The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $(G^{\prime }/G^2)-$expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $(G^{\prime }/G^2)-$expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.

Based on the Hirota bilinear form of a (2+1)-dimensional equation, breathers and resonant multiple waves as well as complexiton solutions are considered in this paper. First, the breather waves are constructed via employing the extend homoclinic test method. By calculation, two kinds of solutions are obtained. Through analysis, three pairs of breathers consisting of hyperbolic functions and trigonometric functions are derived. Furthermore, a rouge wave solution is deduced by applying the Taylor expansion method to a obtained breather wave. In addition, related figures are plotted to illustrate the dynamical features of these obtained solutions. Then, two types of the resonant multi-soliton solutions are obtained by applying the linear superposition principle to the the Hirota bilinear form. At the same time, 3D profiles and 2D density plots are presented to depict the intersection progression of wave motion. Finally, the complexiton solutions are constructed according to the yielded resonant multi-soliton solutions by further utilizing the linear superposition principle. By considering different domain fields, several types of complexiton solutions including the positive ones are derived. Moreover, related 3D and 2D figures are plotted for the obtained results in order to vividly exhibit their dynamics properties.

In this paper, a (3+1)-dimensional nonlinear evolution equation is considered. First, its bilinear formalism is derived by introducing dependent variable transformation. Then, its breather wave solutions are obtained by employing the extend homoclinic test method and related figures are presented to illustrate the dynamical features of these obtained solutions. Next, its resonant multi-soliton solutions are obtained by using the linear superposition principle. Meanwhile, 3D profiles and contour plots are presented to exhibit the process of wave motion. Finally, M-breather wave solutions such as one-breather, two-breather, three-breather and hybrid solutions between breathers and solitons are constructed by applying the complex conjugate method to multi-soliton solutions. Furthermore, their evolutions are shown graphically by choosing suitable parameters.