Related papers: Approximate Peregrine Solitons in Dispersive Nonli…
We study the conditions for soliton-like wave propagation in the Photorefractive (PR) and electro-optic (i.e., Pockels) material, by using Nonlinear Schrodinger (NLS) equation. The complete NLS equation is solved analytically and…
The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in…
The effective long-time dynamics of solitary wave solutions of the nonlinear Schr\"odinger equation in the presence of rough nonlinear perturbations is rigorously studied. It is shown that, if the initial state is close to a slowly…
We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations…
We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional…
The Peregrine breather, today widely regarded as a prototype for spatio-temporally localized rogue waves on the ocean caused by nonlinear focusing, is analyzed by direct numerical simulations based on two-phase Navier-Stokes equations. A…
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use…
This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein--Gordon equations and the Maxwell--Lorentz system. The interest here is in…
Peregrine soliton (PS) is widely regarded as a prototype nonlinear structure capturing properties of rogue waves that emerge in the nonlinear propagation of unidirectional wave trains. As an exact breather solution of the one-dimensional…
It is known that the Swift-Hohenberg equation $\partial u/\partial t = -(\partial_x^2 + 1)^2u + \varepsilon (u-u^3)$ can be reduced to the Ginzburg-Landau equation (amplitude equation) $\partial A/\partial t = 4\partial_x^2 A + \varepsilon…
It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i. e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking…
We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several…
We propose an approximation to general relativity that captures the main gravitational effects of dynamical importance in supernovae. The conceptual link between this formalism and the Newtonian limit is such that it could likely be…
Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical…
We construct and discuss a semi-rational, multi-parametric vector solution of coupled nonlinear Schr\"odinger equations (Manakov system). This family of solutions includes known vector Peregrine solutions, bright-dark-rogue solutions, and…
Breather solutions are considered to be generally accepted models of rogue waves. However, breathers are not localized, while wavefields in nature can generally be considered as localized due to the limited spatial dimensions. Hence, the…
By employing a mapping to classical anharmonic oscillators, we explore a class of solutions to the Nonlinear Schrodinger Equation (NLSE) in 1+1 dimensions and, by extension, asymptotically in general dimensions. We discuss a possible way…
We study a class of discrete focusing nonlinear Schr{\"o}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the…
This paper presents a new approach for addressing the singularly perturbed nonlinear Schr\"odinger (NLS) equation: \begin{equation} -\varepsilon^2\Delta v + V(x) v =f(v),\ v>0,\ \lim_{|x|\to \infty} v(x)=0, \end{equation} where $V$…
In this paper, we study a non-integrable discrete lattice model which is a variant of an integrable discretization of the standard Hopf equation. Interestingly, a direct numerical simulation of the Riemann problem associated with such a…