Lyapunov Inequalities for Bound States of Linear Hamiltonian Systems and Laplace Transform With Measure Preserving

Abstract

We study the Lyapunov inequalities for the time scales, for discrete linear Hamiltonian systems on the time scales. We show the strong instability and stability standing waves for nonlinear Klein – Gordon eqations and Klein – Gordan- Zakharov system. We also show the stability and instability of standing waves for one- dimensional nonlinear Schrodinger equation with multiple – power nonlinearity and stability of bound states of Hamiltonian partial differential equations in the degenerated cases. The moment identities for Skorohod integrals, random Hermite polynomials with Girsanov identities on the Wiener space are characterized. The transfer principle method from Wiener to Poisson space and measure invariance on the Lie- Wiener path space are discussed. We show the Laplace transform identities and describe the measure – preserving transformations on the Lie- Wiener – Poisson spaces.