Numerical study of the generalised Klein-Gordon equations

24 pages, 10 figures, 56 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets

Energy conservation issues in the numerical solution of the semilinear wave equation

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur

New degeneration of Fay's identity and its application to integrable systems

In this paper we prove a new degenerated version of Fay's trisecant identity. The new identity is applied to construct new algebro-geometric solutions of the multi-component nonlinear Schr\"odinger equation. This approach also provides an independent derivation of known algebro-geometric solutions to the Davey-Stewartson equations

LONG TIME DYNAMICS OF THE KLEIN-GORDON EQUATION IN THE NON-RELATIVISTIC LIMIT

In this thesis I study the non-relativistic limit ($c \to \infty$) of the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$, namely $$\frac{1}{c^2} u_{tt} - \Delta u + c^2 u + \lambda |u|^{2(l-1)}u = 0, \; \; t \in \R, x \in M$$ where $\lambda = \pm 1$, $l \geq 2$. The aim of the present work is to discuss the convergence of solutions of the NLKG to solutions of a suitable nonlinear Schr\"odinger (NLS) equation, and to study whether such convergence may hold for large (namely, of size$O(c^r)$ with $r \geq 1$) timescales. In particular I obtain the following results: (1) when $M$ is a general manifold, I show that the solution of NLS describes well the solution of the original equation up to times of order \cO(1); (2) when $M=\mathbb{R}^d$,$d \geq 3$, I consider higher order approximations of NLKG and prove that small radiation solutions of the approximating equation describe well solutions of NLKG up to times of order $O(c^{2r})$ for any $r \geq 1$; (3) when $M=[0,\pi] \subset \mathbb{R}$ I consider the NLKG equation with aconvolution potential and prove existence for long times of solutionsin $H^s$ uniformly in $c$, which however has to belong to a set of large measure. I also get some new dispersive estimates for a Klein-Gordon type equation with a potential

Symmetries in Quantum Mechanics and Statistical Physics

This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed