## Modulational approach to stability of non-topological solitons in semilinear wave equations.(English)Zbl 1158.35389

The stability properties of a class of solitary wave solutions of the equation $$\square \phi+m^2\phi=\beta(|\phi|)\phi$$, where $$\phi:\mathbb{R}^{1+n}\to\mathbb{C}$$, are considered in this paper, where solitary waves of the form $$e^{i\omega t}f(x)$$ are called non-topological solitons. A modulational approach is developed similar to that used with the nonlinear Schrödinger equation, and novel features are displayed in connection with the $$\omega$$-dependence of the stability condition. It is shown that a stability condition can be derived precisely from the Gagliardo-Nirenberg inequality, while the main result of the paper is a strengthening, for this equation, of existing very general stability results for waves in Hamiltonian systems.

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35B35 Stability in context of PDEs 35Q51 Soliton equations
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