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Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation

  • Byungsoo Moon EMAIL logo and Chao Yang
Published/Copyright: September 24, 2024
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

Keywords: solitary waves; shock waves; Degasperis-Procesi-Kadomtsev-Petviashvili equation; Degasperis-Procesi equation
MSC 2010: 35Q53; 35G25; 76B15; 76B25

1 Introduction

In this work, we show the existence of peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation

(1.1) ( u t − u x x t + κ u x + 4 u u x − ( 3 u x u x x + u u x x x ) ) x + u y y = 0 ,

where κ is a real parameter and the real function u ( t , x , y ) depends on the spatial variables x , y ∈ R and the temporal variable t > 0 . Equation (1.1) may be viewed as the Degasperis-Procesi (DP) counterpart of the two-dimensional generalization of the Korteweg-de Vries (KdV) equation [19] in much the same way as Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equation [15], which was formally derived from the incompressible and irrotational three-dimensional shallow water under the CH regime. Local well-posedness, formation of singularities, existence of traveling wave solutions, and the Liouville-type property for CH-KP equation were presented in [15]. The recent related results for other models with respect to transverse effects can be found in [3,13,16].

In the case of no y -dependence, equations (1.1) and CH-KP equation take the form of the integrable DP equation [9]

(1.2) u t − u x x t + κ u x + 4 u u x − ( 3 u x u x x + u u x x x ) = 0

and the Camassa-Holm (CH) shallow-water equation [2,12]

(1.3) u t − u x x t + κ u x + 3 u u x − ( 2 u x u x x + u u x x x ) = 0 ,

respectively.

During the last decade, the DP and CH equations have attracted much attention due to their integrable structure. Both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The DP and CH equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahony (BBM) [1] and KdV equations [19]. They accommodate wave breaking phenomena (i.e., wave profile remains bounded while its slope becomes unbounded in finite time [30]). From a mathematical physics viewpoint, it is interesting to study these equations sharing the peaked solitary wave solution on the line and circle in the limiting case of vanishing linear dispersion ( κ = 0 )

(1.4) u ( t , x ) = c e − ∣ x − c t ∣ , c > 0 ,

(1.5) u ( t , x ) = c cosh 1 2 cosh ( x − c t ) − [ x − c t ] − 1 2 , x ∈ R , t ≥ 0 , c > 0 .

Here the notation [ x ] denotes the greatest integer part of the real number x ∈ R . The orbital stability of peaked solitary wave solutions (1.4) and (1.5) of the CH and DP equations has been established [7,21,23]. It was shown [27] that peakons and a train of peakons for CH equation are asymptotically stable (see also [18] for DP equation). Note that the peaked solitary wave solutions (1.4) and (1.5) are not classical solutions due to the fact that they have a peak at their crest (a feature that explains why they are called “peakons”). It is also worth pointing out that one of the most relevant motivations to look for peaked solitary waves (or periodic) is the fact that the governing equations for irrotational water waves do admit peaked solitary waves (or periodic), namely, the celebrated Stokes waves of greatest height [5,6,29]. There has recently been a growing interest in finding various CH-type equations with higher-order nonlinearity admitting peaked solitary wave solutions (e.g. [4,17,25,28,31,32] and references therein). It is worth mentioning that the recent research works [14,25,28] where peaked waves were shown to be unstable in the dynamical evolution of the CH-related equations are important for the complete picture about (in)stability of peaked waves in the CH-related equations.

Although the DP equation has a similarity to the CH equation, these two equations have truly different features. An important difference between the DP and the CH equation for κ = 0 is that the DP equation also enjoys shock peaked waves [24]

(1.6) u ( t , x ) = − 1 t + c sgn ( x ) e − ∣ x ∣ , c > 0

and periodic shock waves [11]

(1.7) u ( t , x ) = cosh 1 2 sinh 1 2 t + c − 1 sinh x − [ x ] − 1 2 sinh 1 2 , x ∈ R \ Z , c > 0 , 0 , x ∈ Z .

It is unsure that whether such a shock waves (or periodic) (1.6) and (1.7) is stable or not in proper setting. On the other hand, DP equation has the third-order equation in the Lax pair for the isospectral problem [8]

ψ x − ψ x x x − λ m ψ = 0 , m = u − u x x ,

while CH equation has the second-order equation for the isospectral problem [2]

ψ x x − 1 4 ψ − λ m ψ = 0 , m = u − u x x .

Moreover, the DP and CH equations have entirely different forms of conservation laws due to the fact that there is no simple transformation of the DP equation into the CH equation [2,8]. Furthermore, the CH equation has several nice geometric formulations [26], while only a non-metric geometric derivation of the DP equation is available [10]. Their similarities and differences can be found in related recent results [22].

