The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds
Bilinearisation and reduction approach
Da Jun Zhang (Shanghai University)
Shi Min Liu (Shanghai University)
Xiao Deng (TU Delft - Mathematical Physics)
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Abstract
In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schrödinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions (q0, r0) we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution (q0, r0) as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reversespace nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.