 , 1 − e − ∣ x ∣ , x > 1 .

With the weighted function ϒ K = ϒ ( ⋅ ∕ K ) , we carry out the functionals that describe the energy at the right side of i th peaked wave and is close to ‖ u ( t ) ‖ H 1 ( x > x j ( t ) ) , i.e.,

Λ j , K ( t ) = Λ j , K ( t , u ( t ) ) = ∫ R ( u 2 ( t ) + u x 2 ( t ) ) ϒ j , K ( t ) d x ,

where ϒ j , K ( t , x ) = ϒ K ( x − x j ( t ) ) with x j ( t ) , j = 1 , 2 , … , N defined in equation (3.8).

Now, we claim the following fundamental identity holds based on the property of the weighted functional.

Lemma 3.2

Assume that u 0 ( x ) is the initial data of the mCH-Novikov equation in H s ( R ) , with s > 5 2 . Let T > 0 be the maximal existence time of the corresponding strong solution u ( t , x ) . Then, for arbitrary smooth function Θ ( x ) , the following identity holds:

(3.29) d d t ∫ R ( u 2 + u x 2 ) Θ d x = k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 Θ ′ d x + ∫ R 2 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 − 2 u x 2 m Θ ′ d x + k 2 ∫ R u 2 u x 2 Θ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) Θ ′ d x .

Proof

By the local well-posedness result mentioned in Proposition 2.1, we know u ∈ C ( [ 0 , T ) , H s ( R ) ) ∩ C 1 ( [ 0 , T ) , H s − 1 ( R ) ) , for s > 5 2 . Following the density argument, we assume that u ( t , x ) is smooth. Taking the derivative to equation (2.3) with respect to x gives

(3.30) u t x = − k 1 u 2 − 1 3 u x 2 u x x − k 1 p x x ∗ 2 3 u 3 + u u x 2 − k 1 3 p x ∗ u x 3 − k 2 ( u 2 u x ) x − k 2 p x x ∗ u 3 + 3 2 u u x 2 − k 2 2 p x ∗ u x 3 = − k 1 + 1 2 k 2 u u x 2 + ( k 1 + k 2 ) u 2 u x x − k 1 ( u x 2 u x x ) − 2 3 k 1 + k 2 u 3 − 2 3 k 1 + k 2 p ∗ u 3 + k 1 + 3 2 k 2 p ∗ ( u u x 2 ) + k 1 3 + k 2 2 p x ∗ u x 3 .

Using integrating by parts gives

d d t ∫ R ( u 2 + u x 2 ) Θ d x = 2 ∫ R u m t Θ d x − 2 ∫ R u u t x Θ ′ d x ≔ K 1 + K 2 .

Adopting equation (1.1), one has

K 1 = 2 ∫ R u m t Θ d x = − 2 ∫ R u ( k 1 ( ( u 2 + u x 2 ) m ) x + k 2 ( u 2 m x + 3 u u x m ) ) Θ d x = 2 k 1 ∫ R u x ( u 2 − u x 2 ) m Θ d x + 2 k 1 ∫ R u ( u 2 − u x 2 ) m Θ ′ d x + 2 k 2 ∫ R 3 u 2 u x m Θ d x + 2 k 2 ∫ R u 3 m Θ ′ d x − 6 k 2 ∫ R u 2 u x m Θ d x = K 11 + K 12 + K 13 + K 14 + K 15 .

Moreover, it can be deduced from equation (3.30) that

K 2 = − 2 ∫ R u u t x Θ ′ d x = 2 ∫ R u k 1 + 1 2 k 2 u u x 2 + ( k 1 + k 2 ) u 2 u x x − k 1 ( u x 2 u x x ) − 2 3 k 1 + k 2 u 3 Θ ′ d x + 2 ∫ R u 2 3 k 1 + k 2 p ∗ u 3 + k 1 + 3 2 k 2 p ∗ ( u u x 2 ) + 1 3 k 1 + 1 2 k 2 p x ∗ u x 3 Θ ′ d x = k 1 ∫ R 2 u 2 u x 2 + 2 u 3 u x x − 2 u u x 2 u x x − 4 3 u 4 Θ ′ d x + k 1 ∫ R u p ∗ 4 3 u 3 + 2 u u x 2 Θ ′ d x + k 1 ∫ R u p x ∗ 2 3 u x 3 Θ ′ d x + k 2 ∫ R ( u 2 u x 2 + 2 u 3 u x x − 2 u 4 ) Θ ′ d x + k 2 ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + k 2 ∫ R u p x ∗ ( u x 3 ) ⋅ Θ ′ d x = K 21 + K 22 + K 23 + K 24 + K 25 + K 26 .

Observe that

K 11 + K 12 + K 21 + K 22 + K 23 = k 1 ∫ R 2 u x ( u 2 − u x 2 ) m w d x + ∫ R 2 u ( u 2 − u x 2 ) m ⋅ Θ ′ d x + ∫ R 2 u 2 u x 2 + 2 u 3 u x x − 2 u u x 2 u x x − 4 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 2 u u x 2 Θ ′ d x + ∫ R u ⋅ p x ∗ 2 3 u x 3 Θ ′ d x = k 1 2 u x ( u − u x x ) ( u 2 − u x 2 ) Θ d x + ∫ R 2 u ( u − u x x ) ( u 2 − u x 2 ) Θ ′ d x + ∫ R ( 2 u 2 u x 2 + 2 u 3 u x x − 2 u u x 2 u x x − 4 3 u 4 ) Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 2 u u x 2 Θ ′ d x + ∫ R u ⋅ p x ∗ 2 3 u x 3 Θ ′ d x = k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 Θ ′ d x + ∫ R ( 2 u 4 − 2 u 3 u x x − 2 u 2 u x 2 + 2 u u x 2 u x x ) Θ ′ d x + ∫ R 2 u 2 u x 2 + 2 u 3 u x x − 2 u u x 2 u x x − 4 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 2 u u x 2 Θ ′ d x + ∫ R u ⋅ p x ∗ 2 3 u x 3 Θ ′ d x = k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 Θ ′ d x + ∫ R 2 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 2 u u x 2 Θ ′ d x + ∫ R u ⋅ p ∗ ( − 2 u x 2 m + 2 u u x 2 ) Θ ′ d x = k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 Θ ′ d x + ∫ R 2 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 − 2 u x 2 m Θ ′ d x .