In the case of non-vanishing linear dispersion ( κ ≠ 0 ), the DP and CH equations have localized smooth solitary waves [20,22]. Existence and stability result of smooth solitary waves for the DP and CH equations are established in [7,22].

The aim of this work is to prove whether the DP-KP equation (1.1) has properties for existence of solitary waves and existence of shock waves similar to that of the CH-KP equation and DP equation (1.2), respectively. Recently, it is known from [15] that the CH-KP equation admit peaked solitary wave solution of the form

u ( t , x , y ) = c e − ∣ x + β y − c t ∣ if and only if κ + β 2 = 0 .

Since the CH-KP equation and DP equation (1.2) have peaked solitary waves (or periodic) and shock waves, respectively, one can expect that equation (1.1) possesses peaked solitary waves (or periodic) and shock waves. It is a reasonable expectation due to the fact that DP-KP equation (1.1) was able to transform to DP equation (1.2) (Remark 2.1). Until now, it is unclear whether the DP-KP equation (1.1) is integrable and an inverse scattering method is available for it or not. This will be the objective for future work.

Our work is motivated by the works of Gui et al. [15] and Escher et al. [11]. In [15], the existence of peaked solitary wave solutions for the CH-KP equation has been proven for the first time. On the other hand, in [11], the authors present a periodic shock waves of the DP equation (1.2) with κ = 0 . Combining the ideas from both papers we show existence of periodic shock waves of the DP-KP (1.1).

This work is organized as follows. Section 2 is devoted to the existence and non-existence of peaked solitary wave solutions for the DP-KP equation (1.1) for certain cases. Existence of periodic shock waves are presented in Section 3.

2 Peaked solitary waves

In this section, we prove the existence of peaked solitary waves of the DP-KP equation (1.1) in the following form:

(2.1) u ( t , x , y ) = α 1 e − ∣ x + β 1 y − c t ∣ , c ∈ R

and

(2.2) u ( t , x , y ) = α 2 cosh 1 2 − ( x + β 2 y − c t ) + [ x + β 2 y − c t ] , c ∈ R ,

where [ x ] stands for the greatest integer of x ∈ R . Note that the inverse operator ( 1 − ∂ x 2 ) − 1 can be obtained by convolution with the corresponding Green’s function, so that

(2.3) ( 1 − ∂ x 2 ) − 1 f = G * f for all f ∈ L 2 ( X ) ,

where

(2.4) G ( x ) = 1 2 e − ∣ x ∣ for the non-periodic case X = R ,

while

(2.5) G ( x ) = cosh 1 2 − x + [ x ] 2 sinh 1 2 for the periodic case X = S ,

and the convolution product is defined by

f * g ( x ) = ∫ X f ( y ) g ( x − y ) d y .

We can rewrite (1.1) with the initial data u 0 as the following weak form:

(2.6) u t + u u x + ( 1 − ∂ x 2 ) − 1 ∂ x 3 2 u 2 + κ u + ( 1 − ∂ x 2 ) − 1 v y = 0 , t > 0 , ( x , y ) ∈ X = R 2 or S 2 u y = v x . u ∣ t = 0 = u 0 ( x , y ) , ( x , y ) ∈ X = R 2 or S 2 .

The formulation (2.6) allows us to define the notion of a weak solution as follows.

Definition 2.1

Given initial data u 0 ∈ H 1 ( X ) , a function u ∈ C ( [ 0 , T ) ; H loc 1 ( X ) ) is said to be a weak solution of the initial value problem (2.6) if it satisfies the following identity:

(2.7) ∫ 0 T ∫ X − ∂ t ∂ x ϕ u + ∂ x ϕ u ∂ x u + ∂ x G * 3 2 u 2 + κ u − ∂ y 2 ϕ G * u d x d y d t + ∫ X u 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = 0 ,

for any smooth test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × X ) . If u is a weak solution on [ 0 , T ) for every T > 0 , then it is called a global weak solution.

The following theorem deals with the existence of peaked solitary waves for the DP-KP equation (1.1).

Theorem 2.1

The DP-KP equation (1.1) possesses the peaked solitary wave solutions of the form u ( t , x , y ) = c e − ∣ x + β 1 y − c t ∣ for constant β 1 ∈ R if and only if it satisfies κ + β 1 2 = 0 . These peakons are global weak solution to (2.6) in the sense of Definition 2.1.

Proof

Suppose that

(2.8) u c ( t , x , y ) = c e − ∣ x + β 1 y − c t ∣ .

Then, we have

(2.9) ∂ t u c ( t , x , y ) = c ⋅ sign ( x + β 1 y − c t ) u c ( t , x , y ) , ∂ x u c ( t , x , y ) = − sign ( x + β 1 y − c t ) u c ( t , x , y ) .

