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

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