Stability of line solitons for the KP-II equation in ℝ²

Mizumachi, Tetsu

doi:10.1090/memo/1125

Cited by 38 publications

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“…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 nonlinear stability is open.By using a linearized version of the dressing method from [26], explicit eigenfunctions of the spectral stability problem associated with the periodic waves of the KP-II equation (1.1) were constructed in [21].…”

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confidence: 99%

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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confidence: 99%

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Hărăguş

Jin

Pelinovsky

2017

Commun. Math. Phys.

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“…They also showed [27] that the linear equation around Q c for c ∼ 1 satisfies a convective stability property, based in the similarity of IB with KdV for small speeds. This result has been successfully adapted to a more general setting by Mizumachi in a series of works [22,23]. Whether or not the asymptotic stability results à la Martel-Merle [20,21] can be applied to this case, is a challenging problem.…”

Section: 2mentioning

confidence: 99%

Decay in the one dimensional generalized Improved Boussinesq equation

Maulén

Muñoz

2020

SN Partial Differ. Equ. Appl.

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We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space H 1 × H 2 , existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm L 2 × H 1 , for any p > 1, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time H 1 × H 2 solutions, and (ii) decay on compact sets of arbitrarily large bounded in time H 1 × H 2 solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.Here the plus sign denotes the good Boussinesq system, which is locally and globally wellposed in standard Sobolev spaces [8,9], and the minus sign represents the "bad" equation originally derived by Boussinesq [4], which is strongly linearly ill-posed. Precisely, motivated Key words and phrases. Improved Boussinesq, decay, virial. (Ch.M.) Partially funded by Chilean research grants FONDECYT 1150202 and CONICYT PFCHA/DOCTORADO NACIONAL/2016-21160593. (Cl.M.) Partially funded by Chilean research grants FONDECYT 1150202 and 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. Part of this work was done while Cl.M. was visiting the CMLS at

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“…Recall that the KdV soliton is transversally L 2 stable with respect to weak transverse perturbations described by the KP II equation ( [195,194]). This result is unconditional since it was established in [207] that the Cauchy problem for KP II is globally well-posed in H s (R × T), s ≥ 0, or for all initial data of the form u 0 + ψ c where u 0 ∈ H s (R 2 ), s ≥ 0 and ψ c (x − ct, y) is a solution of the KP-II equation such that for every σ ≥ 0, (1 − ∂ 2…”

Section: Variamentioning

confidence: 99%

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Saut

2019

Fields Institute Communications

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This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE techniques and also results more connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line with only a few comments on the periodic case. We will also briefly discuss some close relevant problems in particular the higher order extensions and the two-dimensional (KP like) versions of the BO and ILW equations.Remark 3. The above derivation was performed for purely gravity waves. Surface tension effects result in adding a third order dispersive term in the asymptotic models. One gets for instance the so-called Benjamin equation (see [30,31]):where δ > 0 measures the capillary effects. This equation, which is in some sense close to the KdV equation, is not known to be integrable. Its solitary waves, the existence of which was proven in [30,31] by the degree-theoretic approach, present oscillatory tails. We refer to [145] for the Cauchy problem in L 2 and to [12,21] for further results on the existence and stability of solitary wave solutions and to [46] for numerical simulations.In presence of surface tension, the ILW equation has to be modified in the same way. We are not aware of mathematical results on the resulting equation.Remark 4. The BO equation was fully justified in [219] as a model of long internal waves in a two-fluid system by taking into account the influence of the surface

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Stability of line solitons for the KP-II equation in ℝ²

Cited by 38 publications

References 28 publications

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Decay in the one dimensional generalized Improved Boussinesq equation

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

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