Related papers: Fractional Integrable Nonlinear Soliton Equations
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie…
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related…
We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schroedinger equations. We derive a nonlinear dispersion relation.…
Fractional calculus is an effective tool in incorporating the effects of non-locality and memory into physical models. In this regard, successful applications exist rang- ing from signal processing to anomalous diffusion and quantum…
We show that an integro-differential equation model for pulse propagation in optical transmission lines with dispersion management, is integrable at the {\it leading nonlinear order}. This equation can be transformed into the nonlinear…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons…
We obtain new exact classes of solutions for the nonlinear fractional Fokker-Planck-like equation partial_t rho = partial_x{D(x) partial^{mu -1}_x rho^{nu} - F(x) rho} by considering a diffusion coefficient D = D|x|^{-theta} (theta in R and…
We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and…
In the paper we study nonlocal functionals whose kernels are homogeneous generalized functions. We also use such functionals to solve the Korteweg-de Vries , the nonlinear Schr\"odinger and the Davey-Stewartson equations.
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial…
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed,…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
Fully describing light propagation in a rotating, anisotropic medium with thermal nonlinearity requires modeling the interplay between nonlinear refraction, birefringence, and the nonlinear group index. Incorporating these factors into a…
A new nonlinear Schroedinger equation is obtained explicitly from the fractal Brownian motion of a massive particle with a complex-valued diffusion constant. Real-valued energy (momentum) plane wave and soliton solutions are found in the…
New technique of integration of certain types of partial differential equations is developed. For this purpose non-commutative integration over Cayley-Dickson algebras is used. Applications to non-linear vector partial differential…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave…
In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…