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D.E. Pelinovsky, E.A. Ruvinskaya, O.E. Kurkina, and B. Deconinck

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Short-wave transverse instabilities of line solitons of the
two-dimensional hyperbolic nonlinear Schrodinger equation

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Theoretical Mathematical Physics 179, 452-461 (2014)

**Abstract:**

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schrodinger
equation are unstable with respect to transverse perturbations of arbitrarily small periods,
i.e., short waves. The analysis is based on the construction of Jost functions
for the continuous spectrum of Schrodinger operators,
the Sommerfeld radiation conditions, and the Lyapunov-Schmidt decomposition. Precise asymptotic
expressions for the instability growth rate are derived in the limit of short periods.

**Keywords**:

Water waves, solitary wave, transverse instability, Fermi golden rule, Lyapunov-Schmidt reductions,
Sommerfeld radiation condition.