Communications in Mathematical Sciences

Volume 15 (2017)

Number 3

On the linearized log-KdV equation

Pages: 863 – 880

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n3.a13

Author

Dmitry E. Pelinovsky (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada; and Department of Applied Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia)

Abstract

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log-KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution of the linearized log-KdV equation is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of Gaussian solitary waves. We also introduce a convolution representation of solutions to the linearized log-KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.

Keywords

logarithmic KdV equation, Gaussian solitary waves, Hermite functions, Jacobi difference equation, semi-groups, linear orbital stability, spectral stability

2010 Mathematics Subject Classification

35P10, 35Q53, 37K35, 39A70

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Published 24 February 2017