Resonant solitons from the $3\times 3$ operator

TL;DR

This paper investigates resonant solitons arising from a $3\times 3$ operator, revealing their algebraic structure, asymptotic behavior, and nonlinear resonance conditions, with implications for parametric interactions like up- and down-conversion.

Contribution

It introduces the concept of resonant solitons from a $3\times 3$ operator, analyzing their algebraic structure and resonance conditions, which was not previously understood.

Findings

Resonant solitons occur when two transmission coefficients have equal eigenvalues.

Asymptotic analysis shows these solitons include parametric interactions such as up- and down-conversion.

The equality of eigenvalues acts as a nonlinear resonance condition.

Abstract

Resonant solitons of the $3\times 3$ operator are studied. The scattering data of this operator contains four transmission coefficients, two in each half complex $\zeta$-plane, where $\zeta$ is the spectral parameter. For anti-hermitian symmetry of the potential, the two transmission coefficients in the lower half plane (LHP) become equal to the complex conjugates of those in the upper half plane (UHP). The bound state scattering data for this operator consists in part of the zeros of these two transmission coefficients. Of particular interest is that class of soliton solutions when the two transmission coefficients have exactly equal eigenvalues, which gives rist to "resonant solitons". They arise from a bifurcation which is caused by the algebraic structure of the $3\times 3$ scattering matrix. We detail the asymptotics of this solution, showing that the latter contains the well known parametric interactions of "up-conversion" and "down-conversion". Lastly, we explain how this equality of eigenvalues in different transmission coefficients can be seen to be a nonlinear resonance condition.