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

Abstract

We investigate the instability index of the spectral problem $$-c^2{y}^{\prime \prime }+b^{2}y+V(x)y=-\mathrm{i}z{y}^{\prime }$$ on the line $\mathbb{R}$, where $V\in L_{\mathrm{loc}}^1(\mathbb{R})$ 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 Schrödinger operator $H_V=-c^2\frac{d^2}{d{x}^2}+b^2+V(x)$.

15 pages; to appear in Proc. Amer. Math. Soc

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