Extended Gram-type determinant, wave and rational solutions to two (3+1)-dimensional nonlinear evolution equations

New exact Grammian determinant solutions to two (3+1)-dimensional nonlinear evolution equations are derived. Extended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented. Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems. The bilinear Bäcklund transformations are also constructed for the equations. Also, as an application of the bilinear Bäcklund transforms, a new class of wave and rational solutions to the equations are explicitly computed. Many phenomena in the nonlinear sciences and in the physics can be modeled by a various classes of integrable nonlinear partial differential equations. Consequently, construction of wave solutions of nonlinear evolution equations plays a crucial role in the study of nonlinear sciences and nonlinear phenomena. To obtain the traveling wave solutions for these equations, especially for the higher-dimensional and the coupled nonlinear evolution equations, can make people know the described physical phenomena. Besides traveling wave solutions, another set of interesting multi-exponential wave solutions [10,11] is a linear combination of exponential waves. Nowadays, with the rapid software technology development, solving nonlinear partial differential equations via explicit and symbolic computation is taking an increasing role due to its accuracy, efficiency and its restrained use. To this end, in the open literature, a set of systematic methods have been developed to obtain explicit solutions for nonlinear evolution equation, such as tanh–coth function, sine–cosine function, Jacobi elliptic function method, symmetry method, Weierstrass function method, the F-expansion method, Homotopy perturbation method, variational iteration method [31–33] and so on. However, all methods mentioned above have some restrictions in their applications. On the other hand, the Hirota bilinear formalism [1,2] has been successfully used in the search for exact solutions of continuous and discrete systems [20–22], and also in the search for new integrable equations by testing for multisoliton solutions or Bäcklund transformations, and even used in constructing N-soliton solutions for integrable couplings by perturbation [8]. It is now believed that most integrable systems if not all, could be transformed into bilinear forms by dependent variable transformations. Therefore, one would expect to study most integrable systems within the bilinear formalism. Moreover, various methods have been presented in the last four decades to construct exact solutions for many nonlinear evolution equations, such as Grammian determinant approach [1,23], and Wronskian determinant approach [4,15]. Wronskian determinant, Grammian determinant and Pfaffian solutions to the (3+1)-dimensional generalized KP and BKP equations were constructed in …