Similarly, substituting m = u − u x x into the expressions of K 1 and K 2 , we have

K 13 + K 14 + K 15 + K 24 + K 25 + K 26 = k 2 ∫ R 6 u 2 u x m w d x + ∫ R 2 u 3 m Θ ′ d x + ∫ R − 6 u 2 u x m w d x + ∫ R ( u 2 u x 2 + 2 u 3 u x x − 2 u 4 ) Θ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) Θ ′ d x = k 2 ∫ R 2 u 4 Θ ′ d x − ∫ R 2 u 3 u x x Θ ′ d x + ∫ R ( u 2 u x 2 + 2 u 3 u x x − 2 u 4 ) Θ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) Θ ′ d x = k 2 ∫ R u 2 u x 2 Θ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) Θ ′ d x .

Therefore, combining K 1 and K 2 , one has

d d t ∫ R ( u 2 + u x 2 ) Θ d x = k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 Θ ′ d x + ∫ R 2 3 u 4 Θ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 − 2 u x 2 m Θ ′ d x + k 2 ∫ R u 2 u x 2 Θ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) Θ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) Θ ′ d x ,

which proves equation (3.29).□

Consider the weighted function ϒ j , K defined in equations (3.27) and (3.28). For j = 2 , … , N , we introduce

Λ j , K ( t ) = ∫ R ( u 2 ( t , x ) + u x 2 ( t , x ) ) ϒ j , K ( t , x ) d x .

In the following lemma, we shall show that the functional Λ j , K ( t ) representing the energy at the right of the j − 1 th bump of the solution u ( t , x ) is almost decreasing with respect to time. In the sequel, denote

χ 0 = 1 4 min { c 1 , c 2 − c 1 , … , c N − c N − 1 } .

Lemma 3.3

Consider k 1 > 0 and k 2 > 0 . For s > 5 2 , given the initial data u ( 0 , x ) = u 0 ( x ) ∈ H s ( R ) . Suppose that u ( t , x ) is the corresponding strong solution of the mCH-Novikov equation. There exist κ 0 > 0 and R 0 > 0 depending on k 1 , k 2 , and ( c i ) i = 1 N , such that for 0 < κ < κ 0 and 0 < R 0 < R , if u 0 ( x ) satisfies

u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ( R ) ≤ κ 2

and u ( t , x ) satisfies equation (3.3) on [ 0 , t * ] , then, for each j ∈ { 2 , … , N } , t ∈ [ 0 , t * ] , and 4 ≤ K = O R 1 2 , there holds

(3.31) Λ j , K ( t ) − Λ j , K ( 0 ) ≤ C χ 0 ‖ u 0 ‖ H 1 ( R ) 4 e − R 8 K .

Proof

The proof of this lemma is carried out by the approach developed in the study by El Dika and Molinet [9]. Fix the index j ∈ { 2 , … , N } and take Θ = ϒ j , K in equation (3.29). For t ∈ [ 0 , t * ] , with the fact that d d t ϒ j , K ( t , x ) = − x ˙ j ( t ) ∂ x ϒ j , K ( t , x ) , one obtains

d d t Λ j , K ( t ) = d d t ∫ R ( u 2 + u x 2 ) ϒ j , K d x = − x ˙ j ( t ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K ( t , x ) + k 1 − 1 2 ∫ R ( u 2 + u x 2 ) 2 ϒ ′ d x + ∫ R 2 3 u 4 ϒ ′ d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 − 2 u x 2 m ϒ ′ d x + k 2 ∫ R u 2 u x 2 ϒ ′ d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) ϒ ′ d x + ∫ R u ⋅ p x ∗ ( u x 3 ) ϒ ′ d x .

Recall the bound of r ˙ ( t ) as equation (3.6) in Lemma 3.1. Then, for x i ( t ) = ( r ˜ i − 1 ( t ) + r ˜ i ( t ) ) ∕ 2 , we have the following estimate:

− x ˙ j ( t ) = − r ˜ ˙ j ( t ) − c j 2 − r ˜ ˙ j − 1 ( t ) − c j − 1 2 − c j + c j − 1 2 ≤ − c j − 1 + c j 2 + O ( κ ) + O ( R − 1 ) < − 1 2 c 1 .

Now, we are going to estimate d d t Λ j , K ( t ) . With the sign preservation of momentum density function m ( t , x ) and the fact that

∂ x ϒ j , K ( t , x ) = 1 K ϒ ′ x − x j ( t ) K > 0 ,

it is deduced that

(3.32) d d t Λ j , K ( t ) = − x ˙ j ( t ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K ( t , x ) d x + k 1 ∫ R − 1 2 ( u 2 − u x 2 ) 2 ∂ x ϒ j , K d x + ∫ R 2 3 u 4 ∂ x ϒ j , K d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 − 2 u x 2 m ∂ x ϒ j , K d x + k 2 ∫ R u 2 u x 2 ∂ x ϒ j , K d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) ∂ x ϒ j , K d x + ∫ R u ⋅ p x ∗ ( u x 3 ) ∂ x ϒ j , K d x ≤ − c 1 2 ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + k 1 ∫ R 2 3 u 4 ∂ x ϒ j , K d x + ∫ R u ⋅ p ∗ 4 3 u 3 + 4 u u x 2 ∂ x ϒ j , K d x ︸ I 1 + I 2 + k 2 ∫ R u 2 u x 2 ∂ x ϒ j , K d x + ∫ R u ⋅ p ∗ ( 2 u 3 + 3 u u x 2 ) ∂ x ϒ j , K d x + ∫ R u ⋅ p x ∗ ( u x 3 ) ∂ x ϒ j , K d x ︸ I 3 + I 4 + I 5

For further discussion, we divide R by R = D j ∪ D j c , where

D j = r ˜ j − 1 ( t ) + R 4 , r ˜ j ( t ) − R 4 .

