Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schroedinger Equation with Self-steepening and Self-frequency Shift

Abstract By making use of the generalized sine-Gordon equation expansion method, we lind cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive and the quintic nonlinear Schroedinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.

Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.

Bright and dark optical solitons in the nonlinear Schrdinger equation with fourth-order dispersion and cubic-quintic nonlinearity

By the use of an auxiliary equation, we find bright and dark optical soliton and other soliton solutions for the higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion (FOD), cubic-quintic terms, self-steepening, and nonlinear dispersive terms. Moreover, we give the formation condition of the bright and dark solitons for this higher-order NLSE.

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrdinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrdinger(HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and secondorder rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation on the rogue waves is discussed with the help of graphical simulation.

The Riemann Problem for the Dispersive Nonlinear Shallow Water Equations

The complete analytical solution of the Riemann problem for the homo-geneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii andBrocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for both wet-bed anddry-bed conditions. Moreover, such a set of hyperbolic and dispersive depth-averagedequations shows an interesting resonance phenomenon in the wave pattern of the solu-tion and we define conditions for the occurrence of resonance and present an algorithmto capture it. As an indirect check on the analytical solution we have carried out a de-tailed comparison with the numerical solution of the government equations obtainedfrom a dissipative method that does not make explicit use of the solution of the localRiemann problem.

Dispersive Blow-Up Ⅱ.Schrdinger-Type Equations,Optical and Oceanic Rogue Waves

Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly dispersive equations or systems of equations. The present essay deals with linear and nonlinear Schrdinger equations, a class of fractional order Schrdinger equations and the linearized water wave equations, with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.