Combining (2.9) with the fact ∂ x G ( x ) = − 1 2 sign ( x ) e − ∣ x ∣ for x ∈ R , for any test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × R 2 ) , we have

(2.10) ∫ 0 ∞ ∫ R 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ R 2 ∂ x ϕ ∂ t u c + u c ∂ x u c + ∂ x G * 3 2 u c 2 − ( κ + β 1 2 ) u c d x d y d t = ∫ 0 ∞ ∫ R 2 ∂ x ϕ [ c 2 ⋅ sign ( x + β 1 y − c t ) e − ∣ x + β 1 y − c t ∣ − c 2 ⋅ sign ( x + β 1 y − c t ) e − 2 ∣ x + β 1 y − c t ∣ ] d x d y d t + ∫ 0 ∞ ∫ R 2 ∂ x ϕ ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ d x d y d t ,

where we use the fact G * ∂ y u c = β 1 ∂ x G * u c .

Using the explicit form (2.4) and the convolution product, we deduce

(2.11) ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ = − 1 2 ∫ − ∞ ∞ sign ( x + β 1 y − z ) e − ∣ x + β 1 y − z ∣ 3 2 c 2 e − 2 ∣ z − c t ∣ − ( κ + β 1 2 ) c e − ∣ z − c t ∣ d z .

For x + β 1 y > c t , we can split the right-hand side of (2.11) into the following three parts:

(2.12) ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ = − 1 2 ∫ − ∞ c t + ∫ c t x + β 1 y + ∫ x + β 1 y ∞ sign ( x + β 1 y − z ) e − ∣ x + β 1 y − z ∣ 3 2 c 2 e − 2 ∣ z − c t ∣ − ( κ + β 1 2 ) c e − ∣ z − c t ∣ d z . ≕ J 1 + J 2 + J 3 .

A direct calculation yields

(2.13) J 1 = ∫ − ∞ c t e − ( x + β 1 y − z ) − 3 4 c 2 e 2 ( z − c t ) + 1 2 ( κ + β 1 2 ) c e ( z − c t ) d z = − 1 4 c 2 + 1 4 ( κ + β 1 2 ) c e − ( x + β 1 y − c t ) ,

(2.14) J 2 = ∫ c t x + β 1 y e − ( x + β 1 y − z ) − 3 4 c 2 e − 2 ( z − c t ) + 1 2 ( κ + β 1 2 ) c e − ( z − c t ) d z = 3 4 c 2 e − 2 ( x + β 1 y − c t ) + − 3 4 c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e − ( x + β 1 y − c t ) ,

and

(2.15) J 3 = ∫ x + β 1 y ∞ e ( x + β 1 y − z ) 3 4 c 2 e − 2 ( z − c t ) − 1 2 ( κ + β 1 2 ) c e − ( z − c t ) d z = 1 4 c 2 e − 2 ( x + β 1 y − c t ) − 1 4 ( κ + β 1 2 ) c e − ( x + β 1 y − c t ) .

Plugging the above equalities J 1 − J 3 into (2.12), we find that for x + β 1 y > c t ,

(2.16) ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ = c 2 e − 2 ( x + β 1 y − c t ) + − c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e − ( x + β 1 y − c t ) .

While for x + β 1 ≤ c t , we can also split the right-hand side of (2.11) into three parts:

(2.17) ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ = − 1 2 ∫ − ∞ x + β 1 y + ∫ x + β 1 y c t + ∫ c t ∞ sign ( x + β 1 y − z ) e − ∣ x + β 1 y − z ∣ 3 2 c 2 e − 2 ∣ z − c t ∣ − ( κ + β 1 2 ) c e − ∣ z − c t ∣ d z . ≕ J ˜ 1 + J ˜ 2 + J ˜ 3 .

Applying a similar computation as J 1 , J 2 , J 3 , we find that

(2.18) J ˜ 1 = ∫ − ∞ x + β 1 y e − ( x + β 1 y − z ) − 3 4 c 2 e 2 ( z − c t ) + 1 2 ( κ + β 1 2 ) c e ( z − c t ) d z = − 1 4 c 2 e 2 ( x + β 1 y − c t ) + 1 4 ( κ + β 1 2 ) c e ( x + β 1 y − c t ) ,

(2.19) J ˜ 2 = ∫ x + β 1 y c t e ( x + β 1 y − z ) 3 4 c 2 e 2 ( z − c t ) − 1 2 ( κ + β 1 2 ) c e ( z − c t ) d z = − 3 4 c 2 e 2 ( x + β 1 y − c t ) + 3 4 c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e ( x + β 1 y − c t )

and

(2.20) J ˜ 3 = ∫ c t ∞ e ( x + β 1 y − z ) 3 4 c 2 e − 2 ( z − c t ) − 1 2 ( κ + β 1 2 ) c e − ( z − c t ) d z = 1 4 c 2 e ( x + β 1 y − c t ) − 1 4 ( κ + β 1 2 ) c e ( x + β 1 y − c t ) .

