Related papers: Fractional Integrable and Related Discrete Nonline…
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…
The main goal of this paper is to set up the coarse-grained formulation of a fractional Schr\"odinger equation that incorporates a higher (spatial) derivative term which accounts for relativistic effects at a lowest order. The corresponding…
In this article, we study wave dynamics in the fractional nonlinear Schr\"odinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal…
In this paper, we consider a general form of nonlinear Schr\"{o}dinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schr\"{o}dinger equation is identified by…
We generate hierarchies of derivative nonlinear Schr\"odinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established…
The Nonlinear Schr\"odinger (NLS) equation is widely used in everywhere of natural science. Various nonlinear excitations of the NLS equation have been found by many methods. However, except for the soliton-soliton interactions, it is very…
Static soliton bound states in nonlinear systems are investigated analytically and numerically in the framework of the parametrically driven, damped nonlinear Schr\"odinger equation. We find that the ordinary differential equations, which…
We investigate the spectral and dynamical properties of the fractional nonlinear Schr\"odinger (fNLS) equation with harmonic confinement. In this setting, the classical Laplacian is replaced by its fractional power…
The long-time asymptotic behavior of solutions to the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the…
The Korteweg-deVries (KdV) equation with step boundary conditions is considered, with an emphasis on soliton dynamics. When one or more initial solitons are of sufficient size they can propagate through the step; in this case the phase…
We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then…
We show how to derive noncommutative versions of integrable partial difference equations using Darboux transformations. As an illustrative example, we use the nonlinear Schr\"odinger (NLS) system. We derive a noncommutative nonlinear…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
This paper concerns with some of the results related to the singular solutions of certain types of non-linear integrable differential equations (NIDE) and behavior of the singularities of those equations. The approach heavily relies on the…
We characterize the soliton solutions of the nonlinear Schroedinger equation on the half line with linearizable boundary conditions. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering…
Using a new infinite-dimensional linking theorem, we obtained nontrivial solutions for strongly indefinite periodic Schr\"odinger equations with sign-changing nonlinearities.
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.
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…
We consider the relation between the discrete coupled nonlinear Schr\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\"{o}dinger equation. In the same way…
This paper deals with the initial value problem for the multi-term fractional differential equation. The fractional derivative is defined in the Caputo sense. Firstly the initial value problem is transformed into a equivalent Volterra-type…