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R. Carles and D. Pelinovsky

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On the orbital stability of Gaussian solitary waves in the log-KdV equation

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Nonlinearity 27 (2014), 3185-3202

**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 H^{1}(R) with conserved L^{2} norm and energy,
we construct a weak global solution of the log-KdV equation in
a subset of H^{1}(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.
Nevertheless, using numerical approximations, we show that
the Gaussian initial data do not spread out and preserve their spatial Gaussian decay in the time
evolution of the linearized log-KdV equation.

**Keywords:**

Discrete FPU lattice, granular crystals, log-KdV equation, Gaussian solitary waves, linear stability.