Improved -expansion and first integral methods are used to construct exact solutions of the -dimensional Eckhaus-type extension of the dispersive long wave equation. The -expansion method is based on the assumptions that the travelling wave solutions can be expressed by a polynomial in and the first integral method is based on the theory of commutative algebra in which Division Theorem is of concern. It is worth mentioning that these methods are used for different systems and those two different systems can both be reduced to a system that will be mentioned in this paper. To recapitulate, this investigation has resulted in the exact solutions of the given systems. 1. Introduction The investigation of exact solutions to nonlinear evolution has become an interesting subject in nonlinear science field, since the time when the soliton concept was first introduced by Zabusky and Kruskal in 1965 [1]. It was not until the mid-1960s when applied scientists began to use modern digital computers to study nonlinear wave propagation that the soundness of Russell’s early ideas began to be appreciated. He viewed the solitary wave as a self-sufficient dynamic entity; a “thing” displaying many properties of a particle. From the modern perspective it is used as a constructive element to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, and from the elementary particles of matter to the elementary particles of thought. For a more detailed and technical account of the solitary wave, see [2]. In recent years, other methods have been developed, such as the Backlund transformation method [3], Darboux transformation [4], tanh method [5–7], extended tanh function method [8], the generalized hyperbolic function [9], and variable separation method [10]. The celebrated -dimensional dispersive long wave equation [11, 12] plays an important role in nonlinear physics; many properties of (1) have been reported [12, 13]. It is interesting to study the extensions of (1) in higher-dimensional spaces. To date, there exist two prototypical of (1) to cover the situation of wide channel or open seas. In 1987, Boiti et al. [14] presented the following -dimensional extension related to (1): For (2), the Backlund transformation, soliton solutions are given [14, 15]. In 1985, Eckhaus [16] presented another different two-dimensional extension of (1): which was obtained in the appropriate approximation from the basic equations of hydrodynamics. It is easy to see that if one makes the transformation ,
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