Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

  • Authors:

  • Q. Sheng;A.Q. M. Khaliq;D. A. Voss

  • Affiliations:

  • Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA;Department of Mathematics, University of Wisconsin, La Crosse, WI 54601, USA;Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA

  • Venue:
  • Mathematics and Computers in Simulation
  • Year:
  • 2005

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Abstract

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.