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Stability of Solitary Waves (Workshop)

Orbital stability of Gaussian solitary waves for the logarithmic KdV equation

speaker: Dmitry Pelinovsky (McMaster University, Canada. )

abstract: We consider the logarithmic Korteweg--de Vries (log--KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in $H¹(\mathbb{R})$ with conserved $L^2$ norm and energy, we construct a weak global solution of the log--KdV equation in a subset of $H¹(\mathbb{R})$. This construction yields conditional orbital stability of Gaussian solitary waves of the log--KdV equation, provided uniqueness and continuous dependence of the constructed solution holds.

Furthermore, we study the linearized log--KdV equation at the Gaussian solitary wave and prove that the associated linearized operator has a purely discrete spectrum consisting of simple purely imaginary eigenvalues in addition to the double zero eigenvalue. The eigenfunctions, however, do not decay like Gaussian functions but have algebraic decay.

timetable:
Tue 27 May, 11:15 - 12:00, Aula Dini
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