Related papers: Polygons for finding exact solutions of nonlinear …
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
Nonlinear Sobolev-Burgers PDEs are considered. Their solutions are investigated. A technique of noncommutative line integration is utilized for their description. A new method of PDEs solution with the help of Cayley-Dickson algebras is…
In this letter we apply a method recently devised in \cite{aapla03} to find precise approximate solutions to a certain class of nonlinear differential equations. The analysis carried out in \cite{aapla03} is refined and results of much…
Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are…
A new method named rational expansion method of exponent function is presented to find exact traveling wave solutions of differential-difference equations. This method generalizes the so-called tanh-method and other similar methods. Some…
I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
The algorithm for generation of exact solutions of the nonlinear equation in partial derivatives of a divergent type which is included in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schr\"odinger…
We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a…
In this short note we are presenting a method of finding particular solutions of nonhomegeneous linear equations. This approach is different from methods of undetermined coefficients or variation of parameters presented in virtually every…
We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider…
The nonlinear diffusion equation $u_t = (u^{- 4/3} u_x)_x$ is reduced by the substitution $u = v^{- 3/4}$ to an equation with quadratic nonlinearities possessing a polynomial invariant linear subspace of the maximal possible dimension equal…
The family of fifth order nonlinear evolution equations is studied. Some traveling wave elliptic solutions are found. The classification of these exact solutions is given.
A generalized geometric method is developed for constructing exact solutions of gravitational field equations in Einstein theory and generalizations. First, we apply the formalism of nonholonomic frame deformations (formally considered for…
This note shows that in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility…
We describe exact kink soliton solutions to nonlinear partial differential equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx} = A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality allows…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…