Reduction to modified KdV and its KP-like generalization via phase modulation

Daniel J. Ratliff; Thomas J. Bridges

doi:10.1088/1361-6544/aabfab

Reduction to modified KdV and its KP-like generalization via phase modulation

Daniel J. Ratliff, Thomas J. Bridges

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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Abstract

The main observation of this paper is that the modified Korteweg-de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2 + 1 suggests that a modified Kadomtsev-Petviashvili (or a Konopelchenko-Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2 + 1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schrödinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2 + 1 CQNLS equation.

Original language	English
Pages (from-to)	3794-3813
Number of pages	20
Journal	Nonlinearity
Volume	31
Issue number	8
DOIs	https://doi.org/10.1088/1361-6544/aabfab
Publication status	Published - 9 Jul 2018
Externally published	Yes

Keywords

Lagrangian fields
nonlinear waves
phase dynamics

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10.1088/1361-6544/aabfab

DJR-TJB_Nonlinearity_April2018Accepted author manuscript, 491 KBLicence: CC BY-NC-ND

http://epubs.surrey.ac.uk/846279/Licence: CC BY-NC-ND

Cite this

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abstract = "The main observation of this paper is that the modified Korteweg-de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2 + 1 suggests that a modified Kadomtsev-Petviashvili (or a Konopelchenko-Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2 + 1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schr{\"o}dinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2 + 1 CQNLS equation.",

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Reduction to modified KdV and its KP-like generalization via phase modulation

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Keywords

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