Substituting (2.18)–(2.20) in (2.12) yields for x + β 1 y ≤ c t ,

(2.21) ∂ x G * 3 2 c 2 e − 2 ∣ x + β 1 y − c t ∣ − ( κ + β 1 2 ) c e − ∣ x + β 1 y − c t ∣ = − c 2 e 2 ( x + β 1 y − c t ) + c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e ( x + β 1 y − c t ) ,

which together with (2.16) leads to

(2.22) ∂ x G * 3 2 u c 2 − ( κ + β 1 2 ) u c ( t , x , y ) = c 2 e − 2 ( x + β 1 y − c t ) + − c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e − ( x + β 1 y − c t ) , x + β 1 y > c t , − c 2 e 2 ( x + β 1 y − c t ) + c 2 + 1 2 ( κ + β 1 2 ) c ( x + β 1 y − c t ) e ( x + β 1 y − c t ) , x + β 1 y ≤ c t .

On the other hand, one can deduce

(2.23) ∂ t u c + u c ∂ x u c = c 2 ⋅ sign ( x + β 1 y − c t ) e − ∣ x + β 1 y − c t ∣ − c 2 ⋅ sign ( x + β 1 y − c t ) e − 2 ∣ x + β 1 y − c t ∣ = − c 2 e − 2 ( x + β 1 y − c t ) + c 2 e − ( x + β 1 y − c t ) , x + β 1 y > c t , c 2 e 2 ( x + β 1 y − c t ) − c 2 e − ( x + β 1 y − c t ) , x + β 1 y ≤ c t .

Thanks to (2.22), (2.23), and (2.10), we conclude that for any ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , ∞ ) × R 2 ) ,

(2.24) ∫ 0 ∞ ∫ R 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ R 2 ∂ x ϕ ∂ t u c + u c ∂ x u c + ∂ x G * 3 2 u c 2 − ( κ + β 1 2 ) u c d x d y d t = 0

if and only if κ + β 1 2 = 0 . This completes the proof of Theorem 2.1.□

Next we give a concrete example of weak solutions of the DP-KP equation (1.1), which may be considered as periodic peaked solitary waves.

Theorem 2.2

Let

(2.25) u ( t , x , y ) = c cosh 1 2 cosh 1 2 − ( x + β 2 y − c t ) + [ x + β 2 y − c t ] , x ∈ R , β 2 ∈ R , t ≥ 0 .

The DP-KP equation (1.1) possesses the periodic peaked solitary waves of the form u ( t , x , y ) if and only if it satisfies κ + β 2 2 = 0 . These peakons are global weak solution to (2.6) in the sense of Definition 2.1.

Proof

Assume that

(2.26) u c ( t , x , y ) = c cosh 1 2 cosh 1 2 − ( x + β 2 y − c t ) + [ x + β 2 y − c t ] .

Then, we have

(2.27) ∂ x u c ( t , x , y ) = − c sinh χ cosh 1 2 , ∂ t u c ( t , x , y ) = c 2 sinh χ cosh 1 2 ,

where χ = 1 2 − ( x + β 2 y − c t ) + [ x + β 2 y − c t ] . Using the fact ∂ x G ( x ) = − sinh 1 2 − x + [ x ] 2 sinh 1 2 , x ∈ S , for any test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × S 2 ) , we have

(2.28) ∫ 0 ∞ ∫ S 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ S 2 ∂ x ϕ ∂ t u c + u c ∂ x u c + ∂ x G * 3 2 u c 2 − ( κ + β 1 2 ) u c d x d y d t = ∫ 0 ∞ ∫ S 2 ∂ x ϕ c 2 cosh 1 2 sinh χ − c 2 cosh 2 1 2 cosh χ sinh χ d x d y d t + ∫ 0 ∞ ∫ S 2 ∂ x ϕ ∂ x G * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ d x d y d t .

Here we use the relation G * ∂ y u c = β 2 ∂ x G * u c . We compute

(2.29) ∂ x G ( x ) * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ ( t , x , y ) = − 1 2 sinh 1 2 ∫ S sinh 1 2 − ( x + β 2 y − q ) + [ x + β 2 y − q ] 3 c 2 2 cosh 2 1 2 cosh 2 1 2 − ( q − c t ) + [ q − c t ] − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 − ( q − c t ) − [ q + c t ] d q .

For x + β 2 y > c t , we divide (2.29) as follows:

∂ x G ( x ) * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ ( t , x , y ) = − 1 2 sinh 1 2 ∫ 0 c t + ∫ c t x + β 2 y + ∫ x + β 2 y 1 sinh 1 2 − ( x + β 2 y − q ) + [ x + β 2 y − q ]

× 3 c 2 2 cosh 2 1 2 cosh 2 1 2 − ( q − c t ) + [ q − c t ] − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 − ( q − c t ) − [ q + c t ] d q ≕ K 1 + K 2 + K 3 .

