a b s t r a c t In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.

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Based on the Bäcklund method and the multilinear variable separation approach (MLVSA), this paper nds a general solution including two arbitrary functions for the (2+1)-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for (2+1)-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions (which contains two arbitrary functions). Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave (called semi-localized) patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations (two-dromion solution).

In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the generalized auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.