One may note that if x ∈ D j c , then

∣ x − x j ( t ) ∣ ≥ 1 2 r ˜ j ( t ) − R 4 − r ˜ j − 1 ( t ) − R 4 ,

which, when combined with equation (3.7) and the definition of χ 0 , infers

∣ x − x j ( t ) ∣ ≥ χ 0 t + R 8 .

Moreover, with the fact that K = O ( R ) , one has

∣ x − x j ( t ) ∣ K ≥ χ 0 t + R 8 R > 1 ,

which together with the definition of ϒ gives rise to, for x ∈ D j c ,

∂ x ϒ j , K ( t , x ) = 1 K ϒ ′ x − x j ( t ) K ≤ 1 K e − 1 K χ 0 t + R 8 .

Therefore, with restriction on D j c , inspired by the conservation law ℰ ( u ) = ‖ u ‖ H 1 2 , one has

(3.33) ∫ D j c u 4 ∂ x ϒ j , K d x ≤ C K ‖ u 0 ‖ H 1 4 e − 1 K χ 0 t + R 8 .

On the other hand, for any x ∈ D j and the index i ∈ { 1 , … , N } , it is observed that

(3.34) ∣ x − r ˜ i ( t ) ∣ > R 4 .

Due to the decay property of φ c i ( ⋅ − r ˜ i ( t ) ) and the estimate (3.5), one has

(3.35) ‖ u ‖ L ∞ ( D j ) ≤ u ( t , ⋅ ) − ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) L ∞ ( D j ) + ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) L ∞ ( D j ) ≤ 2 2 u ( t , ⋅ ) − ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) H 1 ( R ) + ∑ i = 1 N ‖ φ c i ( ⋅ − r ˜ i ( t ) ) ‖ L ∞ ( D j ) ≤ O ( κ ) + O e − R 8 .

Hence, combining the estimates on two subregions together, i.e., (3.33) and (3.35), for 0 < κ < κ 0 ≪ 1 and R > R 0 ≫ 1 , one has

(3.36) I 1 = 2 3 ∫ D j c u 4 ∂ x ϒ j , K d x + 2 3 ∫ D j u 4 ∂ x ϒ j , K d x ≤ C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 + 2 3 ‖ u ‖ L ∞ 2 ∫ D j u 2 ⋅ ∂ x ϒ j , K d x ≤ C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 + c 1 16 ( k 1 + k 2 ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x .

Moreover, by the Sobolev embedding theorem and the properties of the kernel function, we know

∫ D j c u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x ≤ C sup x ∈ D j c ∣ ∂ x ϒ j , K ( t , x ) ∣ ⋅ ‖ u ‖ L ∞ ( R ) ∫ R p ∗ ( u 3 + u u x 2 ) d x ≤ C sup x ∈ D j c ∣ ∂ x ϒ j , K ( t , x ) ∣ ⋅ ‖ u ‖ L ∞ ( R ) 2 ∫ R p ∗ ( u 2 + u x 2 ) d x ≤ C sup x ∈ D j c ∣ ∂ x ϒ j , K ( t , x ) ∣ ⋅ ‖ u ‖ L ∞ ( R ) 2 ∫ R e − ∣ x ∣ d x ∫ R ( u 2 + u x 2 ) d x ≤ C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 ,

where σ 1 and σ 2 are arbitrary constants. According to the definition of the weighted function ϒ j , K ( t , x ) , we know

( 1 − ∂ x 2 ) ∂ x ϒ j , K ( t , x ) = ∂ x ϒ j , K ( t , x ) − 1 K 3 ϒ ′ ′ ′ x − x j ( t ) K ≥ 1 − 10 K 2 ∂ x ϒ j , K ( t , x ) .

Picking K ≥ 4 , the above inequality infers

( 1 − ∂ x 2 ) − 1 ∂ x ϒ j , K ( t , x ) ≤ 1 − 10 K 2 − 1 ∂ x ϒ j , K ( t , x ) ,

where ∂ x ϒ j , K ( t , x ) > 0 . Therefore, on the subregion D j , we have

∫ D j u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x ≤ C ‖ u ‖ L ∞ ( D j ) ∫ R ( u 3 + u u x 2 ) p ∗ ∂ x ϒ j , K d x ≤ C ‖ u ‖ L ∞ ( D j ) ‖ u ‖ H 1 ( R ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x ,

where σ 1 and σ 2 are arbitrary constants. With the estimate for v in equation (3.5) and the application of conservation law ℰ ( u ) , one has

‖ u ‖ H 1 ( R ) = ‖ u 0 ‖ H 1 ( R ) ≤ ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( 0 ) ) H 1 ( R ) + ‖ v ( 0 ) ‖ H 1 ( R ) ≤ ∑ i = 1 N ‖ φ c i ‖ H 1 ( R ) + O ( κ ) ≤ ∑ i = 1 N 2 a i + O ( κ ) ,

which along with equation (3.35) gives

∫ D j u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x ≤ O ( κ ) + O e − R 8 ⋅ ∑ i = 1 N 2 a i + O ( κ ) ⋅ ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x .

