Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation

1 Introduction

The method introduced by Killip et al. [43], which was based on the power series expansion of the perturbation determinant associated with the Lax pair, provides a powerful technique for obtaining the low regularity conservation laws for nonlinear integrable partial differential equations (PDEs). By utilizing this method, we develop a new technique in Fourier analysis and complex analysis to study the Fokas-Lenells (FL) equation

where u : ℳ × R → C ( ℳ = R or T = R ∕ Z ), γ and ν are two real parameters, and σ = ± 1 . This equation was first derived by Fokas applying the bi-Hamiltonian method in [28]. In optics, considering higher order linear and nonlinear optical effects, the FL equation is used as a model to describe the femtosecond pulse propagation through single mode optical silica fiber [50].

Incidentally, the following Camassa-Holm (CH) equation on R or T is also considered:

where u : ℳ × R → R and ω is a positive constant. The CH equation describes unidirectional propagation of wave in shallow water. It is derived physically by Camassa and Holm [7] by approximating the Hamiltonian for Euler’s equations in the shallow water regime. In fact, this equation first appeared in the paper of Fuchssteiner and Fokas [30] on hereditary symmetries, they proved that (1.2) is formally integrable as a bi-Hamiltonian generalization of KdV. But it came to be remarkable in the work of Camassa and Holm [7] where the peakon was described. The CH equation also arises in the study of axially symmetric waves motion in hyperelastic rods [21,23]. Its high-frequency limit models nematic liquid crystals [4,39].

Integrable equations enjoy several special properties including the existence of a Lax pair, infinitely many conserved quantities, as well as a bi-Hamiltonian formulation. Actually, the last property provides an important algorithmic approach of constructing integrable equations. Suppose that for all values of the parameter λ the operator θ 1 + λ θ 2 is Hamiltonian, then

gives an integrable equation. If we define

when α = 0 , β = 1 , γ = 2 , then (1.3) yields the KdV equation

If we define q = u − u x x , and replace θ 1 = ∂ x by θ 1 = ∂ x − ∂ x 3 , then (1.3) with α = 2 ω , β = 0 , γ = 1 yields the CH equation (1.2).

If we define r = σ q ¯ , and the compatible pair of Hamiltonian operators

then the system

reduces to the cubic nonlinear Schrödinger (NLS) equation

Furthermore, if we define q = u + i ν u x , and replace σ 3 by θ 1 = σ 3 + i ν I ∂ x , where I is the 2 × 2 identity matrix, then the system

gives the FL equation (1.1). Therefore, the CH equation is considered as an integrable generalization of the KdV equation, and the FL equation is an integrable generalization of the NLS equation.

Utilizing the bi-Hamiltonian structure, the first few conservation laws and the Lax pair of the FL system were constructed by Lenells and Fokas [53]. Moreover, the Lax pair was used to solve the initial value problem on the line with inverse scattering methods under the assumption that the initial data have sufficient smoothness and decay. We may assume that ν has the same sign as γ in equation (1.1), otherwise we just replace u ( x , t ) with u ( − x , t ) . Let β = γ ∕ ν > 0 and μ = 1 ∕ ν . Then, the transformation

converts (1.1) into

The method in our work is applicable to both σ = ± 1 , and the procedures are the same except for the signs in certain places, so we only consider the case σ = 1 . Furthermore, we take β = μ = 1 for simplicity, hence (1.6) can be written as

There are infinitely many conservation laws for the FL flow (1.7) and the first three (see [53] Section 2) are

(1.7) admits a Lax pair with operator

The FL equation is also closely related to the derivative nonlinear Schrödinger (DNLS) equation, i.e., the nonlinearity ∣ q ∣ 2 q in (1.5) is replaced by ( ∣ q ∣ 2 q ) x . It is integrable and belongs to the DNLS hierarchy [53], which is related to the Kaup-Newell spectral problem. In fact, it is the first negative flow of the integrable hierarchy of the DNLS equation [50]. Actually, we cannote that the operator L ( t ; q ) in (1.9) has the same form as the x -part of the Lax pair for the DNLS equation derived in the pioneering work of Kaup and Newell [41].

