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Asymptotic stability of solitary waves in the Benney-Luke model of water waves. (English) Zbl 1289.37043

The authors study the stability of a solitary wave solution of the nonlinear dispersive wave equation \[ \partial _t^2u-\partial _x^2u+a\partial _x^2\partial _t^2u+(\partial _tu)(\partial _x^2u)+2(\partial _xu)(\partial _x\partial _tu)=0, \] with any speed \(c>1\) and \(0<a<b\). In the first part, the authors show that, in the case of the absence of nonzero eigenvalues with nonnegative real part, spectral stability implies linear stability with an exponential decay rate in the exponentially weighted norm for perturbations orthogonal to the adjoint neutral-mode space generated by variations of the phase and wave amplitudes. The proof uses a suitable comparison of a reduced resolvent operator with the corresponding one for the KdV solitons.
In the second part, it is proved that the nonlinear stability follows from the spectral stability.

MSC:

37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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