The patterns of traveling wave on shallow water modeled by Green-Naghdi equation

The Green-Naghdi equations are a shallow water waves model which play important roles in nonlinear wave fields. By using the trial equation method and the Complete discrimination system for the polynomial we obtained the classification of travelling wave patterns. Among those patterns, new singular patterns and double periodic patterns are obtained in the first time. And we draw the graphs which help us to understand the dynamics behaviors of the Green-Naghdi model intuitionally.

Exact single travelling wave solutions to the fractional perturbed Gerdjikov–Ivanov equation in nonlinear optics

In this paper, exact single travelling wave solutions to the nonlinear fractional perturbed Gerdjikov–Ivanov equation are captured by the complete discrimination system for polynomial method and the trial equation method. In the classification, we can find out the original equation has rational function solutions, solitary wave solutions, triangular function periodic solutions, and elliptic function periodic solutions, which are normally very difficult to be obtained by other methods. In particular, the concrete parameters are set to show that the solutions in the classification can be realized in almost all cases.

Optical propagation patterns in nonlocal parabolic law medium

The model of optical propagation in nonlocal parabolic law medium is described by nonlinear Schrödinger equation with high-order nonlinear terms. Exact optical propagation patterns are constructed by proposing a general complex trial equation method. These results show rich optical propagation patterns of the model.

Soliton solutions of coupled higgs field equation via the trial equation method

Since a few recent decades, investigation of nonlinear evolution equations (NLEEs) is becoming an important area of research as they have a variety of applications in various branches of social and scientific disciplines like Ecology, Social Dynamics, Financial Mathematics, Engineering and many branches of Physics such as Biophysics, Chemical Physics, Fibre Optics, Fluid Mechanics, Neuro-physics, Particle Physics, Solid State Physics and many more. Many powerful and efficient methods of finding exact solutions of NLEEs have been proposed so far and the Trial Equation Method [ 1 - 5] is one of them. Many authors have successfully used the method in finding exact solutions of a number of NLEEs. In the present paper, soliton solutions of the Coupled Higgs Field Equation [ 6 - 10 ] are being obtained using the Trial Equation Method. The Coupled Higgs Field Equation describes system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. This coupled equation has applications in the studies of Field Theory and Electromagnetic waves as well. This coupled equation introduces the Higgs field to illustrate the mechanism of generation of mass for Gauge Bosons. The Coupled Higgs Field Equation is generally expressed as the following pair of NLEEs (3) and (2) Here, x and t are spatial and temporal variables respectively, the function is a complex scalar nucleon field, the function is a real scalar meson field, are arbitrary real constants and the subscripts denote partial differentiations with respect to them.Using the Trial Equation Method, the above coupled NLEE is to be solved to obtain some soliton solutions.