On the orbital stability of Gaussian solitary waves in the logKdV equation
Abstract
We consider the logarithmic Kortewegde Vries (logKdV) 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 logKdV equation in a subset of H^1({R}) . This construction yields conditional orbital stability of Gaussian solitary waves of the logKdV equation, provided that uniqueness and continuous dependence of the constructed solution holds.
Furthermore, we study the linearized logKdV 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. Using numerical approximations, we show that the Gaussian initial data do not spread out but produce visible radiation at the left slope of the Gaussianlike pulse in the time evolution of the linearized logKdV equation.
 Publication:

Nonlinearity
 Pub Date:
 December 2014
 DOI:
 10.1088/09517715/27/12/3185
 arXiv:
 arXiv:1401.1738
 Bibcode:
 2014Nonli..27.3185C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 21 pages