IST Versus PDE: A Comparative Study

Klein, Christian; Saut, Jean‐Claude

doi:10.1007/978-1-4939-2950-4_14

Cited by 34 publications

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“…Spatial decay conditions on the data are not explicitly stated in the work of Deift-Zhou on the defocusing mKdV equation [9], but the condition above is used in the application of direct and inverse scattering in [1,40]. We also refer to [32] for a recent survey.…”

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confidence: 99%

Asymptotic stability of solitons for mKdV

Germain

Pusateri

Rousset

2016

Advances in Mathematics

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We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from such data tend uniformly, on the real line, to another solitary wave as time goes to infinity. We describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior. This latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mKdV equation. R u dx.(1.2) As we will remark later, these are not needed to prove the small data result, which also applies to more general versions of (mKdV). 1 2 PIERRE GERMAIN, FABIO PUSATERI, AND FRÉDÉRIC ROUSSETMoreover, we note that solutions of (mKdV) enjoy the scaling symmetry u −→ λu(λ 3 t, λx), which is generated by the vector field S = 1 + x∂ x + 3t∂ t . Known results.Global well-posedness and asymptotic behavior. There is a vast body of literature dealing with the mKdV equation, and in particular with the local and global well-posedness of the Cauchy problem. Without trying to be exhaustive, we mention the early works on the local and global well-posedness by Kenig-Ponce-Vega [29] and Kato [27]. Global well-posedness in low regularity spaces, and in particular in the energy space H 1 , was established in the seminal work of Kenig-Ponce-Vega [30]. In this latter paper the authors considered the wider class of generalized KdV (gKdV) equations ∂ t u + ∂ 3x u + ∂ x u p = 0, p ≥ 2, which includes (mKdV) and the KdV equation (p = 2). Sharp, up to the end-point, global well-posedness in H s for s > 1/4 was proved in the work of Colliander-Keel-Staffilani-Takaoka-Tao [7], for both the focusing and defocusing mKdV equation on the line (and for s ≥ 1/2 in the periodic case). These results are complemented by several ill-posedness results; see for example Christ-Colliander-Tao [6] and references therein. 2 Besides global regularity, another fundamental question for dispersive PDEs concerns the asymptotic behavior for large times. The first proof of global existence with a complete description of the asymptotic behavior of solutions of (mKdV) in the defocusing case, is due to Deift and Zhou [9], who used a steepest descent approach to oscillatory Riemann-Hilbert problems and the inverse scattering transform [52,2]. In [9], thanks to the complete integrability of the defocusing mKdV equation, the authors were able to treat suitably localized initial data with arbitrary size. A proof of global existence and a (partial) derivation of the asymptotic behavior for small localized solutions, without making use of complete integrability, was later given by Hayashi and Naumkin [18,19], following the ideas introduced in the context of the 1d nonlinear Schrödinger (NLS) equation in [17]. Recently, an alternative proof of the results in [19], with a precise derivation of asymptotics and a proof of asymptotic completeness, was given by Har...

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Section: )mentioning

confidence: 99%

Asymptotic stability of solitons for mKdV

Germain

Pusateri

Rousset

2016

Advances in Mathematics

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“…elastically scattering and shape preserving wave packets. Well-known examples occur in the Korteweg-de Vries equation [28][29][30], Sine-Gordon equation [28,29,31] and the nonlinear Schrödinger/Gross-Pitaevskii equation (GPE) [29,30,32]. The nonlinearity in all these wave equations appears as a local term that only depends on the wavefunction at the same point.…”

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confidence: 99%

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

Plestid

O’Dell

2019

Phys. Rev. E

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The Hamiltonian Mean Field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in 1D and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE) which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wavefunction and the depth of the meanfield potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the meanfield potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.

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“…Since IST methods provide so far rigorous results on the Cauchy problem only for small enough initial data, one cannot use them to prove the soliton resolution conjecture (see [253]) as in the KdV case (see eg [242] and the references therein). However we strongly believe that it holds true for the BO and ILW equations, as suggested for instance by the following numerical simulations in [129] (see also [228,187] for other illuminating simulations and [229] for a theoretical analysis of the spectral method in [228]). 10 We first show the formation of solitons from localized initial data for the BO equation in Fig.…”

Section: 4mentioning

confidence: 74%

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

Saut

2019

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

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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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IST Versus PDE: A Comparative Study

Abstract: We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.

Cited by 34 publications

References 208 publications

Asymptotic stability of solitons for mKdV

Asymptotic stability of solitons for mKdV

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

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

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