The DNLS equation admits the scaling symmetry and L 2 is the scaling critical space. It is worth noting that the best global well-posedness in L 2 ( R ) for this equation has been obtained by Harrop-Griffiths et al. [34] by the second-generation method of commuting flows. Before this work, multiple authors sought to obtain well-posedness in H s for s as small as possible. We now review some but not exhaustive results. Local well-posedness in H 1 ∕ 2 was obtained by Takaoka [67] on the real line and Herr [36] on the circle by fixed point arguments. Biagioni and Linares [3] showed that the data-to-solution mapping fails to be locally uniformly continuous on the real line below H 1 ∕ 2 . This indicates that s = 1 ∕ 2 is the optimal index of proving local well-posedness via the contraction mapping principle. By employing exact or approximate conservation laws, one can extend local-in-time solutions to global solutions. However, the conservation laws (including mass, Hamiltonian, etc.) for the DNLS equation are not coercive for large initial data, specifically, when ‖ q 0 ‖ L 2 is large. Thus, the subsequent global results were obtained under the restriction of bounded initial data ‖ q 0 ‖ L 2 . Wu [71,72] proved the global well-posedness in energy space with the initial data q 0 ∈ H 1 ( R ) and ‖ q 0 ‖ L 2 < 2 π , and Mosincat and Oh got a similar result on the torus [60]. Subsequently, Guo and Wu [33] proved global well-posedness in H 1 ∕ 2 ( R ) for ‖ q 0 ‖ L 2 < 2 π by I-method, and Mosincat [59] showed global well-posedness in the circle case under the same L 2 -smallness condition. Recently, by combining the known well-posedness theory with an in-depth analysis of the transmission coefficient, Bahouri and Perelman [2] got rid of the L 2 -smallness condition and proved that the DNLS equation is globally well-posed in H 1 ∕ 2 ( R ) . For the lower regularity, Guo [31] established a priori estimates for s > 1 ∕ 4 on the real line by short-time Fourier restriction method, and Schippa [64] extended this result to periodic boundary conditions. Recently, Klaus and Schippa [45] established low regularity a priori estimates for the DNLS equation in the full subcritical spaces B r , 2 s ( 0 < s < 1 ∕ 2 ) with small ‖ q 0 ‖ L 2 . The microscopic conservation laws for the Schwartz solutions of the DNLS equation with small mass were also worked out by Tang and Xu [69].

The FL equation has been studied in various interesting manners. For example, solutions of the FL equation can be derived by means of the Riemann-Hilbert method [53], inverse scattering transform [80], dressing chain [51], algebrogeometric method [81], Darboux transformation [35,78], and a variable separation technique [70]. The multisoliton solutions including dark solitons and bright solitons were obtained in [51,54,55]. Fan and his co-authors in [12,75] studied the long-time asymptotics for the FL equation with decaying initial value problem by means of the Riemann-Hilbert approach. The initial-boundary value problems of the FL equation have been studied in [52,73].

Less is known about the well-posedness theory for the FL equation. Fokas and Himonas [29] obtained that the local well-posedness of the periodic initial value problem for the FL equation (1.1) in H s ( T ) for s > 3 ∕ 2 , if 1 ∕ ν is not an integer. When 1 ∕ ν is an integer, they gave the local well-posedness for the nonlocal version of the FL equation in the corresponding homogeneous Sobolev spaces. Cheng and Fan [11] used inverse scattering method based on the Riemann-Hilbert problem to obtain the existence of global solutions to the Cauchy problem for the FL equation on the line without the small-norm assumption on initial data u 0 ∈ H 3 ( R ) ∩ H 2 , 1 ( R ) . Recently, they used a newly modified Darboux transformation to obtain the global well-posedness result in H 3 ( R ) ∩ H 2 , 1 ( R ) when the initial data include solitons [13]. However, there are major technical difficulties to use inverse scattering techniques in unweighted Sobolev spaces. For instance, on the line, the decay of the data is insufficient when inverse scattering method is applied.

The principal goal of this work is to obtain the low regularity conservation laws for the FL equation in the Besov spaces. For the NLS equation on the line, Koch and Tataru [46] used the transmission coefficient to obtain almost conserved H s -energies for all s > − 1 ∕ 2 . Killip et al. [43] developed a method to obtain conservation laws both on the real line R and on the circle T for a general class of completely integrable dispersive equations, such as KdV, cubic NLS and mKdV (which lie within the famous AKNS/ZS framework). Then, they showed the global well-posedness for the KdV equation in H − 1 by using these conservation laws [44]. Talbut [68] used the same approach to obtain conservation laws at negative regularity ( − 1 ∕ 2 < s < 0 ) for the Benjamin-Ono (BO) equation. Killip et al. [42] showed very recently that the BO equation is well-posed, both on the line and on the circle, in the Sobolev spaces H s for s > − 1 ∕ 2 by using a new gauge transformation and a modified Lax pair representation of the full hierarchy. As mentioned above, the conservation laws for the DNLS equation were studied in [45,69].

