Soliton solutions of a coupled Korteweg-de Vries equation

Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.

Nonlinear Partial Differential Equations for Scientists and Engineers

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211208700288X

A high-order spectral method for the study of nonlinear gravity waves

Publisher Summary This chapter reviews some recent progress in the nonlinear dynamics of deep-water gravity waves. It attempts to highlight the major developments in theory and experiment commencing with the finding by Lighthill (1965) that a nonlinear, deep-water gravity wave train is unstable to modulational perturbation, up to the present investigations of various aspects of nonlinear phenomena, including three-dimensional instabilities, bifurcations into new steady solutions, statistical properties of random wave fields, and chaotic behavior in time evolution. The governing equations for inviscid, irrotational, incompressible, free surface flows are given in Section II of the chapter, together with some basic steady solutions of the system. The concept of a wave train is introduced in Section III. The stability and evolutionary properties of a weakly nonlinear wave train in two dimensions are considered in Section IV, based on the nonlinear Schrodinger equation, which is an equation describing the wave envelope. Some interesting phenomena, such as the existence of envelope solitons, and the Fermi-Pasta-Ulam recurrence in time of an unstable wave train, are examined. Section V extends these results to three dimensions, using the three-dimensional nonlinear Schrodinger equation. The results indicate that whereas the nonlinear Schrodinger equation is remarkably successful in describing the two-dimensional dynamics, it is inadequate for treatment of the three-dimensional case.

Nonlinear Dynamics of Deep-Water Gravity Waves

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose–Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.

https://iopscience.iop.org/article/10.1088/1751-8113/43/21/213001/pdf

Dark solitons in atomic Bose–Einstein condensates: from theory to experiments

Formation and Interaction of Sonic-Langmuir Solitons Inverse Scattering Method

Nonlinear evolutions of a two-dimensional ion-acoustic wave system are studied analytically and numerically. The solution describing the resonant interaction of three solitons in two-dimensional space is obtained under the approximation of long wavelength and small but finite amplitude. Numerical solutions reveal an importance of soliton resonances in nonlinear developments of two-dimensional ion-acoustic waves.

Interaction of ion-acoustic solitons in two-dimensional space

A Class of Exactly Solvable Nonlinear Evolution Equations

Three-dimensional interactions of ion-acoustic solitons in collisionless plasmas are studied by using Hirota's method. It is pointed out that under certain conditions the resonant interaction among three solitons is possible.

Interaction of Ion-Acoustic Solitons in Three-Dimensional Space

The reductive perturbation method is extended to apply to a strongly dispersive system, in which mudulated plane waves interact each other through nonlinear interactions. The interaction between two envelope solitons is examined.

Nobuo Yajima

Papers

Formation and Interaction of Sonic-Langmuir Solitons Inverse Scattering Method

Interaction of ion-acoustic solitons in two-dimensional space

A Class of Exactly Solvable Nonlinear Evolution Equations

Interaction of Ion-Acoustic Solitons in Three-Dimensional Space

A perturbation approach to nonlinear systems. II. interaction of nonlinear modulated waves