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 $L2$ 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.