Exact combined traveling wave solutions and multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations

• The homogeneous balance of undetermined coefficients method is firstly proposed to construct not only the exact solutions but also multi-symplectic structures of some types of nonlinear partial differential equations.
• The exact combined traveling wave solutions and a multi-symplectic structure of the variant Boussinesq equations is first presented.
• The definition and a multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations are first proposed in the multi-symplectic sense. In particular, the variant Boussinesq–Whitham–Broer–Kaup type equations can degenerate to the variant Boussinesq equations, the Broer–Kaup equations and the Whitham–Broer–Kaup equations.

The homogeneous balance of undetermined coefficients method (HBUCM) is firstly proposed to construct not only the exact traveling wave solutions, three-wave solutions, homoclinic solutions, N-soliton solutions, but also multi-symplectic structures of some nonlinear partial differential equations (NLPDEs). By applying the proposed method to the variant Boussinesq equations (VBEs), the exact combined traveling wave solutions and a multi-symplectic structure of the VBEs are obtained directly. Then, the definition and a multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations (VBWBKTEs) which can degenerate to the VBEs, the Whitham–Broer–Kaup equations (WBKEs) and the Broer–Kaup equations (BKEs) are given in the multi-symplectic sense. The HBUCM is also a standard and computable method, which can be generalized to obtain the exact solutions and multi-symplectic structures for some types of NLPDEs.