Breather and rogue-wave solutions of the semi-discrete and continuous nonlinear Schrödinger equations on theta-function backgrounds

Abstract A new method to construct localized-wave solutions of the semi-discrete and continuous nonlinear Schrödinger (sd-NLS) equations on theta-function backgrounds simultaneously is developed. (1) The semi-discrete NLS equation is rewritten as the difference-quotient NLS equation so that the continuum limits are easy to calculate. (2) By rewriting the difference-quotient equation in the form of Hiorta bilinear equations, a theta-function seed solution for the difference-quotient NLS equation is derived, in which the theta-function is so general that both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>dn</mml:mi> </mml:mrow> </mml:math> - and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>cn</mml:mi> </mml:mrow> </mml:math> -type seed solutions are its special reductions. (3) Using the Baker–Akhiezer functions on algebraic curves, a systematic method for solving continuous and semi-discrete spectral problems with theta-function potentials is given, from which rogue-wave and breather solutions of the sd-NLS equation on theta-function backgrounds are constructed with the aid of the derived Darboux transformations. (4) By investigating the exact solutions of the difference-quotient NLS equation, it is found that the sd-NLS equation has extremely high rogue waves ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>600</mml:mn> </mml:mrow> </mml:math> times of the background wave), which is not observed in the NLS equation. This method has universality and can be applied to other semi-discrete integrable equations.