Localized and complex soliton solutions to the integrable (4+1)-dimensional Fokas equation

We explore a family of exact traveling and localized soliton solutions of the nonlinear integrable (4+1)-dimensional Fokas equation by different approaches, namely, the Jacobian-function method, the Pades type transformation, He's semi-inverse variational technique and sine-cosine or triangle function approach. Some new exact traveling wave solutions of physical interest involving some constraint conditions are reported. The reported solutions are Jacobi doubly periodic wave solutions, fractional soliton, localized soliton when parameters are taken to be special values of doubly periodic functions and complex Bloch solitons. These obtained solutions may facilitate us in understanding the dynamical behavior of physical phenomenon governed by nonlinear integrable Fokas equation and show the applicability and efficacy of the used approaches that can be applied for higher dimensional nonlinear integrable equations as well as linear ones in mathematical physics. The differences and similarities between the used distinct approaches are discussed. To exhibits, the dynamical behavior of reported solutions, the two and three-dimensional structures are numerically simulated.