Using the identities cosh 2 X = 1 2 cosh 2 X + 1 2 and sinh X cosh Y = 1 2 sinh ( X + Y ) + 1 2 sinh ( X − Y ) , a direct calculation yields

(2.30) K 1 = − 1 2 sinh 1 2 ∫ 0 c t sinh 1 2 − ( x + β 2 y − q ) 3 c 2 2 cosh 2 1 2 cosh 2 1 2 + ( q − c t ) − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 + ( q − c t ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 ∫ 0 c t 1 4 sinh 3 2 − ( x + β 2 y + 2 c t ) + 3 q + 1 4 sinh − 1 2 − ( x + β 2 y − 2 c t ) − q + 1 2 sinh 1 2 − ( x + β 2 y ) + q d q + ( κ + β 2 2 ) c 2 sinh 1 2 cosh 1 2 ∫ 0 c t 1 2 sinh ( 1 − ( x + β 2 y + c t ) + 2 q ) + 1 2 sinh ( − ( x + β 2 y − c t ) ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 1 12 cosh 3 2 − ( x + β 2 y − c t ) − 1 12 cosh 3 2 − ( x + β 2 y + 2 c t ) − 1 4 cosh − 1 2 − ( x + β 2 y − c t ) + 1 4 cosh − 1 2 − ( x + β 2 y − 2 c t ) + 1 2 cosh 1 2 − ( x + β 2 y − c t ) − 1 2 cosh 1 2 − ( x + β 2 y ) + ( κ + β 2 2 ) c 2 cosh 1 2 sinh 1 2 1 4 cosh ( 1 − ( x − c t + β 2 y ) ) − 1 4 cosh ( 1 − ( x + c t + β 2 y ) ) + c t 2 sinh ( − ( x − c t + β 2 y ) ) ,

(2.31) K 2 = − 1 2 sinh 1 2 ∫ c t x + β 2 y sinh 1 2 − ( x + β 2 y − q ) × 3 c 2 2 cosh 2 1 2 cosh 2 1 2 − ( q − c t ) − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 − ( q − c t ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 ∫ c t x + β 2 y 1 4 sinh 3 2 − ( x + β 2 y − 2 c t ) − q + 1 4 sinh − 1 2 − ( x + β 2 y + 2 c t ) + 3 q + 1 2 sinh 1 2 − ( x + β 2 y ) + q d q

+ ( κ + β 2 2 ) c 2 cosh 1 2 sinh 1 2 ∫ c t x + β 2 y 1 2 sinh ( 1 − ( x − c t + β 2 y ) ) + 1 2 sinh ( − ( x + c t + β 2 y ) + 2 q ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 − 1 4 cosh 3 2 − 2 ( x + β 2 y − c t ) + 1 4 cosh 3 2 − ( x + β 2 y − c t ) + 1 12 cosh − 1 2 + 2 ( x + β 2 y − c t ) − 1 12 cosh − 1 2 − ( x − c t + β 2 y ) + 1 2 cosh 1 2 − 1 2 cosh 1 2 − ( x − c t + β 2 y ) + ( κ + β 2 2 ) c 2 sinh 1 2 cosh 1 2 1 2 ( x + β 2 y − c t ) sinh ( 1 − ( x − c t + β 2 y ) )

and

(2.32) K 3 = − 1 2 sinh 1 2 ∫ x + β 2 y 1 sinh − 1 2 − ( x + β 2 y − q ) × 3 c 2 2 cosh 2 1 2 cosh 2 1 2 − ( q − c t ) − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 − ( q − c t ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 ∫ x + β 2 y 1 1 4 sinh 1 2 − ( x + β 2 y − 2 c t ) − q + 1 4 sinh − 3 2 − ( x + β 2 y + 2 c t ) + 3 z + 1 2 sinh − 1 2 − ( x + β 2 y ) + q d q + ( κ + β 2 2 ) c 2 sinh 1 2 cosh 1 2 ∫ x + β 2 y 1 1 2 sinh ( − ( x − c t + β 2 y ) ) + 1 2 sinh ( − 1 − ( x + c t + β 2 y ) + 2 q ) d q = − 3 c 2 4 sinh 1 2 cosh 2 1 2 − 1 4 cosh − 1 2 − ( x + β 2 y − 2 c t ) + 1 4 cosh 1 2 − 2 ( x − c t + β 2 y ) + 1 12 cosh 3 2 − ( x + 2 c t + β 2 y ) − 1 12 cosh − 3 2 + 2 ( x + β 2 y − c t ) + 1 2 cosh 1 2 − ( x + β 2 y ) − 1 2 cosh − 1 2 + ( κ + β 2 2 ) c 2 sinh 1 2 cosh 1 2 1 4 cosh ( 1 − ( x + c t + β 2 y ) ) − 1 4 cosh ( ( x − c t + β 2 y ) − 1 ) + 1 2 ( 1 − ( x + β 2 y ) ) sinh ( − ( x − c t + β 2 y ) ) .

