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

Kostenko, Aleksey; Nicolussi, Noema

doi:10.1090/proc/14048

On the Hamiltonian–Krein index for a non-self-adjoint spectral problem
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by Aleksey Kostenko and Noema Nicolussi PDF

Proc. Amer. Math. Soc. 146 (2018), 3907-3921 Request permission

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 $V\in L^1_\textrm {loc}(\mathbb {R})$ 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 Schrödinger operator $H_V=-c^2\frac {d^2}{dx^2}+b^2 +V(x)$.

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