Modulational instability and homoclinic orbit solutions in vector nonlinear Schr\"odinger equation

TL;DR

This paper develops a method to derive and analyze multi-component rogue wave and breather solutions in nonlinear Schrödinger equations, linking modulational instability to homoclinic orbit solutions.

Contribution

It introduces a general approach for obtaining multi-high-order rogue wave and breather solutions in N-component nonlinear Schrödinger equations, clarifying existence conditions.

Findings

Derived general forms for Akhmediev breather and rogue wave solutions.

Constructed multi-high-order rogue wave and breather solutions.

Linked modulational instability with homoclinic orbit solutions.

Abstract

Modulational instability has been used to explain the formation of breather and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. We develop a method to derive general forms for Akhmediev breather and rogue wave solutions in a $N$-component nonlinear Schr\"odinger equations. The existence condition for each pattern is clarified clearly. Moreover, the general multi-high-order rogue wave solutions and multi-Akhmediev breather solutions for $N$-component nonlinear Schr\"odinger equations are constructed. The results further deepen our understanding on the quantitative relations between modulational instability and homoclinic orbits solutions.