Higher-order modulation instability in a fourth-order Nonlinear Schr\"{o}dinger Equation

TL;DR

This paper provides a comprehensive analysis of higher-order modulation instability in a fourth-order nonlinear Schrödinger equation, revealing geometric structures in parameter space and extending these insights to the entire hierarchy of such equations.

Contribution

It introduces a complete dynamical description of higher-order modulation instability and characterizes the geometric bounds of parameter space for solutions.

Findings

Parameter space bounded by a circle for standard NLS.

Parameter space bounded by an intersecting circle and ellipse for the fourth-order NLS.

Similar geometric structures extend to higher-order equations in the hierarchy.

Abstract

We present a complete dynamical description of the higher-order modulation instability for a fourth-order nonlinear Schr\"{o}dinger equation. For two-breather solutions of this equation, we have identified the locus in a geometrical space where the growth rates for the breathers are equal in parameter space. We show that a circle bounds the entire parameter space for the nonlinear Schr\"{o}dinger equation. In contrast, it is bound by an intersecting circle and an ellipse for the fourth-order equation. We show that, for all the higher-order equations in the nonlinear Schr\"{o}dinger equation hierarchy, the parameter space follows a similar geometric interpretation as for the fourth-order equation.