A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, seeks travelling solutions in form a finite series tanh. present, boundary conditions are implemented this expansion. The associated velocity can then be determined priori, provided solution vanishes at infinity. Hence, exact closed obtained easily various cases.

With the aid of tanh method, nonlinear wave equations are solved in a perturbative way. First, KdVBurgers equation is investigated limit weak dispersion. As result, general shock profile, with solitary-wave contribution superposed, emerges. For particular choice parameters, comparison exact solution made. Further, MKdVBurgers same and similar results obtained.

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution equations from the real exponential underlying linear equations. physical concept is one mixing these elementary through nonlinearities in system. emphasis is, however, on mathematical aspects, i.e. procedure necessary to find single solutions. By means examples it shown how various cases pulse-type kink-type are be obtained by this method. An exhaustive list so treated...

The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and equations has been generalized systematized. class of solitary is extended to analytic (rather than rational) functions the real exponential linearized equation. Expanding in an infinite series these exponentials, exact solution PDE obtained, whenever can be summed. Methods solving recursion relation coefficients summing closed form are discussed. algorithm now suited by any symbolic...

In contrast to overtaking interactions, head-on collisions between two electrostatic solitons can be dealt with only by use of an approximate method, which limits the range validity but offers valuable insights. Treatments in plasma physics literature all assumptions stretching space and time expansion dependent variables that are seldom, if ever, discussed. All models force a separability lowest order, corresponding linear waves opposite equally large velocities. A systematic exposition...

A recursion operator is an integro-differential which maps a generalized symmetry of nonlinear PDE to new symmetry. Therefore, the existence guarantees that has infinitely many higher-order symmetries, key feature complete integrability. Completely integrable PDEs have bi-Hamiltonian structure and Lax pair; they can also be solved with inverse scattering transform admit soliton solutions any order. straightforward method for symbolic computation polynomial operators in (1+1) dimensions...

A direct method for the computation of polynomial conservation laws systems nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The avoids advanced differential-geometric tools. Instead, it solely based on calculus, variational and linear algebra. Densities are constructed as combinations scaling homogeneous terms with undetermined coefficients. derivative (Euler operator) used to compute homotopy operator fluxes. illustrated PDEs describing wave phenomena fluid...

The Harry Dym equation, which is related to the classical string problem, derived in three different ways. An implicit cusp solitary-wave solution constructed via a simple direct method. existing connections between and Korteweg-de Vries equations are uniformised simplified, transformations their respective solutions carried out explicitly. Whenever possible, physical insights provided.

Head-on collisions between two electrostatic solitons are dealt with by the Poincaré-Lighthill-Kuo method of strained coordinates, for a plasma composed number cold (positive and negative) ion species Boltzmann electrons. The nonlinear evolution equations both their phase shift due to collision, resulting in time delays, established. A Korteweg-de Vries description is generic conclusion, except when composition special enough replace quadratic cubic nonlinearity equations, concomitant...

The automation of the traditional Painlevé test in Mathematica is discussed.The package PainleveTest.mallows for testing polynomial systems nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants).Except where limited memory, there no restriction on number independent or dependent variables.The quite robust determining all possible dominant behaviors Laurent series solutions equation.The omission valid a common problem many...