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

By Jianke Yang, Taras I. Lakoba. 2007

256 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/385f10330005d5686df028de0891ed18b8f63754

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In Atomic and Molecular Physics, and Optics

1. Jianke Yang
2. Taras I. Lakoba
3. Sergei K. Turitsyn
4. Yan Rybalko
5. Dmitry Shepelsky
6. Wenxiu Ma
7. Bo Yang

In Statistical and Nonlinear Physics

1. Jianke Yang
2. Taras I. Lakoba
3. Sergei K. Turitsyn
4. Bo Yang
5. Yan Rybalko
6. Dmitry Shepelsky
7. Wenxiu Ma

Citations in the corpus

• Accelerated Imaginary‐time Evolution Methods for the Computation of Solitary Waves (2007, methodology): “It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.”