Parametric Instability in Mathieu Equation for Interaction P-S Waves

P. Mathieu

Cantor Spectrum for the Almost Mathieu Equation

Recently, there has been an explosion of interest in the study of Schrijdinger operators and Jacobi matrices with almost periodic potential (see, e.g., the review [ 161). The general belief is that generically the spectrum is a Cantor set, i.e., a nowhere dense perfect, closed set. Since it is easy to prove that the spectrum is closed and perfect (see, e.g., [2]), the key is to prove that the s...

Instability of Periodic Sign-changed Waves for Sine-gordon Equation

The present paper deals with results of orbital instability of periodic waves related to sine-Gordon equation. Our periodic solutions obtained in our study tend to a kink wave solutions in the infinite wavelength scenario. The approaches used here will be the classical Grillakis, Shatah and Strauss’ theory.

Instability of Solitary Waves for a Nonlinearly Dispersive Equation

Solitary-wave solutions of a nonlinearly dispersive evolution equation are considered. It is shown that these waves are unstable in a certain parameter range.

WKB and Resurgence in the Mathieu Equation

In this paper, based on lectures by the authors at the May 2015 workshop Resurgence, Physics and Numbers, at the Centro di Ricerca Matematica Ennio De Giorgio of the Scuola Normale Superiore in Pisa, we explain the origin of resurgent trans-series in the Mathieu equation spectral problem, using uniform WKB and all-orders (exact) WKB. Exact quantization conditions naturally arise, and their expa...