The homogeneous balance of undetermined coefficients method and its application

Abstract

Abstract:The homogeneous balance of undetermined coefficients method is firstly proposed to solve such nonlinear partial differential equations (PDEs), the balance numbers of which are not positive integers. The proposed method can also be used to derive more general bilinear equation of nonlinear PDEs. The Eckhaus equation, the KdV equation and the generalized Boussinesq equation are chosen to illustrate the validity of our method. The proposed method is also a standard and computable method, which can be generalized to deal with some types of nonlinear PDEs.

Conserved quantities, optimal system and explicit solutions of a (1 + 1)-dimensional generalised coupled mKdV-type system

“…Nonlinear partial differential equations (NLPDEs) have rapidly become indispensable in the quest to conceptualise the world around us [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46] , [47] , [48] , [49] , [50] . We give a few recent studies of NLPDEs presented in the literature.…”

“…Other principal methods for obtaining soliton and periodic solutions of NLPDEs have been developed over the years. These include the Riccati-Bernoulli sub-ODE method [18] , the homogeneous balance of undetermined coefficients method [19] , [20] , [21] , [22] , the first integral method [23] , the bifurcation technique [24] , the generalised unified method [25] , the multiple exp-function method [26] , dynamical system approach [27] , [28] , simplified Hirota’s method [29] , [30] , the ((G′/G))-expansion function method [31] , Kudryashov’s method [32] , Jacobi elliptic function expansion technique [33] , and the power series technique, [34] , [36] , [37] .…”