arXiv subjects

V. Merle

Publications and source records attributed to V. Merle.

3 recordsLinked to original sources

arXiv1997

Asymptotic dynamics of short-waves in nonlinear dispersive models

The multiple-scale perturbation theory, well known for long-waves, is extended to the study of the far-field behaviour of short-waves, commonly called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation can propagates short-waves. This result contradict the Benjamin hypothesis that short-waves tends not to propagate in this model and close a part of the old controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon equations are understood as an all orders asymptotic behaviour of short-waves. It is proved that the antikink solution of phi-4 model which was never obtained perturbatively can be obtained by perturbation expansion in the wave-number k in the short-wave limit.

solv-int↗

arXiv1997

Modified Korteweg-de Vries Hierachies in Multiple-Times Variables and the Solutions of Modified Boussinesq Equations

We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each order of the perturbative scheme. We show that the multiple-time variables needed to obtain a regular perturbative series are completely determined by the associated linear theory in the case of a solitary-wave solution, but requires the knowledge of each order of the perturbative series in the case of a kink-wave solution. These appropriate multiple-time variables allow us to show that the solitary-wave as well as the kink-wave solutions of the modified Botussinesq equation are actually respectively a solitary-wave and a kink-wave satisfying all the equations of suitable modified Korteweg-de Vries hierarchies.

solv-int↗

arXiv1996

Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation

By considering the long-wave limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg-de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple scale method in obtaining uniform perturbative series.

patt-sol↗