Mark J. Ablowitz

Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time asymptotic result is obtained via the method of stationary phase. The asymptotic approximation is found to be in good agreement with the exact solution and reveals hyperbolic spiral structure. Numerical simulations show that these spiral waves persist in the presence of weak nonlinearity. While spiral solutions are frequently found in excitable media governed by reaction-diffusion systems, they comprise a new class of interesting two space one time dimensional solutions in fundamental linear and nonlinear dispersive wave systems.

In recent years, there has been considerable interest in the study of wave propagation in nonlinear photonic lattices. The interplay between nonlinearity and periodicity has led researchers to manipulate light and discover new and interesting phenomena such as new classes of localized modes, usually referred to as solitons, and novel surface states that propagate robustly. A field where both nonlinearity and periodicity arises naturally is nonlinear optics. But there are other areas where waves propagating on background lattices play an important role, including photonic crystal fibers and Bose-Einstein condensation. In this review article the propagation of wave envelopes in one and two-dimensional periodic lattices associated with additional potential in the nonlinear Schrodinger (NLS) equation, termed lattice NLS equations, are studied. A discrete reduction, known as the tight-binding approximation, is employed to find the linear dispersion relation and the equations governing nonlinear discrete envelopes for two-dimensional simple periodic lattices and two-dimensional non-simple honeycomb lattices. In the limit under which the envelopes vary slowly, continuous envelope equations are derived from the discrete system. The coefficients of the linear evolution system are related to the dispersion relation in both the discrete and continuous cases. For simple lattices, the continuous systems are NLS type equations. In honeycomb lattices, in certain cases, the continuous system is found to be nonlinear Dirac equations. Finally, it is possible to realize so-called topological insulator systems in an optical waveguide setting. The modes supported by these systems are associated with spectral topological invariants and, remarkably, can propagate without backscatter from lattice defects.

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t ^α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A ^α , where A is the amplitude of the wave. Fractional extensions of the modified Korteweg–deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion.

A laser-fabricated waveguide array creates a nonlinear medium that supports solitons

Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse...

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schr\"odinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complicated due to the associated branching structure of the associated linear eigenfunctions. With constant amplitude at infinity, four cases are analyzed; they correspond to two different signs of nonlinearity and two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. In terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations and the RST NLS equation is the time dependence of the scattering data. Spatially dependent boundary conditions are also briefly considered.