Using the inverse spectral theory of nonlinear Schrödinger (NLS) equation we correlate development rogue waves in oceanic sea states characterized by Joint North Sea Wave Project (JONSWAP) spectrum with proximity to homoclinic solutions NLS equation. We find numerical simulations that develop for JONSWAP initial data are “near” data, while do not occur “far” from data. show decomposition provides a simple criterium predicting occurrence and strength waves.

We investigate rogue waves in deep water the framework of nonlinear Schrödinger (NLS) and Dysthe equations. Amongst homoclinic orbits unstable NLS Stokes waves, we seek good candidates to model actual waves. In this paper propose two selection criteria: stability under perturbations initial data, persistence model. find that requiring selects maximal dimension. Persistence (a particular) perturbation a orbit dimension all whose spatial modes are coalesced. These results suggest more...

Modulated deep-water 1D Stokes waves are considered experimentally and theoretically. Wave trains modulated in a controlled fashion their evolution is recorded. Data from repeated laboratory experiments reproducible near the wave maker, but diverge away maker. Numerical integration of perturbed nonlinear Schrodinger equation an associated linear spectral problem indicate that under suitable conditions periodic evolve chaotically. Sensitive neighborhood homoclinic manifolds unperturbed found.

The focusing nonlinear Schr\"odinger equation is numerically integrated over moderate to long time intervals. In certain parameter regimes small errors on the order of roundoff grow rapidly and saturate at values comparable main wave. Although constants motion are nearly preserved, a serious phase instability (chaos) develops in numerical solutions. found be associated with homoclinic structures underlying mechanisms apply equally well many Hamiltonian wave systems.

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrödinger Equation (NLS) have been shown to be responsible for generation numerical instabilities and chaos. In this paper we undertake a dynamical systems type approach modeling observed irregular behavior conservative discretization NLS. Using heuristic Mel'nikov methods, existence pair isolated homoclinic orbits is indicated perturbed system. The structure persistent that are predicted by theory possesses same...

Abstract. In this article we conduct a broad numerical investigation of stability breather-type solutions the nonlinear Schrödinger (NLS) equation, widely used model rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently to waves. However, issue whether these robust with respect kind random perturbations occurring physical settings laboratory experiments has just recently begun be addressed. Numerical for spatially periodic one...

In physical regimes described by the cubic, focusing, nonlinear Schrödinger (NLS) equation, 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 solutions include Akhmediev breathers (N = 1), which are among most widely adopted spatially periodic models waves, their multi-mode generalizations > will referred as breathers. Numerical simulations linear stability analysis...

Discretizations and associated numerical computation of solutions certain integrable systems, such as the nonlinear Schrödinger equation (NLS) sine-Gordon (sG) equations with periodic boundary values can lead to instabilities, chaotic spurious results. The chaos be due truncation errors or even roundoff traced fact that these systems are strongly unstable when initial in neighbourhood homoclinic manifolds. By using spectral transform NLS tracking evolution relevant eigenvalues one observe...