Quasi-periodic waves to the defocusing nonlinear Schrödinger equation. (English) Zbl 1537.37093
Summary: A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the \((1+1)\)-dimensional and \((2+1)\)-dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann-Schottky problem for hyperelliptic curves with 2 infinite points.
{© 2024 IOP Publishing Ltd & London Mathematical Society}
{© 2024 IOP Publishing Ltd & London Mathematical Society}
MSC:
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B15 | Almost and pseudo-almost periodic solutions to PDEs |
Keywords:
quasi-periodic wave; defocusing nonlinear Schrödinger equation; theta function; Hirota’s bilinear method; Riemann-Schottky problemReferences:
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