Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation

We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.

On the interaction phenomena to the nonlinear generalized Hietarinta-type equation

In this paper,we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equa-tion for dispersive waves in(2+1)dimension.The various wave formations are retrieved by using Hirota’s bilinear method(HBM)and various test function perspectives.The Hirota method is a widely used and robust mathematical tool for finding soliton solutions of nonlinear partial differential equa-tions(NLPDEs)in a variety of disciplines like mathematical physics,nonlinear dynamics,oceanography,engineering sciences,and others requires bilinearization of nonlinear PDEs.The different wave structures in the forms of new breather,lump-periodic,rogue waves,and two-wave solutions are recovered.In addi-tion,the physical behavior of the acquired solutions is illustrated in three-dimensional,two-dimensional,density,and contour profiles by the assistance of suitable parameters.Based on the obtained results,we can assert that the employed methodology is straightforward,dynamic,highly efficient,and will serve as a valuable tool for discussing complex issues in a diversity of domains specifically ocean and coastal engineering.We have also made an important first step in understanding the structure and physical be-havior of complex structures with our findings here.We believe this research is timely and relevant to a wide range of engineering modelers.The results obtained are useful for comprehending the fundamental scenarios of nonlinear sciences.