New Approaches for Nonlinear Waves

Dear Colleagues,

Theoretical and experimental studies of nonlinear waves constitute a vast area of research that includes various fields of physics, applied mathematics, and engineering. These fields include nonlinear optics and photonics (in particular, plasmonics), quantum matter (in particular, Bose–Einstein condensates (BECs)), hydrodynamics and plasmas, condensed matter and solid state (including lattice dynamics, magnetic media, and superconductivity), molecular and biological physics, chemically reacting media, ecology, electric networks, and many other areas of fundamental and applied phenomenology.

The propagation of nonlinear waves has been studied in continuous and discrete settings alike, both conservative and dissipative. These settings feature single- or multi-component structures. The well-known spatial, temporal, and spatiotemporal modes, which belong to the broad class of nonlinear waves, include solitons (fundamental, topological, and vortical ones, as well as complex self-trapped states in the form of hopfions and skyrmions); various types of periodic, quasi-periodic, and random wavetrains; diverse vortex and multipole patterns; and wave patterns widely studied as shock, domain wall, kink, rogue, and singular types, etc. Modes similar to these have been studied in detail in conservative systems and are known too in dissipative media, where their existence and stability are supported by an intrinsic gain or external pump. Investigations of these modes address their existence and structure, stability, and interactions between two or several modes. In particular, a fundamental problem that has been addressed in many studies is the stabilization of wave modes against the instability caused by critical and supercritical collapse in two- and three-dimensional settings with ubiquitous self-focusing cubic nonlinearity. This stabilization may be provided in various ways, such as lattice (spatially-periodic) potentials, the nonlocality of the nonlinear interactions, competition with higher-order defocusing nonlinearity, and linear effects provided by the spin-orbit coupling (SOC).

Some fundamental models of nonlinear-wave propagation are provided by classical integrable equations, such as the one-dimensional Korteweg–de Vries, nonlinear Schrödinger, sine-Gordon, and Toda lattice equations, and two types of two-dimensional Kadomtsev–Petviashvili equations. In most other cases, the relevant models are not integrable but admit the use of various approximate methods, such as variational, harmonic balance and energy balance methods; this is in addition to systematic numerical simulations. Experimental studies of nonlinear waves are performed using a variety of techniques.

The development of theoretical and experimental studies in the field of nonlinear waves has led to the steady expansion of the area. Relatively recent advances in the field include nonlinear-wave propagation in diverse artificial photonic media, various settings combining the SOC and nonlinearity of two- and multi-component systems occurring in BEC and optics or photonics, “quantum droplets”, maintained by the interplay of mean field and quantum fluctuation effects in BEC, and nonlinear waves in systems with fractional diffraction and/or dispersion.

This Special Issue welcomes the submission of theoretical and experimental papers that present novel results obtained via previously developed methods, as well as the development of new methods that advance various aspects of this vast field. Both original papers and full-length or brief review articles are welcome. All submissions will be subject to the standard review procedure. Accepted contributions will be immediately published online. All published papers belonging to the Special Issue will be collected and summarized shortly after the submission deadline.

Prof. Dr. Boris A. Malomed
Prof. Dr. Kwok W. Chow
Dr. Huimin Yin
Guest Editors

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• solitons
• vortices
• wavetrains
• hopfions
• skyrmions
• kinks
• shock waves
• domain walls
• rogue waves
• singular waves
• Bose-Einstein condensates
• nonlinear optics
• spin-orbit coupling
• quantum droplets
• competing nonlinearities
• lattice potentials
• nonlocal nonlinearity