Abstract
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.
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