Communications in Mathematical Sciences
Volume 15 (2017)
On the linearized log-KdV equation
Pages: 863 – 880
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.
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