Multiple Soliton Solutions for a New Generalization of the Associated Camassa-Holm Equation by Exp-Function Method

The Exp-function method is generalized to construct N-soliton solutions of a new generalization of the associated Camassa-Holm equation. As a result, one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formulae of N-soliton solutions are derived. It is shown that the Exp-function method may provide us with a straightforward, effective, and alternative mathematical tool for generating N-soliton solutions of nonlinear evolution equations in mathematical physics

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.Comment: to appears in Physical Review E. 4 pages, revtex file

Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.Comment: 31 page

Integrable nonlinear evolution equations.

by Zheng Yu-kun.Thesis (Ph.D.)--Chinese University of Hong Kong, 1991.Includes bibliographical references.Preface --- p.1Chapter Chapter 1. --- Gauge Transformation and the Higher Order Korteweg-de Vries Equations --- p.6Chapter 1. --- Higher order KdV equations --- p.6Chapter 2. --- η2-dependent higher order mKdV equations --- p.9Chapter 3. --- η2-dependent Miura transformation and Backlund transformation --- p.13Chapter 4. --- Gauge transformation of the wave function --- p.15Chapter 5. --- Backlund transformation for the η2 -dependent higher order mKdV equation --- p.24Chapter 6. --- Applications --- p.25Chapter 7. --- References --- p.30Chapter Chapter 2. --- Solutions of a Nonisospectral and Variable Coefficient Korteweg-de Vries Equation --- p.31Chapter 1. --- Introduction --- p.31Chapter 2. --- Nonisospectral variable coefficient KdV-type equations --- p.32Chapter 3. --- Invariance of LP under the Crum transformation --- p.34Chapter 4. --- Backlund transformation for the h-t-KdV equation --- p.35Chapter 5. --- Solutions --- p.39Chapter 6. --- References --- p.43Chapter Chapter 3. --- Nonisospectral Variable Coefficient Higher Order Korteweg-de Vries Equations --- p.45Chapter 1. --- Introduction --- p.45Chapter 2. --- Nonisospectral t-ho-KdV equations --- p.47Chapter 3. --- Nonisospectral η2 dependent variable coefficient higher order modified KdV equation --- p.50Chapter 4. --- Backlund transformation and gauge transformation --- p.57Chapter 5. --- Example. Solutions of second order ni-t-KdV equation and its corresponding ni-t-η2-mKdV equation --- p.61Chapter 6. --- References --- p.66Chapter Chapter 4. --- Gauge and Backlund Transformations for the Variable Coefficient Higher-Order Modified Korteweg-de Vries Equation --- p.67Chapter 1. --- Introduction --- p.67Chapter 2. --- The t-ho-mKdV equation --- p.68Chapter 3. --- Some results about the t-ho-KdV equation --- p.74Chapter 4. --- A Backlund transformation for the t-ho-mKdV equation --- p.76Chapter 5. --- Gauge transformat ion and the Backlund transformation --- p.78Chapter 6. --- References --- p.85Chapter Chapter 5. --- Gauge and Backlund Transformat ions for the Generalized Sine-Gordon Equation and Its η Dependent Modified Equation --- p.86Chapter 1. --- Introduction --- p.86Chapter 2. --- Generalized Sine-Gordon equation --- p.87Chapter 3. --- Backlund transformation for the GSGE --- p.92Chapter 4. --- Gauge transformations for AKNS systems --- p.98Chapter 5. --- η dependent modified GSGE and its Backlund transformation --- p.102Chapter 6. --- Summary and example --- p.105Chapter 7. --- References --- p.110Chapter Chapter 6. --- Backlund Transformation for the Nonisospectral and Variable Coefficient Nonlinear Schrodinger Equation --- p.111Chapter 1. --- Introduction --- p.111Chapter 2. --- A generalized NLSE --- p.112Chapter 3. --- Γ Riccati equation system --- p.114Chapter 4. --- Invariance of the Γ-system --- p.116Chapter 5. --- Lax pair corresponding to the GNLSE --- p.119Chapter 6. --- BT´ةs for the Γ evolution equation and the GNLSE --- p.121Chapter 7. --- References --- p.126Chapter Chapter 7. --- Backlund Transformations for the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation and Its λ-Modified Equation --- p.127Chapter 1. --- Introduction --- p.127Chapter 2. --- The CDGSKE and the λ-mCDGSKE --- p.128Chapter 3. --- The general solution for the scattering problem of the CDGSKE --- p.130Chapter 4. --- The BT for the λ-mCDGSKE --- p.135Chapter 5. --- The BT for the CDGSKE --- p.136Chapter 6. --- References --- p.139Summary --- p.14