Constance Schober

Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings

A Hamiltonian difference scheme associated with the integrable nonlinear Schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution. The chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the solutions are not subject to even constraints, the traditional mechanism of crossings of the unperturbed homoclinic orbits/manifolds is not observed.

Numerical simulation of quasi-periodic solutions of the sine-Gordon equation

Abstract Analytic solutions of the sine-Gordon equation corresponding to periodic boundary conditions can be complicated, especially if initial values are chosen in the vicinity of homoclinic orbits. For such initial values it has been demonstrated that numerical solutions develop instabilities and may become chaotic. Because of the analytical complexities one cannot easily calculate the accuracy of the numerical solutions and therefore compare different numerical schemes in a straightforward manner. In this note we evaluate numerical methods in terms of the nonlinear spectrum. In particular, it provides one with a means of comparing symplectic and nonsymplectic integrators for integrable infinite dimensional Hamiltonian systems.

The nonlinear Schro¨dinger equation: asymmetric perturbations traveling waves and chaotic structures

It is well known that for certain parameter regimes the periodic focusing Non-linear Schrodinger (NLS) equation exhibits a chaotic response when the system is perturbed. When even symmetry is imposed the mechanism for chaotic behavior is due to the symmetric subspace being separated by homoclinic manifolds into disjoint invariant regions. For the even case the transition to chaotic behavior has been correlated with the crossings of critical level sets of the constants of motion (homoclinic crossings). Using inverse spectral theory, it is shown here that in the symmetric case the homoclinic manifolds do not separate the full NLS phase space. Consequently the mechanism of homoclinic chaos due to homoclinic crossings is lost. Near integrable dynamics, when no symmetry constraints are imposed, are examined and an example of a temporal irregular solution that exhibits random flipping between left and right traveling waves is provided.

HAMILTONIAN INTEGRATORS FOR THE NONLINEAR SCHROEDINGER EQUATION

Hamiltonian integration schemes for the Nonlinear Schroedinger Equation are examined. The efficiency with respect to accuracy and integration time of an integrable scheme, a standard conservative scheme, and a symplectic method is compared.

Dispersion, group velocity, and multisymplectic discretizations

This paper examines the dispersive properties of multisymplectic discretizations of linear and nonlinear PDEs. We focus on a leapfrog in space and time scheme and the Preissman box scheme. We find that the numerical dispersion relations are monotonic and determine the relationship between the group velocities of the different numerical schemes. The group velocity dispersion is used to explain the qualitative differences in the numerical solutions obtained with the different schemes. Furthermore, the numerical dispersion relation is found to be relevant when determining the ability of the discretizations to resolve nonlinear dynamics.

Homoclinic manifolds and numerical chaos in the nonlinear Schro¨dinger equation

Crossings of homoclinic manifolds is a well known mechanism underlying observed chaos in low dimensional systems. We discuss an analogous situation as it pertains to the numerical simulation of a well known integrable partial differential equation, the nonlinear Schrodinger equation. In various parameter regimes, depending on the initial data, numerical chaos is observed due to either truncation effects or errors on the order of roundoff. The explanation of the underlying cause of the chaos being due to crossing of homoclinic manifolds induced by the numerical errors is elucidated. The nonlinear Schrodinger equation is prototypical of a much wider class of nonlinear systems in which computational chaos can be a significant factor.

Chaos in Symplectic Discretizations of the Pendulum and Sine-Gordon Equations

Homoclinic orbits play a crucial role in the dynamics of perturbations of the pendulum and sine-Gordon equations. In this paper we examine how well the homoclinic structures are preserved by symplectic discretizations. We discuss the property of exponentially small splitting distances between the stable and unstable manifolds for symplectic discretizations of the pendulum equation. A description of the sine-Gordon phase space in terms of the associated nonlinear spectral theory is provided. We examine how preservation of the homoclinic structures (i.e. the nonlinear spectrum) depends on the order of the accuracy and the symplectic property of the numerical scheme.