Motivated by these work, we will show that the determinant

given by

where κ ˜ = − i κ 2 (the tilde may be dropped later on), is conserved for solutions of (1.7). Then, by using this conserved quantity, we obtain the growth of Besov norms of the solution. Now, we state our main result.

Local well-posedness of the CH equation in Sobolev spaces and Besov spaces has been studied [17,24,25,47,48]. It was shown that the CH equation is well-posed in H s for s > 3 ∕ 2 and ill-posed in H s for s < 3 ∕ 2 in the sense that the data-to-solution map fails to be uniformly continuous. In the critical Sobolev space H 3 ∕ 2 , there exists norm inflation and ill-posedness [32]. Xin and Zhang [74] proved the existence of a global-in-time weak solution to the Cauchy problem of the CH equation with initial data in H 1 ( R ) . We refer to [5,19,38] on existence and uniqueness of global weak solutions to the CH equation. Xu and Yang [76] established the local well-posedness of solutions with the low regularity in the periodic setting for a class of one-dimensional generalized Rotation-Camassa-Holm equation.

The inverse scattering transform method is used to study the initial value problem for the CH equation in [15,18,40,49]. Deift and Zhou [26,27] developed a very effective tool called the nonlinear steepest descent method to obtain the long-time asymptotic behavior of systems, see also [57,75].

There are several important features for the CH equation. For example, (1.2) is an infinite-dimensional Hamiltonian system which is completely integrable [14,15]. The CH equation possesses non-smooth peakon-type solitary or periodic traveling wave solutions, in sharp contrast to the solitary waves for the KdV equation, which are always analytic [1,6,8,16]. A peakon is a weak solution in some Sobolev space with corner at its crest. Constantin further showed the stability of peakons and orbital stability of solitary wave solution for the CH equation in [20,22]. Stability of peakons for modified CH equation was obtained in [62]. Moon [58] obtained the existence of peaked traveling wave solution of μ -Novikov equation which is a linear combination of the Novikov equation and CH equation. Xu and his collaborators [10,77] proved local well-posedness, rigidity property, and asymptotic stability of peakons in the energy space H 1 ( R ) for some generalized shallow water equations.

An infinite number of conservation laws of the KdV equation help in deriving the necessary a priori estimates for solutions in H k ( R ) for every k ∈ Z + . However, although (1.2) is integrable and has infinitely many conservation laws, global existence of regular solutions is guaranteed only for a special class of initial data, and only the H 1 ( R ) norm is preserved for smooth solutions to (1.2). In fact, it has been observed by Camassa and Holm [7], Constantin and Escher [17], and McKean [56] that finite-time blow-up of smooth solutions always occurs for a large class of initial data. In particular, McKean [56] gave a necessary and sufficient criterion on the initial data for the finite-time formulation of singularities in a smooth solution to the CH equation.

Without loss of generality, we fix ω = 1 in the CH equation (1.2) and assume that u ( x ) and u 0 ( x ) are in Schwartz space. The Lax pair formulation transcends the distinction between the problem on the line with decaying and periodic data. (1.2) admits the following Lax pair:

where

where q = u − u x x and λ is a spectral parameter.

We consider the perturbation determinant

which formally represents the ratio of the “characteristic polynomials” of Lax operator and that of the Schrödinger operators with no potential. Perturbation determinant plays an important role in perturbation and scattering theory [63,65,79]. After simple calculation, one can see that the determinant of (1.13) can be rewritten as

where R 0 = ( − ∂ x 2 + κ 2 ) − 1 and κ 2 = 1 4 + λ . R 0 is well-defined whenever κ > 0 . Then

The leading term in the Fredholm expansion is

which is actually a conservation quantity of the CH equation (1.2). Obviously, this will not extend to functions u in H 1 . By using a device of Hilbert [37], we consider the regularized 2-determinant to drop the ℓ = 1 term

Furthermore, since R 0 q R 0 is self-adjoint, we consider the following series:

Now, this renormalization allows us to extend the notion of determinant to the class of Hilbert-Schmidt (HS) operators. By a simple computation, we know that the operator R 0 q R 0 is HS if and only if q ∈ H − 1 . Precisely, the HS norm of R 0 q R 0 is comparable to ‖ u ‖ H 1 (see Corollary 3.3). Then, we use the conservation of the perturbation determinant (1.15) associated with the Lax pair to show the following a priori estimates for Schwartz solution to (1.2).