Plugging the above equalities K 1 − K 3 into (2.29) gives us for x + β 2 y > c t ,

(2.33) ∂ x G ( x ) * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ ( t , x , y ) = − 3 c 2 4 sinh 1 2 cosh 2 1 2 2 3 sinh ( 1 ) sinh 1 2 − ( x + β 2 y − c t ) − 2 3 sinh 1 2 sinh ( 1 − 2 ( x + β 2 y − c t ) )

+ ( κ + β 2 2 ) c 2 sinh 1 2 cosh 1 2 1 2 sinh 1 2 − ( x + β 2 y − c t ) cosh 1 2 − cosh 1 2 − ( x + β 2 y − c t ) sinh 1 2 × 1 2 − ( x + β 2 y − c t ) = − c 2 cosh 1 2 sinh 1 2 − ( x + β 2 y − c t ) + c 2 cosh 2 1 2 sinh 1 2 − ( x + β 2 y − c t ) cosh 1 2 − ( x + β 2 y − c t ) + ( κ + β 2 2 ) c 4 sinh 1 2 sinh 1 2 − ( x + β 2 y − c t ) − ( κ + β 2 2 ) c 2 cosh 1 2 cosh 1 2 − ( x + β 2 y − c t ) 1 2 − ( x + β 2 y − c t ) .

While for x + β 2 y ≤ c t , we also divide (2.29) into three parts,

(2.34) ∂ x G ( x ) * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ ( t , x , y ) = − 1 2 sinh 1 2 ∫ 0 x + β 2 y + ∫ x + β 2 y c t + ∫ c t 1 sinh 1 2 − ( x + β 2 y − q ) + [ x + β 2 y − q ] × 3 c 2 2 cosh 2 1 2 cosh 2 1 2 − ( q − c t ) + [ q − c t ] − ( κ + β 2 2 ) c cosh 1 2 cosh 1 2 − ( q − c t ) − [ q + c t ] d q ≕ K ˜ 1 + K ˜ 2 + K ˜ 3 .

Applying a similar computation as K 1 − K 3 to the terms K ˜ 1 − K ˜ 3 on the right-hand side of (2.34), we find that for x + β 2 ≤ c t ,

(2.35) ∂ x G ( x ) * 3 c 2 2 cosh 2 1 2 cosh 2 χ − ( κ + β 2 2 ) c cosh 1 2 cosh χ ( t , x , y ) = c 2 cosh 1 2 sinh 1 2 + ( x + β 2 y − c t ) − c 2 cosh 2 1 2 sinh 1 2 + ( x + β 2 y − c t ) cosh 1 2 + ( x + β 2 y − c t ) − ( κ + β 2 2 ) c 4 sinh 1 2 sinh 1 2 + ( x + β 2 y − c t ) + ( κ + β 2 2 ) c 2 cosh 1 2 cosh ( x + β 2 y − c t ) + 1 2 ( x + β 2 y − c t ) + 1 2 .

For the terms ∂ t u c + u c ∂ x u c , we have

(2.36) ∂ t u c + u c ∂ x u c = c 2 cosh 1 2 sinh χ − c 2 cosh 2 1 2 cosh χ sinh χ

= c 2 cosh 1 2 sinh 1 2 − ( x + β 2 y − c t ) − c 2 cosh 2 1 2 cosh 1 2 − ( x + β 2 y − c t ) sinh 1 2 − ( x + β 2 y − c t ) , x + β 2 y > c t , − c 2 cosh 1 2 sinh 1 2 + ( x + β 2 y − c t ) + c 2 cosh 2 1 2 cosh 1 2 + ( x + β 2 y − c t ) sinh 1 2 + ( x + β 2 y − c t ) , x + β 2 y ≤ c t .

Using the property of linear independence of functions cosh χ and sinh χ and (2.33), (2.35), (2.36), it follows from (2.27) that for every test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × S 2 ) ,

(2.37) ∫ 0 ∞ ∫ S 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ S 2 ∂ x ϕ ( κ + β 2 2 ) c 4 sinh 1 2 sinh χ − ( κ + β 2 2 ) c 2 cosh 1 2 χ cosh χ d x d y d t = 0

if and only if κ + β 2 2 = 0 . This completes the proof of Theorem 2.2.□

Remark 2.1

For the initial value problem equation (1.1) with the initial data u 0 ( x , y ) = F 0 ( x + γ y ) , function u ( x , y , t ) = F ( x + γ y , t ) is a solution to equation (1.1) by the uniqueness. Here the function F ( t , η ) is a solution of the following initial value problem for the DP equation

(2.38) F t − F t η η + ( κ + γ 2 ) F η + 4 F F η − ( 3 F η F η η + F F η η η ) = 0

with initial data F ( 0 , η ) = F 0 ( η ) .

