Dynamics of kink-soliton solutions of the \((2+1)\)-dimensional sine-Gordon equation. (English. Russian original) Zbl 1486.81102
Theor. Math. Phys. 210, No. 1, 68-84 (2022); translation from Teor. Mat. Fiz. 210, No. 1, 80-98 (2022).
Summary: We study the dynamics of explicit solutions of the \((2+1)\)-dimensional (2D) sine-Gordon equation. The Darboux transformation is applied to the associated linear eigenvalue problem to construct nontrivial solutions of the 2D sine-Gordon equation in terms of a ratio of determinants. We obtain a generalized expression for an \(N\)-fold transformed dynamical variable, which enables us to calculate explicit expressions of nontrivial solutions. To explore the dynamics of kink soliton solutions, explicit expressions for one- and two-soliton solutions are derived for particular column solutions. Different profiles of kink-kink and kink-anti-kink interactions are illustrated for different parameters and arbitrary functions. We also present a first-order bound state solution.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35C08 | Soliton solutions |
| 35Q05 | Euler-Poisson-Darboux equations |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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