Exact travelling wave solutionsof some nonlinear evolutionary equations

Abstract

Abstract. Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.

Analytical description of the coherent structures within the hyperbolic generalization of burgers equation

“…Most of these solutions cannot be obtained within the dialects of direct algebraic methods, previously applied to this equation [6]. Let us discuss the solutions obtained in section 2.…”

“…[1,2]), which is not effective in obtaining solutions with the given properties, there are employed techniques based on choosing a proper transformation (or ansatz), turning the problem of finding out exact solutions to the algebraic one [3,4,5]. In our previous study [6] we employed a certain dialect of the the direct algebraic method to the hyperbolic generalization of Burgers equation (GBE). Yet in the following publications [7,8] we showed that within this methodology it is impossible to obtain the the solitary wave solution of GBE, occurring to exist for certain values of the parameters [7].…”

Exact traveling wave solutions for some nonlinear evolution equations

“…In order to better understand the nonlinear phenomena as well as further practical applications, it is important to seek their more exact traveling wave solutions. In the recent years, many powerful methods have been proposed for obtaining traveling solitary wave solutions to nonlinear evolution equations such as Hirota's bilinear method [18], the sine-cosine method [20,21], the exp-function method [16], the Jacobi elliptic function method [12], the auxiliary ordinary differential equation method [14], the direct algebraic method [17,22], and so on.…”

Elliptic solutions of the quintic complex one-dimensional Ginzburg–Landau equation

“…At present methods for construction of special solutions of nonintegrable systems in terms of elementary (more precisely, degenerated elliptic) and elliptic functions are actively developed [2,7,8,9,16,17,21,25,26,27,29,31,32,33,37,40,41,43,49] (see also [30] and references therein). Some of these methods are intended for the search for elliptic solutions only [26], others allow to find either solutions in terms of elementary functions only [7,8,31,49] or both types of solutions [2,9,16,17,21,25,27,29,32,33,37,40,41,43].…”

“…Some of these methods are intended for the search for elliptic solutions only [26], others allow to find either solutions in terms of elementary functions only [7,8,31,49] or both types of solutions [2,9,16,17,21,25,27,29,32,33,37,40,41,43]. Note that the methods [40,29] allow to find multivalued solutions as well.…”