In view of the proof of existence result of smooth solitons of the DP equation in [22], we may also have the following result and detailed proof is omitted.

Theorem 2.3

If κ + γ 2 > 0 , then equation (2.38) possesses a localized smooth solitary wave solution.

3 Periodic shock waves

It is our purpose in this section to show that the DP-KP equation (1.1) admits the periodic shock waves of the form

(3.1) u ( t , x , y ) = cosh 1 2 sinh 1 2 t + c − 1 sinh x + γ y − [ x + γ y ] − 1 2 sinh 1 2 , ( x , y ) ∈ R 2 \ Z 2 , c > 0 , 0 , ( x , y ) ∈ Z 2 .

Such a type of solution is a weak solution in the sense of Definition 2.1. The following result on the existence of the periodic shock waves can be obtained by verifying the definition of weak solution.

Theorem 3.1

The DP-KP equation (1.1) admits a global weak solution in the periodic shock wave form of (3.1) for some constant β 3 ∈ R if and only if it satisfies that κ + β 3 2 = 0 .

Proof

Assume that

(3.2) u c ( t , x , y ) = cosh 1 2 sinh 1 2 t + c − 1 sinh x + β 3 y − [ x + β 3 y ] − 1 2 sinh 1 2 , ( x , y ) ∈ R 2 \ Z 2 , c > 0 , 0 , ( x , y ) ∈ Z 2 .

Then, we have

(3.3) ∂ x u c ( t , x , y ) = cosh 1 2 sinh 1 2 t + c − 1 cosh ζ sinh 1 2

and

(3.4) ∂ t u c ( t , x , y ) = − cosh 1 2 sinh 1 2 t + c − 2 cosh 1 2 sinh ζ sinh 2 1 2 ,

where ζ = ( x + β 3 y ) − [ x + β 3 y ] − 1 2 . Using (3.2), (3.3), (3.4), and integration by parts, we deduce from the Definition 2.1 that for every test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × S 2 ) ,

(3.5) ∫ 0 ∞ ∫ S 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ S 2 ∂ x ϕ ∂ t u c + u c ∂ x u c + ∂ x G * 3 2 u c 2 − ( κ + β 3 2 ) u c d x d y d t = ∫ 0 ∞ ∫ S 2 ∂ x ϕ − cosh 1 2 sinh 1 2 t + c − 2 cosh 1 2 sinh ζ sinh 2 1 2 + cosh 1 2 sinh 1 2 t + c − 2 cosh ζ sinh ζ sinh 2 1 2 d x d y d t + ∫ 0 ∞ ∫ S 2 ∂ x ϕ ∂ x G * cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 ζ 2 sinh 2 1 2 − cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh ζ sinh 1 2 d x d y d t ,

where we use the relation G * ∂ y u c = β 3 ∂ x G * u c . Note that

∂ x G ( x ) = − sinh 1 2 − x + [ x ] 2 sinh 1 2 , x ∈ R \ Z .

It then follows that

(3.6) ∂ x G * cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 ζ 2 sinh 2 1 2 − cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh ζ sinh 1 2 ( t , x , y ) = − 1 2 sinh 1 2 ∫ S sinh 1 2 − ( x + β 3 y − z ) + [ x + β 3 y − z ] × cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 z − [ z ] − 1 2 2 sinh 2 1 2 − cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh z − [ z ] − 1 2 sinh 1 2 d z = ∫ 0 x + β 3 y sinh 1 2 − ( x + β 3 y − z ) × − cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 z − 1 2 4 sinh 3 1 2 + cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh z − 1 2 2 sinh 2 1 2 d z + ∫ x + β 3 y 1 sinh − 1 2 − ( x + β 3 y − z ) × − cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 z − 1 2 4 sinh 3 1 2 + cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh z − 1 2 2 sinh 2 1 2 d z .

