Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution

Molinet, Luc; Saut, Jean‐Claude; Tzvetkov, Nikolay

doi:10.1007/s00220-007-0243-1

Cited by 45 publications

(50 citation statements)

References 35 publications

(71 reference statements)

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“…On the other hand, for the KP-I equation, a conserved higher-order energy functional has been constructed in [24,25]. After transforming this quantity to the variables used in the KP-II equation Similarly to F (u), the y-independent part of H(u) is equivalent to the higher-order energy functional H(u) of the KdV equation (1.2) given by (4.16).…”

Section: Discussionmentioning

confidence: 99%

“…The self-adjoint operator L is related to the Hessian operator of the standard energy functional expanded at the periodic traveling wave. Similarly, the self-adjoint operator K can be found from the Hessian operator of a higher-order energy functional, as for instance the one used in the proof of global well-posedness for the KP-I equation [24,25]. Then the operators JL and JK commute.…”

Section: Introductionmentioning

confidence: 99%

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Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Hărăguş

Jin

Pelinovsky

2017

Commun. Math. Phys.

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We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew-and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev-Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.where the subscripts denote partial derivatives with respect to the spatial variables (x, y) and the temporal variable t. This equation is referred to as the KP-II equation, where the index II stands for the version relevant to the case of negative transverse dispersion. The KP-I equation is obtained by replacing the positive sign in front of the term u yy by a negative sign, and it is relevant to the case of positive transverse dispersion. Both versions of the KP equation are two-dimensional extensions of the KdV equation u t + 6uu x + u xxx = 0, (1.2) that governs one-dimensional nonlinear waves in the longitudinal direction of the x axis. Just like the KdV equation, the KP-II and KP-I equations arise as particular models in the classical water-wave problem, in the cases of small and large surface tension, respectively.The KP equations quickly became very popular due to their integrability properties [26], including a rich family of exact solutions, a bi-Hamiltonian structure and the recursion operator, a countable set of conserved quantities and symmetries, as well as the inverse scattering transform techniques. At the same time, they became popular in the analysis of the stability of nonlinear waves, both relying upon functional-analytic methods and integrability techniques. As a model equation for surface water waves, some of the obtained results were extended to the Euler equations describing the full hydrodynamic problem [5,12,29].Stability properties of traveling waves are quite different for the two versions of the KP equation. While both periodic and solitary waves are transversely unstable in the KP-I equation (e.g., see recent works [8,15,27,28] and the references therein), it is expected that they are transversely stable in the KP-II equation [1,16]. Numerical evidences of these stability properties can be found for instance in [19,20]. For the case of solitary waves, the transverse nonlinear stability has been recently proved for periodic transverse perturbations in [23], and for fully localized perturbations in [22]. In contrast, there are few analytical results for periodic waves for which, in particular, the question of transverse non...

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Hărăguş

Jin

Pelinovsky

2017

Commun. Math. Phys.

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“…For a proof of this lemma, we refer to [2,30] or [23] (Lemma 2, p. 783). The proof on [23] is performed for functions on R 2 but the proof works equally well in the R x × T y setting.…”

Section: Lemma 21mentioning

confidence: 99%

“…The proof on [23] is performed for functions on R 2 but the proof works equally well in the R x × T y setting. In order to motivate the mKP-II equation, we now introduce the Miura transforms that we use in this paper.…”

Section: Lemma 21mentioning

confidence: 99%

Stability of the line soliton of the KP-II equation under periodic transverse perturbations

Mizumachi

Tzvetkov

2011

Math. Ann.

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“…This leads to our study of the Cauchy problem for the ZK equation in H s × . A second possibility (which has been carried out in [13] for similar problems for the KP-I equation) is to consider two-dimensional "localized" perturbations of the one-dimensional solution . This motivates the study of the Cauchy problem (1.5) below.…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

Linares

Pastor

Saut

2010

Communications in Partial Differential Equations

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Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution

Cited by 45 publications

References 35 publications

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Stability of the line soliton of the KP-II equation under periodic transverse perturbations

Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

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