𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On a class of discretizations of Hamiltonian nonlinear partial differential equations

✍ Scribed by P.G Kevrekidis

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
207 KB
Volume
183
Category
Article
ISSN
0167-2789

No coin nor oath required. For personal study only.

✦ Synopsis

We present a new class of discretizations of partial differential equations (PDEs) that preserve a (momentum-like) integral of the motion. This results in an "effective" translational invariance for the dynamical problem and the absence of a Peierls-Nabarro barrier that is usually present in discretizations of Hamiltonian PDEs. A general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential-difference equations are analyzed in a number of different cases for nonlinear Klein-Gordon as well for nonlinear SchrΓΆdinger type systems. While for the former nonintegrability of the dynamical problem is evident, in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems. We also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones.

πŸ“œ SIMILAR VOLUMES

On a certain class of a nonlinear evolut
On a certain class of a nonlinear evolution partial differential equations
✍ S. Mesloub; H.A. Abdusalam πŸ“‚ Article πŸ“… 2008 πŸ› Elsevier Science 🌐 English βš– 159 KB

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered

Invariant Manifolds for a Class of Dispe
Invariant Manifolds for a Class of Dispersive, Hamiltonian, Partial Differential Equations
✍ Claude-Alain Pillet; C.Eugene Wayne πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 255 KB

We construct an invariant manifold of periodic orbits for a class of non-linear Schro dinger equations. Using standard ideas of the theory of center manifolds, we rederive the results of Soffer and Weinstein (Comm. Math. Phys. 133, 119 146 (1997); J. Differential Equations 98, 376 390 (1992)) on the