Related papers: A Generalized Method for the Darboux Transformatio…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
The exactly solvable scalar-tensor potential of the four-component Dirac equation has been obtained by the Darboux transformation method. The constructed potential has been interpreted in terms of nucleon-nucleon and Schwinger interactions…
A ``hybrid method'', dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of…
The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing…
Phase equations describing the evolution of large scale modulation of spatially periodic patterns in two dimensional systems are derived by employing the renormalization group method. A general formula for phase diffusion coefficients is…
We describe a general multiplier method to obtain boundary stabilization of the wave equation by means of a (linear or quasi-linear) Neumann feedback. This also enables us to get Dirichlet boundary control of the wave equation. This method…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
The $N=2 \;a=-2$ supersymmetric KdV equation is studied. A Darboux transformation and the corresponding B\"acklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated…
In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original…
In this paper, We extend the two-component coupled Hirota equation to the three-component one, and reconstruct the Lax pair with $4\times4$ matrixes of this three-component coupled system including higher-order effects such as third-order…
By combining stability analysis of scalar field theories with the Darboux transformation technique, we create models featuring kink-like solutions whose quantum perturbations are all bounded. On the one hand, the stability analysis relates…
We study the Darboux transformation (DT) for Dirac equations with (1+1) potentials. Exact solutions for the adiabatic external field are constructed. The connection between the exactly soluble Dirac (1+1) potentials and the soliton…
For a semi-linear Schr\"{o}dinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude,…
New types of irreducible second order Darboux transformations for the one dimensional Schroedinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the…
We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on…
A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of…
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 introduce a generalized version of the Ablowitz-Ladik model with a power-law nonlinearity, as a discretization of the continuum nonlinear Schr\"{o}dinger equation with the same type of the nonlinearity. The model opens a way to study the…
In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several…
A recently proposed discrete version of the Schrodinger spectral problem is considered. The whole hierarchy of differential-difference nonlinear evolution equations associated to this spectral problem is derived. It is shown that a discrete…