On the Hamiltonian-Krein index for a non-self-adjoint spectral problem

TL;DR

This paper develops a method to determine the instability index of a non-self-adjoint spectral problem related to soliton stability, linking it to spectral properties of a Schrödinger operator.

Contribution

It introduces a formula for the instability index of a non-self-adjoint problem using spectral data from an associated Schrödinger operator.

Findings

Provides a practical formula for the instability index.

Connects non-self-adjoint spectral problems to Schrödinger operator spectra.

Applicable to stability analysis of solitons in nonlinear equations.

Abstract

We investigate the instability index of the spectral problem $$ -c^2y'' + b^2y + V(x)y = -\mathrm{i} z y' $$ on the line $\mathbb{R}$, where $V\in L^1_{\rm loc}(\mathbb{R})$ is real valued and $b,c>0$ are constants. This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). We show how to apply the standard approach in the situation under consideration and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schr\"odinger operator $H_V=-c^2\frac{d^2}{dx^2}+b^2 +V(x)$.