Thus, for 0 < κ < κ 0 ≪ 1 and R > R 0 ≫ 1 , we obtain

(3.37) ∫ R u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x = ∫ D j u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x + ∫ D j c u ( p ∗ ( σ 1 u 3 + σ 2 u u x 2 ) ) ∂ x ϒ j , K d x ≤ O ( κ ) + O e − R 8 ⋅ ∑ i = 1 N 2 a i + O ( κ ) ⋅ ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 .

Hence, we have

(3.38) I 2 ≤ O ( κ ) + O e − R 8 ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8

and

(3.39) I 4 ≤ O ( κ ) + O e − R 8 ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 ,

by choosing σ 1 = 4 3 , σ 2 = 4 and σ 1 = 2 , σ 2 = 3 in equation (3.37), respectively. With the same pattern, discussing the integrals in two subregions, one has

(3.40) I 3 = ∫ D j u 2 u x 2 ∂ x ϒ j , K d x + ∫ D j c u 2 u x 2 ∂ x ϒ j , K d x ≤ ‖ u ‖ L ∞ 2 ∫ D j u x 2 ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 ≤ c 1 16 ( k 1 + k 2 ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 4 e − 1 K χ 0 t + R 8 .

Moreover, by taking advantage of integration by parts and the properties of weighted function, we obtain

I 5 = − ∫ R u x ⋅ p ∗ ( u x 3 ) ⋅ ∂ x ϒ j , K d x − ∫ R u ⋅ p ∗ ( u x 3 ) ⋅ ∂ x 2 ϒ j , K d x ≤ C ∫ R u ( p ∗ u 3 ) ∂ x ϒ j , K d x + C K ∫ R u ( p ∗ u 3 ) ∂ x ϒ j , K d x ≤ C ( 1 + 1 K ) ‖ u ‖ L ∞ ∫ R ( p ∗ u 3 ) ∂ x ϒ j , K d x ≤ C ‖ u ‖ L ∞ 2 ∫ R u 2 ⋅ ∂ x ϒ j , K d x ,

which implies that

(3.41) I 5 ≤ C ‖ u ‖ L ∞ ( D j ) 2 ∫ D j u 2 ⋅ ∂ x ϒ j , K d x + C ‖ u ‖ L ∞ ( D j c ) 2 ∫ D j c u 2 ⋅ ∂ x ϒ j , K d x ≤ c 1 16 ( k 1 + k 2 ) ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 ( R ) 2 e − 1 K χ 0 t + R 8 .

Substituting equations (3.36) and (3.38)–(3.41) into equation (3.32), one has

d d t I j , k ( t ) ≤ − c 1 2 ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + c 1 4 ∫ R ( u 2 + u x 2 ) ∂ x ϒ j , K d x + C K ‖ u 0 ‖ H 1 4 e − 1 K χ 0 t + R 8

for each j ∈ { 2 , … , N } and any t ∈ [ 0 , t * ] with a positive constant C . Finally, with the application of the Gronwall’s inequality, we deduce

I j , k ( t ) − I j , k ( 0 ) ≤ C K ‖ u 0 ‖ H 1 4 ∫ 0 t e − 1 K χ 0 τ + R 8 d τ ≤ C χ 0 ‖ u 0 ‖ H 1 4 e − R 8 K ,

which completes the proof of Lemma 3.3.□

3.3 Local and global energy estimates

With the definition and properties of the weighted functions ϒ j , K ( t , x ) given by equations (3.27) and (3.28), we shall define the bump functions as follows:

Ψ 1 = 1 − ϒ 2 , K , Ψ N = ϒ N , K , Ψ i = ϒ i , K − ϒ i + 1 , K , i = 2 , … , N − 1 .

Note that for x ∈ R and t ∈ [ 0 , t ∗ ] ,

∑ i = 1 n Ψ i ( t , x ) = 1 .

Taking R ≫ 1 and K = R and combining with the exponential decay property of the weighted functions ϒ j , K ( t , x ) , we derive the following properties for the bump functions:

(3.42) ∣ 1 − Ψ i ∣ ≤ 4 e − R 4 K , r ˜ i − R ∕ 4 < x < r ˜ i + R ∕ 4 ∣ Ψ i ∣ ≤ 4 e − R 4 K , r ˜ j − R ∕ 4 < x < r ˜ j + R ∕ 4 , j ≠ i .

With the aforementioned properties of bump functions Ψ i ( t , x ) , we shall introduce the following localized conservation laws of ℰ i and ℱ i , where

ℰ i ( t ) = ∫ R ( u 2 + u x 2 ) Ψ i ( t ) d x , ℱ i ( t ) = ∫ R u 4 + 2 u 2 u x 2 − 1 3 u x 4 Ψ i ( t ) d x .

First, we shall present a global estimate describing the distance between u ( x ) and ∑ i = 1 N φ c i ( ⋅ − y i 0 ) . The proof of the global estimate is almost identical to that in ([18], Lemma 3.1), and hence we omit it.

Lemma 3.4

Consider k 1 > 0 , k 2 > 0 , and y 1 < ⋯ < y N be N spatial translations, where y i − y i − 1 ≥ R ∕ 2 . Assume that u ( t , x ) is the solution of the mCH-Novikov equation given by Proposition 2.1. Then, for fixed time t, there holds

(3.43) u ( x ) − ∑ i = 1 N φ c i ( ⋅ − y i ) H R 2 = ℰ ( u ) − ∑ i = 1 N ℰ ( φ c i ) − 4 ∑ i = 1 N a i ( u ( y i ) − a i ) + O R − 1 4 ,

where the constants involving in O R − 1 4 depends on k 1 , k 2 , and { c i } i = 1 N .

