Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared‐operator method, the modified squared‐operator method, and the power‐conserving squared‐operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equati...

Full description

Saved in:
Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) 2007-02, Vol.118 (2), p.153-197
Main Authors: Yang, Jianke, Lakoba, Taras I.
Format: Article
Language:English
Subjects:
Constants
Differential equations
Energy conservation
Iterative methods
Mathematical analysis
Methods
Nonlinearity
Solitary waves
Solitons
Studies
Wave equations
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Three new iteration methods, namely the squared‐operator method, the modified squared‐operator method, and the power‐conserving squared‐operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast‐converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared‐operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher‐gap vortex solitons in the two‐dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared‐operator method delivers the best performance among the methods proposed in this article.
ISSN:0022-2526
1467-9590
DOI:10.1111/j.1467-9590.2007.00371.x