Constance M. Schober

Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrodinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

The cubic nonlinear Schrodinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The nonlinear spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete.

Dynamical criteria for rogue waves in nonlinear Schrödinger models

We investigate rogue waves in deep water in the framework of the nonlinear Schrodinger (NLS) and Dysthe equations. Amongst the homoclinic orbits of unstable NLS Stokes waves, we seek good candidates to model actual rogue waves. In this paper we propose two selection criteria: stability under perturbations of initial data, and persistence under perturbations of the NLS model. We find that requiring stability selects homoclinic orbits of maximal dimension. Persistence under (a particular) perturbation selects a homoclinic orbit of maximal dimension all of whose spatial modes are coalesced. These results suggest that more realistic sea states, described by JONSWAP power spectra, may be analyzed in terms of proximity to NLS homoclinic data. In fact, using the NLS spectral theory, we find that rogue wave events in random oceanic sea states are well predicted by proximity to homoclinic data of the NLS equation.

Dispersive properties of multisymplectic integrators

Multisymplectic (MS) integrators, i.e. numerical schemes which exactly preserve a discrete space-time symplectic structure, are a new class of structure preserving algorithms for solving Hamiltonian PDEs. In this paper we examine the dispersive properties of MS integrators for the linear wave and sine-Gordon equations. In particular a leapfrog in space and time scheme (a member of the Lobatto Runge-Kutta family of methods) and the Preissman box scheme are considered. We find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved. The group velocity dispersion (GVD) is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes. Further, the numerical dispersion relations for the linearized sine-Gordon equation provides information on the ability of the MS integrators to capture the sine-Gordon dynamics. We are able to link the numerical dispersion relations to the total energy of the various methods, thus providing information on the coarse grid behavior of MS integrators in the nonlinear regime.

Conformal conservation laws and geometric integration for damped Hamiltonian PDEs

Conformal conservation laws are defined and derived for a class of multi-symplectic equations with added dissipation. In particular, the conservation laws of energy and momentum are considered, along with those that arise from linear symmetries. Numerical methods that preserve these conformal conservation laws are presented in detail, providing a framework for proving a numerical method exactly preserves the dissipative properties considered. The conformal methods are compared analytically and numerically to standard conservative methods, which includes a thorough inspection of numerical solution behavior for linear equations. Damped Klein-Gordon and sine-Gordon equations, and a damped nonlinear Schrodinger equation, are used as examples to demonstrate the results.

Rogue Waves in Higher Order Nonlinear Schrödinger Models

We discuss physical and statistical properties of rogue wave generation in deep water from the perspective of the focusing Nonlinear Schrodinger equation and some of its higher order generalizations. Numerical investigations and analytical arguments based on the inverse spectral theory of the underlying integrable model, perturbation analysis, and statistical methods provide a coherent picture of rogue waves associated with nonlinear focusing events. Homoclinic orbits of unstable solutions of the underlying integrable model are certainly candidates for extreme waves, however, for more realistic models such as the modified Dysthe equation two novel features emerge: (a) a chaotic sea state appears to be an important mechanism for both generation and increased likelihood of rogue waves; (b) the extreme waves intermittently emerging from the chaotic background can be correlated with the homoclinic orbits characterized by maximal coalescence of their spatial modes.

Observable and reproducible rogue waves

In physical regimes described by the cubic, focusing, nonlinear Schrodinger (NLS) equation, the N-dimensional homoclinic orbits of a constant amplitude wave with N unstable modes appear to be good candidates for experimentally observable and reproducible rogue waves. These homoclinic solutions include the Akhmediev breathers (Nxa0=xa01), which are among the most widely adopted spatially periodic models of rogue waves, and their multi-mode generalizations (Nxa0>xa01), and will be referred to as multi-mode breathers. Numerical simulations and a linear stability analysis indicate that the breathers with a maximal number of modes (maximal breathers) are robust with respect to rather general perturbations of the initial data in a neighborhood of the unstable background.

Efficiency of exponential time differencing schemes for nonlinear Schrödinger equations

The nonlinear Schrodinger (NLS) equation and its higher order extension (HONLS equation) are used extensively in modeling various phenomena in nonlinear optics and wave mechanics. Fast and accurate nonlinear numerical techniques are needed for further analysis of these models. In this paper, we compare the efficiency of existing Fourier split-step versus exponential time differencing methods in solving the NLS and HONLS equations. Soliton, Stokes wave, large amplitude multiple mode breather, and N-phase solution initial data are considered. To determine the computational efficiency we determine the minimum CPU time required for a given scheme to achieve a specified accuracy in the solution u(x, t) (when an analytical solution is available for comparison) or in one of the associated invariants of the system. Numerical simulations of both the NLS and HONLS equations show that for the initial data considered, the exponential time differencing scheme is computationally more efficient than the Fourier split-step method. Depending on the error measure used, the exponential scheme can be an order of magnitude more efficient than the split-step method.

Conservation properties of multisymplectic integrators

Recent results on the local and global properties of multisymplectic discretizations of Hamiltonian PDEs are discussed. We consider multisymplectic (MS) schemes based on Fourier spectral approximations and show that, in addition to a MS conservation law, conservation laws related to linear symmetries of the PDE are preserved exactly. We compare spectral integrators (MS versus non-symplectic) for the nonlinear Schrodinger (NLS) equation, focusing on their ability to preserve local conservation laws and global invariants, over long times. Using Lax-type nonlinear spectral diagnostics we find that the MS spectral method provides an improved resolution of complicated phase space structures.

Numerical investigation of the stability of the rational solutions of the nonlinear Schrdinger equation

HightlightsA broad numerical investigation involving large ensembles of perturbed initial data indicates the Peregrine and second order rational solutions of the NLS equation are linearly unstable.A highly accurate Chebyshev pseudo-spectral method is developed for solving the NLS equation that uses the map x=cot in combination with the Fast Fourier Transform to approximate uxx on an infinite domain.A modified Fourier spectral method that can treat initial data with discontinuous derivatives over periodic domains is developed for solving the NLS equation, of which the resolution of the Peregrine solution is a particular example. The rational solutions of the nonlinear Schrdinger (NLS) equation have been proposed as models for rogue waves. In this article, we develop a highly accurate Chebyshev pseudo-spectral method (CPS4) to numerically study the stability of the rational solutions of the NLS equation. The scheme CPS4, using the map x=cot and the FFT to approximate uxx, correctly handles the infinite line problem. A broad numerical investigation using CPS4 and involving large ensembles of perturbed initial data, indicates the Peregrine and second order rational solutions are linearly unstable.Although standard Fourier integrators are often used in current studies of the NLS rational solutions, they do not handle solutions with discontinuous derivatives correctly. Using standard Fourier pseudo-spectral method (FPS4) for Peregrine initial data yields tiny Gibbs oscillations in the first steps of the numerical solution. These oscillations grow to O(1), providing further evidence of the instability of the Peregrine solution. To resolve the Gibbs oscillations we modify FPS4 using a spectral-splitting technique which significantly improves the numerical solution.