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The nonlinear stage of the modulational (Benjamin - Feir) instability of unidirectional deep water surface gravity waves is simulated numerically by the firth-order nonlinear envelope equations. The conditions of steep and breaking waves…
In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic-quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by…
In this paper we study a system of coupled nonlinear Schrodinger equations modelling a quantum degenerate mixture of bosons and fermions. We analyze the stability of plane waves, give precise conditions for the existence of solitons and…
We consider a system of two discrete nonlinear Schr\"{o}dinger equations, coupled by nonlinear and linear terms. For various physically relevant cases, we derive a modulational instability criterion for plane-wave solutions. We also find…
We investigate the modulation instability of multiple four-wave mixing arising from a dual-frequency pump in a single-mode fiber or waveguide. By applying the Floquet theory on account of the periodic nature of four-wave mixing, we reveal a…
The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized,…
Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in…
Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects…
Formation of turbulence of capillary waves is studied in laboratory experiments. The spectra show multiple exponentially decreasing harmonics of the parametrically excited wave which nonlinearly broaden with the increase in forcing.…
Sea ice attenuates waves propagating from the open ocean. Here we model the evolution of energetic unidirectional random waves in the marginal ice zone with a nonlinear Schr\"{o}dinger equation, with a frequency dependent dissipative term…
Nonlinearity in the Schr\"odinger equation gives rise to rich phenomena such as soliton formation, modulational instability, and self-organization in diverse physical systems. Motivated by recent advances in engineering nonlinear gauge…
We consider a nonlinear Schr\"odinger equation with double power nonlinearity \begin{align*} i\partial_t u+\Delta u-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the…
We explore a prototypical two-dimensional model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis,…
In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schr\"odinger equation with mixed dispersion. The existence of solitary waves can be determined both as minimizers…
The present paper is the first part of a project devoted to the fractional nonlinear Schr\"{o}dinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some…
For the double power one dimensional nonlinear Schr{\"o}dinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by…
We consider the implementation of the split-step method where the linear part of the nonlinear Schr\"odinger equation is solved using a finite-difference discretization of the spatial derivative. The von Neumann analysis predicts that this…
We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the…
Understanding, predicting, and controlling physical processes often relies on the analysis of the dynamics of partial differential equations (PDEs). In this context, the present study offers an in-depth investigation into the nonlinear…
The modulation instability is a focusing mechanism responsible for the formation of strong wave localizations not only on the water surface, but also in a variety of nonlinear dispersive media. Such dynamics is initiated from the injection…