In the next lemma, we provide the relationship between the localized conservation laws of ℰ i and ℱ i and the localized maximum value u i ( x ) , where u i ( x ) = max x ∈ I i u ( x ) . For the sake of completeness, its proof will be given in Appendix A.1.

Lemma 3.5

Consider k 1 > 0 and k 2 > 0 . Let r ˜ 1 < ⋯ < r ˜ N be N spatial translations, where r ˜ i − r ˜ i − 1 ≥ 3 R ∕ 4 . Consider the interval ℐ i = [ x i , x i + 1 ] defined in equation (3.8). Assume that u ( t , x ) is the solution of the mCH-Novikov equation given by Proposition 2.1. For i = 1 , … , N , denote the local maximum value of u i ( x ) by

(3.44) u ( r i ∗ ) = max x ∈ ℐ i u ( x ) ≔ M i ,

where ∣ r i ∗ − r ˜ i ∣ < R ∕ 12 . Then, there holds

(3.45) ℱ i ( u ) ≤ 4 3 M i 2 ℰ i ( u ) − 4 3 M i 4 + O R − 1 2 .

Furthermore, we shall estimate the difference between the local maximum of the solution u ( t , x ) and the maximum of each single peaked solitary wave. The approach is inspired by the study of Liu et al. [18], and the detailed proof is given in Appendix A5.

Lemma 3.6

Consider k 1 > 0 and k 2 > 0 . For s > 5 2 , given the initial data u ( 0 , x ) = u 0 ( x ) ∈ H s ( R ) . Suppose that u ( t , x ) is the corresponding strong solution of the mCH-Novikov equation. There exist κ 0 > 0 and R 0 > 0 depending on k 1 , k 2 , and ( c i ) i = 1 N , such that for 0 < κ < κ 0 and 0 < R 0 < R , u 0 ( x ) satisfies

u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ( R ) ≤ κ 2 , y i 0 − y i − 1 0 > R ,

and u ( t , x ) satisfies equation (3.3) on [ 0 , t * ] . For i ∈ { 1 , 2 , … , N }

M i ( t ) = max x ∈ ℐ i ( x ) u ( t , x ) = u ( t , r i ∗ ( t ) ) ∀ t ∈ [ 0 , t * ] ,

where the interval ℐ i ( t ) is given in equation (3.8). Then, one derives

(3.46) ∑ i = 1 N a i ∣ M i ( t ) − a i ∣ ≤ O ( κ ) + O R − 1 4 ∀ t ∈ [ 0 , t * ] ,

where the constants in O ( ⋅ ) depend on k 1 , k 2 , { c i } i = 1 N , and ‖ u 0 ‖ H s .

3.4 Proof of Theorem 1.1

According to the argument established in equations (3.1) and (3.2), to complete the proof of Theorem 1.1, it is sufficient to show that, for fixed time t ∗ , there exists a constant C > 0 independent of A such that

u ( t * , x ) − ∑ i = 1 N φ c i ( x − y i ) H 1 ≤ C κ + R − 1 8 ,

where y 1 < y 2 < ⋯ < y N with y i − y i − 1 > 2 R ∕ 3 .

Indeed, for i = 1 , … , N , consider y i = r i ∗ ( t * ) ∈ ℐ i ( t * ) , where r i ∗ ( t * ) is defined in equation (3.9), namely,

M i ( t * ) = max x ∈ ℐ i ( t * ) u ( t * , x ) ≔ u ( t * , r i ∗ ( t * ) ) .

According to the estimates given by equations (3.7) and (3.10), we obtain

r i ∗ ( t * ) − r i − 1 ∗ ( t * ) ≥ r ˜ i ( t * ) − r ˜ i − 1 ( t * ) − ∣ r i ∗ ( t * ) − r ˜ i ( t * ) ∣ − ∣ r i − 1 ∗ ( t * ) − r ˜ i − 1 ( t * ) ∣ ≥ 3 R 4 − R 6 > 2 R 3 .

Then, taking advantage of the global estimate (3.43) and the local estimate (3.46), it is deduced that

u ( t * , x ) − ∑ i = 1 N φ c i ( x − r i ∗ ( t * ) ) H 1 ( R ) 2 = ℰ ( u ( t * ) ) − ∑ i = 1 N ℰ ( φ c i ) − 4 ∑ i = 1 N a i ( M i − a i ) + O e − R 4 ≤ ℰ ( u 0 ) − ∑ i = 1 N ℰ ( φ c i ) + 4 ∑ i = 1 N ∣ a i ∣ ∣ M i ( t * ) − a i ∣ + O e − R 4 ≤ O ( κ 2 ) + O ( κ ) + O e − R 4 ≤ O ( κ ) + O R − 1 4 .

In consequence, for 0 < κ < κ 0 ≪ 1 and R > R 0 ≫ 1 depending on k 1 , k 2 , and { c i } i = 1 N , we conclude that

u ( t * , x ) − ∑ i = 1 N φ c i ( x − r i ∗ ( t * ) ) H 1 ( R ) ≤ C κ + R − 1 8 ,

where the C depends only on k 1 , k 2 , { c i } i = 1 N , and ‖ u 0 ‖ H s , not on A . Hence, we may choose C = A ∕ 2 , which then completes the proof of Theorem 1.1.

  1. Funding information: This research is supported by the National Natural Science Foundation of China under Grant No. 12001491 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ20A010006.

