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D.E. Pelinovsky

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On the linearized log-KdV equation

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Communications in Mathematical Sciences 15, 863-880 (2017)

**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 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 solitary waves. We also
introduce a convolution representation of solutions to the 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:**

Log-KdV equation, Gaussian solitary waves, linear and spectral stability, Hermite functions,
Jacobi difference equation, energy estimates.