There are several new ingredients in this study. First, the method introduced by Killip, Vişan, and Zhang for completely integrable dispersive PDEs is applied to general completely integrable equations. These equations, like KdV, cubic NLS, DNLS, and BO, enjoy the invariant scaling symmetry, but the FL equation and the CH equation do not. Second, we use the Fourier transform to compute the trace of operators in the Schatten class (Lemma 2.1), and develop a new approach in Fourier analysis and complex analysis to estimate the leading term of α ( κ ; q ( t ) ) . This method is not limited to the explicit kernel of the operators, but also applicable to general kernels, see the proof of Proposition 3.1 and Proposition 3.2 for more details. Finally, the proof of conservation of perturbation determinant for the FL equation is non-trivial. In order to prove Theorem 4.2, the key point is to prove

where R − = ( ∂ − κ ) − 1 and R + = ( ∂ + κ ) − 1 . Formally, the term ( R − q x R + q ¯ x ) R − q R + q ¯ x in the right-hand side is comparable to R − ∣ q ∣ 2 q R + q ¯ x because R − and R + can be treated as integral operators. However, the term in the left-hand side is R − ∣ q ∣ 2 q x R + q ¯ x . They differ by one-order derivative. In fact, we utilize the symmetry of operators’ trace to obtain some cancellation.

Organization of the paper. In Section 2, we present the notations and recall some basic facts about operator traces. In Section 3, we estimate the leading term of the perturbation determinants α ( κ ; q ( t ) ) associated with the FL and CH equations. In Section 4, we prove that the perturbation determinants α ( κ ; q ( t ) ) are conserved. Finally, by utilizing the conservation laws, we obtain the a priori estimates and complete the proof of Theorems 1.1 and 1.4 in Section 5.

2 Notations and preliminaries

In this section, we present the notations and several results that play a major role in the following estimates. For a detailed presentation, one can also refer to [43,45,65,66].

The Fourier transform is defined as

for functions on the line and

for functions on the circle T = R ∕ Z .

For s ∈ R , 1 ≤ r ≤ ∞ , we define Sobolev norms and Besov norms by

and with the usual interpretation when r = ∞ .

We recall some basic facts about operator traces and perturbation determinant. Denote I p the Schatten class of compact operators on L 2 with ℓ p -summable singular values. I p is complete and an embedded subalgebra of bounded operators. One can easily obtain the embedding I p ⊂ I q from ℓ p ⊂ ℓ q for p < q . I ∞ is the space of compact operators. I 1 is the space with elements of trace-class. If A is an operator on L 2 with continuous integral kernel K ( x , y ) , then the trace of operator A is defined by

We refer to elements of I 2 by HS operators. An operator A on L 2 is HS class ( I 2 ) if and only if it admits an integral kernel K ( x , y ) ∈ L 2 , and define

Products of two or more HS operators A j with kernels K j ( x , y ) are of trace class, and the operator trace can be computed as

By Fubini’s theorem, this allows us to cycle the trace

Specially,

Moreover, if A is a trace class operator and B is a bounded operator on L 2 , then A B and B A are of trace class, and

The class I p is two-sided ideals in the algebra of bounded linear operators on L 2 . Furthermore, we have

and the Hölder-like estimate

provided that 1 ∕ p = 1 ∕ q + 1 ∕ r , 1 ≤ p , q , r ≤ ∞ . For more details on operator theory and trace ideals, we refer to [65],[66, Chapter 3].

Next lemma tells us that one can consider the trace of operator from the Fourier transform point of view.

Lemma 2.1 also holds in the circle setting. The following results are elementary and will be used frequently later.

This lemma indicates that α ( t ) is differentiable and submits to term-by-term differentiation. If A ( t ) is self-adjoint, then α ( t ) is comparable to HS norm of A ( t ) .

3 Main estimates for the leading term

From Lemma 2.3 we know that the HS norm of operator A is almost conserved as long as the perturbation determinant is conserved under the assumption that A is self-adjoint (like the CH equation, A = R 0 q R 0 ). If A is not self-adjoint (like the FL equation), by studying the relation between the leading term in expansion of the perturbation determinant and the Sobolev norm of operator A , we can also obtain the increment of Sobolev norm of the solution.

In this section, we develop a new technique in Fourier analysis and complex analysis to calculate the leading term of the series expansion

and

even though they have been shown in [43,45,61] by using the explicit kernels of the free resolvent ( κ ± ∂ ) − 1 and R 0 = ( − ∂ 2 + κ 2 ) − 1 . Our method is based on Fourier transform and the residue theorem in complex analysis. It is not limited to the explicit kernel of the operators, but also applicable to general operators, for instance, the higher order operators.