  2. Conflict of interest: The author declares that there is no conflict of interest.

Appendix A Proofs of Lemmas 3.5 and 3.6

A.1 Proof of Lemma 3.5

Proof

Fix i ∈ { 1 , 2 , … , N } and time t and take M i = max x ∈ ℐ i u ( x ) . Consider the auxiliary function ρ ( x ) defining by

ρ ( x ) = u − u x , x < r i ∗ , u + u x , x > r i ∗ .

Then, one has

∫ R ρ 2 ( x ) Ψ i ( x ) d x = ∫ ( u 2 + u x 2 ) Ψ i d x − 2 ∫ − ∞ r i ∗ u u x Ψ i d x + 2 ∫ r i ∗ + ∞ u u x Ψ i d x .

With an application of integration by parts and the localized energy function, one has

(A1) ∫ R ρ 2 ( x ) Ψ i ( x ) d x = ℰ i ( u ) − u 2 Ψ i ∣ − ∞ r i ∗ + ∫ − ∞ r i ∗ u 2 Ψ i , x d x + u 2 Ψ i ∣ r i ∗ + ∞ − ∫ r i ∗ + ∞ u 2 Ψ i , x d x ,

which, when combined with the exponential decay property of the bump function Ψ i , gives rise to

(A2) ∫ R ρ 2 ( x ) Ψ i ( x ) d x = ℰ i ( u ) − 2 M i 2 Ψ i ( r i ∗ ) + ∫ − ∞ r i ∗ u 2 Ψ i , x d x − ∫ r i ∗ + ∞ u 2 Ψ i , x d x .

Furthermore, to connect the two localized conservation laws, we introduce another auxiliary function ζ ( x ) in the following form:

ζ ( x ) = u 2 ( x ) + 2 3 u ( x ) u x ( x ) − 1 3 u x 2 ( x ) , x > r i ∗ , u 2 ( x ) − 2 3 u ( x ) u x ( x ) − 1 3 u x 2 ( x ) , x < r i ∗ .

It satisfies

∫ ρ 2 ( x ) ζ ( x ) Ψ i ( x ) d x = ∫ r i ∗ + ∞ ( u 2 + 2 u u x + u x 2 ) u 2 + 2 3 u u x − 1 3 u x 2 Ψ i d x + ∫ − ∞ r i ∗ ( u 2 − 2 u u x + u x 2 ) u 2 − 2 3 u u x − 1 3 u x 2 Ψ i d x ≔ I I 1 + I I 2 .

Using integrating by parts and exponential decay Ψ i , one has

I I 1 = ∫ r i ∗ + ∞ u 4 + 2 u 2 u x 2 − 1 3 u x 4 Ψ i d x + 8 3 ∫ r i ∗ + ∞ u 3 u x Ψ i d x = ∫ r i ∗ + ∞ u 4 + 2 u 2 u x 2 − 1 3 u x 4 Ψ i d x − 2 3 ∫ r i ∗ + ∞ u 4 Ψ i , x d x − 2 3 M i 4 Ψ i ( r i ∗ ) .

In the same fashion, we derive

I I 2 = ∫ − ∞ r i ∗ u 4 + 2 u 2 u x 2 − 1 3 u x 4 Ψ i d x + 2 3 ∫ − ∞ r i ∗ u 4 Ψ i , x d x − 2 3 M i 4 Ψ i ( r i ∗ ) ,

which, when combined with the expression of I I 1 , gives rise to

(A3) ∫ R ρ 2 ( x ) ζ ( x ) Ψ i ( x ) d x = ℱ i ( u ) − 4 3 M i 4 Ψ i ( r i ∗ ) − 2 3 ∫ r i ∗ + ∞ u 4 Ψ i , x d x + 2 3 ∫ − ∞ r i ∗ u 4 Ψ i , x d x .

On the other hand, with the sign preservation of momentum density, we know u > ∣ u x ∣ for any x ∈ R , which implies

ζ ( x ) = u 2 ( x ) ± 2 3 u ( x ) u x ( x ) − 1 3 u x 2 ( x ) ≤ 4 3 u 2 ( x ) .

Therefore, together with the definition of local maximum value, one shall obtain

∫ R ρ 2 ( x ) ζ ( x ) Ψ i ( x ) d x ≤ 4 3 ∫ R u 2 ( x ) ρ 2 ( x ) Ψ i ( x ) d x ≤ 4 3 M i 2 ∫ R ρ 2 ( x ) Ψ i ( x ) d x + 4 3 ∫ ℐ i c ρ 2 ( x ) u 2 ( x ) Ψ i ( x ) d x ,

which carries out the following estimate with the identity (A2)

(A4) ∫ R ρ 2 ( x ) ζ ( x ) Ψ i ( x ) d x ≤ 4 3 M i 2 E i ( u ) − 8 3 M i 4 Ψ i ( r i ∗ ) + 4 3 M i 2 ∫ − ∞ r i ∗ u 2 Ψ i , x d x − 4 3 M i 2 ∫ r i ∗ + ∞ u 2 Ψ i , x d x + 4 3 ∫ ℐ i c u 2 ( x ) ρ 2 ( x ) Ψ i ( x ) d x .

Considering both equations (A3) and (A4), we obtain that

ℱ i ( u ) ≤ 4 3 M i 2 ℰ i ( u ) − 4 3 M i 4 Ψ i ( r i ∗ ) + 4 3 M i 2 ∫ − ∞ r i ∗ u 2 Ψ i , x d x − 4 3 M i 2 ∫ r i ∗ + ∞ u 2 Ψ i , x d x + 2 3 ∫ r i ∗ + ∞ u 4 Ψ i , x d x − 2 3 ∫ − ∞ r i ∗ u 4 Ψ i , x d x + 4 3 ∫ ℐ i c u 2 ( x ) ρ 2 ( x ) Ψ i ( x ) d x .

