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

Abstract

Summary:Three new iteration methods, namely the squared-operator method, the modified squaredoperator 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.

A mode elimination technique to improve convergence of iteration methods for finding solitary waves

“…(3.3) and (3.4). In Example 4.1 in Section 4, we demonstrated that this technique can be superior to an alternative, squared-operator, technique [8] when applied to finding lowest-order nonfundamental solutions. However, for finding higher-order solutions, the technique of Ref.…”

“…However, for finding higher-order solutions, the technique of Ref. [8] appears to be more practical.…”

“…We will also compare the performance of this method with that of a universally-convergent method proposed in [8].…”

Newton-conjugate-gradient methods for solitary wave computations

“…So far, a number of numerical methods have been developed. Examples include the Newton's method [1,2], the shooting method [3], the Petviashvili-type methods [4][5][6], the accelerated imaginary time evolution methods [7,8], the squared-operator iteration methods [9], etc. The Newton's method is a classical iteration method.…”

Bulk vortices and half-vortex surface modes in parity-time-symmetric media

“…The stationary solution with real propagation constant b is looked for as q(ξ, η, ζ) = u(ξ, η)e ibζ , where complex function u(ξ, η) obeys equation To find the stationary self-trapping solutions, we used numerical simulations with the modified squaredoperator method [48]. While the existence and stability of the simplest single-beam solitons in the present setting is quite evident, as the first step of the analysis we produce double-beam self-trapped states.…”