Abstract
The nonlinear Schrödinger equation has wide range of applications in physics with spatial scales that vary from microns to kilometres. Consequently, its solutions are also universal and can be applied to water waves, optics, plasma and Bose-Einstein condensate. The most remarkable solution presently known as the Peregrine solution describes waves that appear from nowhere. This solution describes unique events localized both in time and in space. Following the language of mariners they are called “rogue waves”. As thorough mathematical analysis shows, these waves have properties that differ them from any other nonlinear waves known before. Peregrine waves can serve as ‘elementary particles’ in more complex structures that are also exact solutions of the nonlinear Schrödinger equation. These structures lead to specific patterns with various degrees of symmetry. Some of them resemble “atomic like structures”. The number of particles in these structures is not arbitrary but satisfies strict rules. Similar structures may be observed in systems described by other equations of mathematical physics: Hirota equation, Davey-Stewartson equations, Sasa-Satsuma equation, generalized Landau-Lifshitz equation, complex KdV equation and even the coupled Higgs field equations describing nucleons interacting with neutral scalar mesons. This means that the ideas of rogue waves enter nearly all areas of physics including the field of elementary particles.
Highlights
The nonlinear Schrödinger equation (NLSE) has wide range of applications in physics with spatial scales that vary from microns to kilometres and even light years
Being a practical introduction to a special issue, this article provides a basic review of mathematical results on NLSE that are important for understanding the nonlinear phenomena in general
Multi-rogue waves are well studied [32,33,34] but, perhaps, their physics is less understood [35,36,37] than the physics of any other solutions of the NLSE
Summary
The nonlinear Schrödinger equation (NLSE) has wide range of applications in physics with spatial scales that vary from microns to kilometres and even light years. Being a practical introduction to a special issue, this article provides a basic review of mathematical results on NLSE that are important for understanding the nonlinear phenomena in general It leaves aside the complexities of inverse scattering technique [13], Darboux transformation [18], theta functions [19] and other sophistications of modern mathematics [20]. The eigenvalues corresponding to the individual contributions are located at the same point of the complex plane Despite this complication, multi-rogue waves are well studied [32,33,34] but, perhaps, their physics is less understood [35,36,37] than the physics of any other solutions of the NLSE. We will concentrate on the Peregrine wave, its analogs and its higher-order combinations This is a very small subset of the whole set of multi-parameter families of solutions of the NLSE. Eq 4 provides an additional tool for adjustment of initial conditions to the required levels in optical and hydrodynamic experiments
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