Related papers: Factorization of the nonlinear Schroedinger equati…
We consider a factorization of the non-stationary Schrodinger operator based on the parabolic Dirac operator introduced by Cerejeiras/ Kahler/ Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct…
We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain…
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati…
In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*}…
In flat spacetime, the Dirac equation is the "square root" of the Klein-Gordon equation in the sense that by applying the square of the Dirac operator to the Dirac spinor, one recovers the Klein-Gordon equation duplicated for each component…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
We present a generalized study and characterization of the integrability properties of the derivative non-linear Schr\"odinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the…
The present work addresses the study and characterization of the integrability of three famous nonlinear Schr\"odinger equations with derivative-type nonlinearities in 1+1 dimensions. Lax pairs for these three equations are successfully…
The nonlocal Darboux transformation of the stationary axially symmetric Schr\"odinger equation is considered. It is shown that a special case of the nonlocal Darboux transformation provides the generalization of the Moutard transformation.…
In the present study a particular case of Gross-Pitaevskii or non-linear Schr\"odinger equation is rewritten to a form similar to a hydrodynamic Euler equation using the Madelung transformation. The obtained system of differential equations…
A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schroedinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansaetze…
We obtain novel nonlinear Schr\"{o}dinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential…
This paper is devoted to a factorization of the higher dimensional Schrodinger operator in the framework of Clifford analysis.
Considered is the Schr\"odinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting…
The nonlocal Darboux transformation of the two - dimensional stationary Schr\"odinger equation is considered and its relation to the Moutard transformation is established. It is shown that a special case of the nonlocal Darboux…
We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant…
This work aims to initiate a discussion on finding solutions to non-homoge\-neous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon…
It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schr\"odinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to…
This paper is concerned with an alternative analytical solution of time-fractional nonlinear Schrodinger equation and nonlinear coupled Schrodinger equation obtained by employing fractional reduced differential transform method. The…
We show that the Dirac factorization method can be successfully employed to treat problems involving operators raised to a fractional power. The technique we adopt is based on an extension of the Pauli matrices and the properties of the…