On the relationship between the N-soliton solution of the modified Korteweg-de Vries equation and the KdV equation solution

Perel'man, T. L.; Fridman, A.Kh.; El'yashevich, M.M.

doi:10.1016/0375-9601(74)90185-6

Cited by 31 publications

(6 citation statements)

References 3 publications

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“…An important difference between positive and negative β that in fact provides an explanation for our numerical results is that the negative β mKdV has kink solutions, while the positive β case does not. As shown in previous studies (see reference [41]) when a soliton interacts with a kink, the interaction causes the soliton to flip signs, thereby becoming an antisoliton. Therefore, given the numerical results, we argue that the background for the β < 0 mKdV equation becomes a traveling kink-antikink.…”

Section: Role Of Mkdv Solitons In the Fput Recurrencesupporting

confidence: 52%

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The β Fermi-Pasta-Ulam-Tsingou recurrence problem

Pace

Reiss

Campbell

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We perform a thorough investigation of the first FPUT recurrence in the β-FPUT chain for both positive and negative β. We show numerically that the rescaled FPUT recurrence time Tr = tr/(N + 1) 3 depends, for large N , only on the parameter S ≡ Eβ(N + 1). Our numerics also reveal that for small |S|, Tr is linear in S with positive slope for both positive and negative β. For large |S|, Tr is proportional to |S| −1/2 for both positive and negative β but with different multiplicative constants. In the continuum limit, the β-FPUT chain approaches the modified Korteweg-de Vries (mKdV) equation, which we investigate numerically to better understand the FPUT recurrences on the lattice. In the continuum, the recurrence time closely follows the |S| −1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions which exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for Tr which depends only on S. In the latter regime, we show that Tr ∝ |S| −1/2 is predicted by the soliton theory in the continuum limit. We end by discussing the striking differences in the amount of energy mixing as well as the existence of the FPUT recurrences between positive and negative β and offer some remarks on the thermodynamic limit.

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Section: Role Of Mkdv Solitons In the Fput Recurrencesupporting

confidence: 52%

“…The speed, v, of the soliton given in equation (D1) has been expressed in terms of the eigenvalue E in the corresponding Schrödinger equation given by equation (25) through the N -soliton solutions as [41,64]…”

Section: Appendix D: Solitons In the Mkdv Equationmentioning

confidence: 99%

The β Fermi-Pasta-Ulam-Tsingou recurrence problem

Pace

Reiss

Campbell

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…for u 0 ∈ H 2 (R). mKdV appears as a good approximation of some physical problems as ferromagnetic vortices [44], fluid mechanics [22], electrodynamics [36], plasma physics [9,37], etc.…”

Section: Setting Of the Problemmentioning

confidence: 99%

Orbital stability of a sum of solitons and breathers of the modified Korteweg–de Vries equation

Semenov

2022

Nonlinearity

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In this article, we prove that a sum of solitons and breathers of the modified Korteweg–de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in H 2. More precisely, we will show that if a solution of mKdV is close enough to a sum of solitons and breathers with distinct velocities at t = 0 in the H 2 sense, then it stays close to this sum of solitons and breathers for any time t ⩾ 0 in the H 2 sense, up to space translations for solitons or space and phase translations for breathers, provided the condition that the considered solitons and breathers are sufficiently decoupled from each other and that the velocities of the considered breathers are all positive, except possibly one. The constants that appear in this stability result do not depend on translation parameters. From this, we deduce the orbital stability of any multi-breather of mKdV, provided the condition that the velocities of the considered breathers are all positive, except possibly one (the condition about the decoupling of the considered solitons and breathers between each other is not required in this setting). The constants that appear in this stability result depend on translation parameters of the considered solitons and breathers.

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“…Hirota [1] found the N-soliton solution for 0165-2125/$ -see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2007.11.003 the focusing mKdV equation, while both Perelman et al [2] and Ono [3] developed the N-soliton solution for the defocusing case. Of particular interest in this case is the elastic interaction of a soliton with a dissipationless shock wave.…”

Section: Introductionmentioning

confidence: 98%

Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Marchant

2008

Wave Motion

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On the relationship between the N-soliton solution of the modified Korteweg-de Vries equation and the KdV equation solution

Cited by 31 publications

References 3 publications

The β Fermi-Pasta-Ulam-Tsingou recurrence problem

The β Fermi-Pasta-Ulam-Tsingou recurrence problem

Orbital stability of a sum of solitons and breathers of the modified Korteweg–de Vries equation

Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

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