On the orbital stability of Gaussian solitary waves in the log-KdV equation
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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 $H^1(\mathbb{R})$ with conserved $L^2$ norm and
energy, we construct a weak global solution of the log-KdV equation in a subset
of $H^1(\mathbb{R})$. 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. 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.