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Exact solutions with solitary patterns for the Zakharov-Kuznetsov equations with fully nonlinear dispersion. (English) Zbl 1129.35450

Summary: In this paper, nonlinear dispersive Zakharov-Kuznetsov \(\mathrm{ZK}(m,n,k)\) equations \[ u_t+a(u^m)_x+b(u^n)_{xxx}+c(u^k)_{yyx}=0,\quad mnk\neq 0, \] (where \(a\), \(b\) and \(c\) are arbitrary constants and \(m\), \(n\) and \(k\) are integers) are solved exactly by using the Adomian decomposition method. Two special cases, \(\mathrm{ZK}(2,2,2)\) and \(\mathrm{ZK}(3,3,3)\), are chosen to illustrate the concrete scheme of the decomposition method. General formulas for solutions of \(\mathrm{ZK}(m,n,k)\) equations are established.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
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