相关论文: Variable coefficient nonlinear Schr\"odinger equat…
An exactly solvable position-dependent mass Schr\"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems…
We apply an extension of a new method of group classification to a family of nonlinear wave equations labelled by two arbitrary functions, each depending on its own argument. The results obtained confirm the efficiency of the proposed…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.
The point symmetry group is studied for the generalized Webster-type equation describing non-linear acoustic waves in lossy channels with variable cross sections. It is shown that, for certain types of cross section profiles, the admitted…
We suggest a method for integrating sub-families of a family of nonlinear {\sc Schr\"odinger} equations proposed by {\sc H.-D.~Doebner} and {\sc G.A.~Goldin} in the 1+1 dimensional case which have exceptional {\sc Lie} symmetries. Since the…
We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups.…
We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one,…
As it has been proven, the determination of general one-dimensional Schr\"odinger Hamiltonians having third-order differential ladder operators requires to solve the Painlev\'e IV equation. In this work, it will be shown that some specific…
The nonlinear Schroedinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon…
This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable…
Using the one dimensional free particle symmetries, the quantum finance symmetries are obtained. Namely, it is shown that Black-Scholes equation is invariant under Schr\"odinger group. In order to do this, the one dimensional free…
Wave Kinetic Equations (WKEs) are often used to describe the evolution of ensemble averaged wave amplitudes for nonlinear wave systems. In the present manuscript we describe a new approach to direct numerical simulation of solutions to…
In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schr\"odinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate…
The main contribution of our paper is to give a partial classification of the quasi-exactly solvable Lie algebras of first order differential operators in three variables, and to show how this can be applied to the construction of new…
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…
We discuss mathematical methods to derive Nonlinear Schr\"odinger equations (NLS) in "low dimensional" settings, i.e. the 3-dimensional physical space e.g. to 2 or 1 space dimensions. Beside from the case the system exhibits an internal…
A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional…
We investigated the analytical solution of fractional order K(m,n) type equation with variable coefficient which is an extended type of KdV equations into a genuinely nonlinear dispersion regime. By using the Lie symmetry analysis, we…
We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is…
Equations of fluid mechanics with N=1 Schrodinger supersymmetry are formulated within the method of nonlinear realizations of Lie groups.