Related papers: Nonlinear Schr\"odinger Equation: Generalized Darb…
In the present work, a nonlocal nonlinear Schr\"odinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This…
We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a…
The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather…
The method of Darboux transformation, which is applied on cnoidal wave solutions of the sine-Gordon equation, gives solitons moving on a cnoidal wave background. Interesting characteristics of the solution, i.e., the velocity of solitons…
General high-order rogue waves of the NLS-Boussinesq equation are obtained by the KP-hierarchy reduction theory. These rogue waves are expressed with the determinants, whose entries are all algebraic forms. It is found that the fundamental…
\rm We obtain the global smooth effects for the solutions of the linear Schr\"odinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical…
Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super B\"{a}cklund transformation and is…
Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the…
In this work, we prove the existence of wave operator for the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_x^2 u +i |u|^{2\sigma}\partial_x u=0, \end{align*} with…
The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential…
Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schr\"odinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics and plasmas, exhibits…
According to classical non-relativistic Schr\"odinger equation, any local perturbation of wave function instantaneously affects all infinite region, because this equation is of parabolic type, and its solutions demonstrate infinite speed of…
We construct higher order rogue wave solutions for the Gerdjikov-Ivanov equation explicitly in term of determinant expression. Dynamics of both soliton and non-soliton solutions is discussed. A family of solutions with distinct structures…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
A generalized derivative nonlinear Schr\"odinger equation, \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x + (\gamma-1)(\gamma-2)|q|^4 q = 0 , is studied by means of Hirota's bilinear formalism. Soliton solutions are…
The Fokas-Lenells (FL) equation arises as a model eqution which describes for nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order (in the leading asymptotic order the nonlinear…
Using Lie group theory we construct explicit solitary wave solutions of coupled nonlinear Schrodinger systems with spatially inhomogeneous nonlinearities. We present the general theory, use it to construct different families of explicit…
Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrowband in frequency but not necessarily with narrow angular distributions the developed asymptotic…
Using the elementary axioms of special relativity and quantum mechanics we construct a wave equation which generalizes the Schrodinger equation. We also solve the general second and some higher order differential equations.
In this paper, we investigate smooth positon and breather-positon solutions of a generalized nonlinear Schr\"{o}dinger (GNLS) equation which contains higher order nonlinear effects. With the help of generalized Darboux transformation (GDT)…