Abstract
We consider the logarithmic Korteweg–de Vries (log–KdV) equation, which models soli-tary 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 con-served 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 soli-tary 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 eigen-functions, however, do not decay like Gaussian functions but have algebraic decay. Using numerical approximations, we show that the Gaussian initial data do not spread out but pro-duce visible radiation at the left slope of the Gaussian-like pulse in the time evolution of the linearized log–KdV equation.