A simple computation shows that

(3.7) − 3 4 sinh 3 1 2 cosh 1 2 sinh 1 2 t + c − 2 ∫ 0 x + β 3 y sinh 1 2 − ( x + β 3 y − z ) sinh 2 z − 1 2 d z + κ + β 3 2 2 sinh 2 1 2 cosh 1 2 sinh 1 2 t + c − 1 ∫ 0 x + β 3 y sinh 1 2 − ( x + β 3 y − z ) sinh z − 1 2 d z = − 3 4 sinh 3 1 2 cosh 1 2 sinh 1 2 t + c − 2 cosh 1 2 sinh 2 x + β 3 y − 1 2 − cosh 1 2 − ( x + β 3 y ) sinh 2 − 1 2 + 1 3 sinh 1 2 sinh ( 2 ( x + β 3 y ) − 1 ) − 1 3 sinh 1 2 − ( x + β 3 y ) sinh ( − 1 ) − 2 3 cosh 1 2 cosh ( 2 ( x + β 3 y ) − 1 ) + 2 3 cosh 1 2 − ( x + β 3 y ) cosh ( − 1 ) + κ + β 3 2 2 sinh 2 1 2 cosh 1 2 sinh 1 2 t + c − 1 1 2 sinh ( x + β 3 y ) − 1 2 cosh ( 1 − ( x + β 3 y ) ) ( x + β 3 y )

and

(3.8) − 3 4 sinh 3 1 2 cosh 1 2 sinh 1 2 t + c − 2 ∫ x + β 3 y 1 sinh 1 2 − ( x + β 3 y − z ) sinh 2 z − 1 2 d z + κ + β 3 2 2 sinh 2 1 2 cosh 1 2 sinh 1 2 t + c − 1 ∫ x + β 3 y 1 sinh 1 2 − ( x + β 3 y − z ) sinh z − 1 2 d z = − 3 4 sinh 3 1 2 cosh 1 2 sinh 1 2 t + c − 2 cosh 1 2 − ( x + β 3 y ) sinh 2 1 2 − cosh − 1 2 sinh 2 ( x + β 3 y ) − 1 2 + 1 3 sinh 1 2 − ( x + β 3 y ) sinh ( 1 ) − 1 3 sinh − 1 2 sinh ( 2 ( x + β 3 y ) − 1 ) − 2 3 cosh 1 2 − ( x + β 3 y ) cosh ( 1 ) + 2 3 cosh − 1 2 cosh ( 2 ( x + β 3 y ) − 1 ) + κ + β 3 2 2 sinh 2 1 2 cosh 1 2 sinh 1 2 t + c − 1 1 2 sinh ( 1 − ( x + β 3 y ) ) + 1 2 cosh ( − ( x + β 3 y ) ) ( x + β 3 y − 1 ) .

By (3.6) and (3.7), we have

(3.9) ∂ x G ( x ) * cosh 1 2 sinh 1 2 t + c − 2 3 sinh 2 ζ 2 sinh 2 1 2 − cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) sinh ζ sinh 1 2 ( t , x , y ) = cosh 1 2 sinh 1 2 t + c − 2 − sinh x + β 3 y − 1 2 cosh x + β 3 y − 1 2 sinh 2 1 2 + cosh 1 2 sinh x + β 3 y − 1 2 sinh 2 1 2 + cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) 2 sinh 1 2 − cosh 1 2 cosh x + β 3 y − 1 2 4 sinh 2 1 2 + ( κ + β 3 2 ) x + β 3 y − 1 2 sinh x + β 3 y − 1 2 2 sinh 1 2 .

It is found from (3.4), (3.5), and (3.8) that for every test function ϕ ( t , x , y ) ∈ C c ∞ ( [ 0 , T ) × S 2 ) ,

∫ 0 ∞ ∫ S 2 − ∂ t ∂ x ϕ u c + ∂ x ϕ u c ∂ x u c + ∂ x G * 3 2 u c 2 + κ u c − ∂ y 2 ϕ G * u c d x d y d t + ∫ R 2 u c , 0 ( x , y ) ∂ x ϕ ( 0 , x , y ) d x d y = ∫ 0 ∞ ∫ S 2 ∂ x ϕ cosh 1 2 sinh 1 2 t + c − 1 ( κ + β 3 2 ) 2 sinh 1 2 − cosh 1 2 cosh ζ 4 sinh 2 1 2 + ( κ + β 3 2 ) ζ sinh ζ 2 sinh 1 2 d x d y d t = 0

if and only if κ + β 3 2 = 0 . This completes the proof of Theorem 3.1.□

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that helped to improve and clarify the article greatly.

  1. Funding information: The work of Moon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant No. 2020R1F1A1A01048468). The work of Yang was supported by the China Scholarship Council (No. 202306680038) and the Fundamental Research for the Central Universities (3072022GIP2403).

  2. Author contributions: Both authors contributed equally and significantly to this article.

  3. Conflict of interest: C. Yang is a member of the Editorial Board of ANONA, but this did not affect the final decision for the article.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed in the current study.

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Received: 2023-10-27
Revised: 2024-06-16
Accepted: 2024-08-10
Published Online: 2024-09-24

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

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https://doi.org/10.1515/anona-2024-0040
Keywords for this article
solitary waves; shock waves; Degasperis-Procesi-Kadomtsev-Petviashvili equation; Degasperis-Procesi equation
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Articles in the same Issue

  1. Regular Articles
  2. Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
  3. Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  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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