Based on the construction of the bump function Ψ i , one has

∣ ∂ x Ψ i ∣ ≤ 1 K ϒ ′ ≤ O ( R − 1 ∕ 2 ) .

Moreover, with the estimation of equation (3.10) in Lemma 3.1, i.e., ∣ r i ∗ − r ˜ i ∣ < R 12 , and the properties of the bump function (3.42), i.e.,

∣ 1 − Ψ i ( r i ∗ ) ∣ ≤ 4 e − R 4 K ≤ O R − 1 2 ,

one concludes from the Sobolev embedding theorem that

ℱ i ( u ) ≤ 4 3 M i 2 ℰ i ( u ) − 4 3 M i 4 + O L − 1 2 ,

which completes the proof of Lemma 3.5.□

A.2 Proof of Lemma 3.6

Proof

Suppose that u ( t , x ) is the corresponding strong solution of the mCH-Novikov equation. With the properties of the bump function Ψ i , we have

ℰ ( u ) = ∑ i = 1 N ℰ i ( u ) , ℱ ( u ) = ∑ i = 1 N ℱ i ( u ) .

From the fact that

(A5) u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ( R ) ≤ κ 2 ,

it is deduced from the Minkowski inequality that

ℰ ( u 0 ) − ℰ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) ≤ ‖ u 0 ‖ H 1 + ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ‖ u 0 ‖ H 1 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ≤ u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 + 2 ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 u 0 − ∑ i = 1 N φ c i ( ⋅ − y i 0 ) H 1 ≤ κ 2 + 2 ∑ i = 1 N ‖ φ c i ‖ H 1 ( R ) κ 2 ≤ O ( κ 2 ) ,

which together with equation (3.3) implies that

ℰ ( u ( t ) ) = ℰ ( u 0 ) ≤ ℰ ( u 0 ) − ℰ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) + ℰ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) ≤ ∑ i = 1 N ℰ ( φ c i ) + O ( κ 2 ) + O e − R 4 .

By the relationship between the localized conservation laws of ℰ i and ℱ i and the localized maximum value u i ( x ) , for fixed t ∈ [ 0 , t * ] , one derives

(A6) M i 4 ( t ) − M i 2 ( t ) ℰ i ( u ( t ) ) + 3 4 ℱ i ( u ( t ) ) ≤ O R − 1 2 , i = 1 , … , N .

In order to estimate the difference between the local maximum of the solution u ( t , x ) and the maximum of each single peaked solitary wave, we define a polynomial function P i ( λ ) by

(A7) P i ( λ ) = λ 4 − ℰ i ( u ( t ) ) λ 2 + 3 4 ℱ i ( u ( t ) ) .

With conservation laws (equation (3.1)), for ℰ ( φ c i ( ⋅ − y i 0 ) ) = 2 a i 2 and ℱ ( φ c i ( ⋅ − y i 0 ) ) = 4 3 a i 4 , we obtain

(A8) P 0 i ( λ ) = λ 4 − 2 a i 2 λ 2 + a i 4 = ( λ 2 − a i 2 ) 2 = ( λ − a i ) 2 ( λ + a i ) 2 .

Combining equations (A7) and (A8), one has

(A9) P 0 i ( λ ) = P i ( λ ) + ( ℰ i ( u ( t ) ) − ℰ ( φ c i ) ) λ 2 − 3 4 ( ℱ i ( u ( t ) ) − ℱ ( φ c i ) ) .

Let λ = M i ( t ) , then

P 0 i ( M i ) = ( M i − a i ) 2 ( M i + a i ) 2 ≥ a i 2 ( M i − a i ) 2 ,

which deduces from equation (A6) that

a i 2 ( M i − a i ) 2 ≤ P i ( M i ) + M i 2 ( ℰ i ( u ( t ) ) − ℰ ( φ c i ) ) − 3 4 ( ℱ i ( u ( t ) ) − ℱ ( φ c i ) ) ≤ M i 2 ( ℰ i ( u ( t ) ) − ℰ ( φ c i ) ) − 3 4 ( ℱ i ( u ( t ) ) − ℱ ( φ c i ) ) + O R − 1 2 .

Taking summation with respect to i from 1 to N for the above identity, one obtains

∑ i = 1 N a i 2 ( M i ( t ) − a i ) 2 ≤ ∑ i = 1 N M i 2 ( t ) ( ℰ i ( u ( t ) ) − ℰ ( φ c i ) ) − 3 4 ∑ i = 1 N ( ℱ i ( u ( t ) ) − ℱ ( φ c i ) ) + O R − 1 2 ≤ ∑ i = 1 N M i 2 ( t ) ( ℰ i ( u ( t ) ) − ℰ i ( u 0 ) ) + ∑ i = 1 N M i 2 ( t ) ( ℰ i ( u 0 ) − ℰ ( φ c i ) ) ︸ I I I 1 + I I I 2 − 3 4 ∑ i = 1 N ( ℱ i ( u ( t ) ) − ℱ ( φ c i ) ) ︸ I I I 3 + O R − 1 2 .

