Related papers: On classification of higher-order integrable nonli…
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
We find new solutions, including soliton-like ones, for a special case of non-Abelian loop Toda equations associated with complex general linear groups. We use the method of rational dressing based on an appropriate block-matrix…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
In this paper, a Hirota method is developed for applying to the nonlinear Schr\"odinger equation with arbitrary time-dependent linear potential which denotes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein…
The dressing chain equations for factorizing operators of a spectral problem are derived. The chain equations itselves yield nonlinear systems which closure generates solutions of the equations as well as of the nonlinear system if both…
We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schr\"{o}dinger equations. The integrability of the new models is established by…
We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the…
We propose a systemic method of applying the auxiliary systems of original equations to find the high order nonlocal symmetries of nonlinear evolution equation. In order to validate the effectiveness of the method, some examples are…
We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. It is appropriate to call these systems of equations "higher discriminants".
We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two…
Physically relevant soliton solutions of the resonant nonlinear Schrodinger (RNLS) equation with nontrivial boundary conditions, recently proposed for description of uniaxial waves in a cold collisionless plasma, are considered in the…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of…
We discuss an extension of the modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations. The extension includes the possibility for use of: (i) more than one simplest…
In this paper we shall study differential equations in the complex domain. The method of indeterminate coefficients and the majorant method lead to a proof of the existence and uniqueness of meromorphic solution of differential equations.…
We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main…
In this work, the $\overline{\partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}):…
Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…