First, with the application of Sobolev embedding theorem, for 0 < κ < κ 0 ≪ 1 and R > R 0 ≫ 1 , one has

(A10) M i 2 ( t ) ≤ ‖ u ( t , x ) ‖ L ∞ ( R ) 2 ≤ 1 2 ℰ ( u 0 ) ≤ 1 2 ∑ i = 1 N ℰ ( φ c i ) + O ( ε 2 ) + O e − R 4 ≤ ∑ i = 1 N a i 2

for κ 0 and R 0 both depending on k 1 , k 2 , and ( c i ) i = 1 N . Then, it follows from equation (A5), the exponential decay of peaked solitary wave φ c i , and bump function Ψ i , as well as the definition of localized energy function ℰ i , that

∑ i = 1 N ( ℰ i ( u 0 ) − ℰ ( φ c i ) ) ≤ ∑ i = 1 N ∣ ∫ ℐ i ( 0 ) ( u 0 2 + u 0 , x 2 ) Ψ i d x − ‖ φ c i ‖ H 1 ( ℐ i ( 0 ) ) 2 ∣ + O R − 1 2 ≤ ∑ i = 1 N ∣ ‖ u 0 ‖ H 1 ( ℐ i ( 0 ) ) 2 − ‖ φ c i ‖ H 1 ( ℐ i ( 0 ) ) 2 ∣ + O R − 1 2 ≤ ∑ i = 1 N u 0 − ∑ j = 1 N φ c i ( ⋅ − y j 0 ) H 1 ( R ) + 2 ∑ j = 1 N 2 a j ⋅ u 0 − ∑ j = 1 N φ c i ( ⋅ − y j 0 ) H 1 ( ℐ i ( 0 ) ) + ∑ j ≠ i j = 1 N ‖ φ c j ‖ H 1 ( ℐ i ( 0 ) ) + O R − 1 2 ≤ ∑ i = 1 N κ 2 + 2 ∑ k = 1 N 2 a i κ 2 + O R − 1 2 + O R − 1 2 ≤ O ( κ 2 ) + O R − 1 2 ,

which, when combined with equation (A10), yields

I I I 2 ≤ ∑ i = 1 N M i 2 ( t ) ∣ ℰ i ( u 0 ) − ℰ ( φ c i ) ∣ ≤ O ( κ 2 ) + O R − 1 2 .

Second, for I I I 3 , following with a same fashion developed in Lemma 3.5 in [25], we obtain

ℱ ( u 0 ) − ℱ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) ≤ O ( κ 2 ) .

Then, according to the spatial translation relation y i 0 − y i − 1 0 > R 2 and the conservation laws, one deduces that

I I I 3 ≤ 3 4 ℱ ( u 0 ) − ℱ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) + 3 4 ℱ ∑ i = 1 N φ c i ( ⋅ − y i 0 ) − ∑ i = 1 N F ( φ c i ) ≤ O ( κ 2 ) + O ( R − R 4 ) .

Third, based on the definition of the bump function Ψ i and the Abel transform, we have,

I I I 1 = M N 2 ∑ i = 1 N ( ℰ i ( u ( t ) ) − ℰ i ( u 0 ) ) − ∑ j = 1 N − 1 ( M j + 1 2 − M j 2 ) ∑ i = 1 j ( ℰ i ( u ( t ) ) − ℰ i ( u 0 ) ) = ∑ j = 1 N − 1 ( M j + 1 2 ( t ) − M j 2 ( t ) ) ( Λ j + 1 , K ( t ) − Λ j + 1 , K ( 0 ) ) .

According to equation (3.10) given in Lemma 3.1, one knows

∣ r i ∗ ( t ) − r ˜ i ( t ) ∣ < R 12 , ∀ t ∈ [ 0 , t * ] ,

which along with equation (3.5) yields

u ( t , ⋅ ) − ∑ i = 1 N φ c i ( ⋅ − r i ∗ ( t ) ) H 1 ( R ) ≤ u ( t , ⋅ ) − ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) H 1 ( R ) + ∑ i = 1 N φ c i ( ⋅ − r i ∗ ( t ) ) − ∑ i = 1 N φ c i ( ⋅ − r ˜ i ( t ) ) H 1 ( R ) ≤ ‖ v ( t , x ) ‖ H 1 ( R ) + ∑ i = 1 N ‖ φ c i ( x − r i ∗ ( t ) ) − φ c i ( x − r ˜ i ( t ) ) ‖ H 1 ( R ) ≤ O ( κ ) + O e − R 4 .

Then, adopting equations (3.5), (3.7), and (3.10) and the exponential decay of φ c i , there holds

∣ u ( t , r i ∗ ( t ) ) − 2 a i ∣ ≤ u ( t , r i ∗ ( t ) ) − ∑ j = 1 N φ c j ( r i ∗ ( t ) − r j ∗ ( t ) ) + ∑ j ≠ i j = 1 N φ c i ( r i ∗ ( t ) − r j ∗ ( t ) ) ≤ ‖ u ( t , x ) − ∑ j = 1 N φ c j ( x − r j ∗ ( t ) ) ‖ L ∞ ( R ) + O e − R 4 ≤ O ( κ ) + O e − R 4 ,

which yields

∣ M i ( t ) − 2 a i ∣ ≤ O ( κ ) + O e − R 4 .

According to the relationship between { c i } i = 1 N and { a i } i = 1 N , for 0 < κ ≪ 1 and R ≫ 1 , there appears that

0 < M 1 ( t ) < ⋯ < M N − 1 ( t ) < M N ( t ) ,

which implies that

M j + 1 2 ( t ) − M j 2 ( t ) ≥ 0 .

Hence, we derive from equation (3.31) that

I I I 1 ≤ O R − 1 2 .

Finally, according to the estimates on I I I 1 , I I I 2 , and I I I 3 , we conclude that

∑ i = 1 N a i 2 ( M i − a i ) 2 ≤ O ( κ 2 ) + O R − 1 2 ,

which then infers that

∑ i = 1 N a i ∣ M i − a i ∣ ≤ O ( κ ) + O R − 1 4 .

This completes the proof of Lemma 3.6.□

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Received: 2023-05-04
Revised: 2023-09-27
Accepted: 2024-01-09
Published Online: 2024-02-01

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2023-0124
Keywords for this article
orbital stability; modified Camassa-Holm-Novikov equation; shallow water equation; multi-peakons
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Articles in the same Issue

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  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
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  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
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  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
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