Patterns Of Traveling Waves Of Nonlinear Wave Motions And Diffusion And Non-newton Fluid

In the paper,we use the analytic methods to study systematically traveling wave patterns for five typical nonlinear models in petroleum engineering field,including the nonlinear Schrodinger equation describing the deep water wave motion and Kd V6 equation for shallow water,ZK equation with fractional power and other related models,for plasma,nonlinear reaction-diffusion equation with variable coefficients for the ocean oil pollution,and a kind of coupled nonlinear partial differential equations system for the non-Newtonian micro-polar fluid.By using powerful mathematical tools and proposing new methods,we solve completely the construction and classification of traveling wave patterns for five typical nonlinear models,obtain the criterion of topological stability of patterns and adjustment and control mechanism of parameters,and give the physical realization and three-dimensional graphs to show the structures of each pattern.These analytic results show the diversity of traveling wave patterns in the above models,and the similarity and differences between these different kinds of models.The concrete results are as follows.By the complete discrimination system for polynomial method,we obtain the complete classification of all single traveling wave patterns for the deep-water nonlinear Schrodinger equation,and give clearly the conditions of existence of every pattern.By taking the concrete parameters,we prove that these patterns can be realized in physics.Three dimensional graphs show the structure of each pattern,such as singular pattern,periodic pattern,double periodic pattern and so forth.According to the parameter’s conditions,three patterns are topological stable,and others are topological unstable or semi-stable.Moreover,a rational solution with denominator being a positive polynomial describes a kind of ocean rogue wave from another view.Then we propose an improved trial equation method by which we obtain a series of traveling wave patterns for the shallow water Kd V6 equation,among those,there are three new solutions.These patterns show abundant dynamical behaviors of the high order model.The three-dimensional graphs show the structures of some typical patterns.In addition,by the Morrison’s formula and Cai et al’s simple method,we give the formula of force of wave action on the piles in platform based on the rational solution and solitary wave for the Kd V6model.By using the traveling wave reduction and the complete discrimination system for polynomial,we obtain the classification of traveling wave patterns for the ZK equations with fractional powerγ=1/2和γ=3/2and fractional order JM and ZK equations.Under the concrete parameters,all patterns can be realized in physics and show abundant space structures of wave propagation.According to the dynamics,the double periodic patterns are stable and others are unstable or semi-stable.As some physical parameters change,the unstable patterns will become the stable patterns in general.By proposing a novel method namely scale transformation method,we obtain the double periodic traveling wave pattern and other exact solutions for the nonlinear reaction-diffusion equation.The method revealed the deep symmetry of the model,and can be used easily in practice.The results show a variety of dynamical behaviors.Especially,the elliptic function solution means the periodic structure.In addition,since the velocity is the function of time,the properties of wave patterns will change as the time passing.From the graphs,we can see that these patterns are very special and show the complicated time-space structures.We also discuss the mechanism of periodic pattern in diffusion of ocean pollution.By the mathematical models,we discuss the effects of the third order nonlinear term on periodic pattern.We study a kind of mic-polar non-Newtonian fluid model and give a complete solution.Under the traveling wave transformation,the model is reduced into an ordinary differential equation.Further,we obtain the complete classification of all single traveling wave patterns.These patterns show the typical topological stability.In general case,the unstable patterns will change into stable patterns,and only two stable patterns can be observed.The parameters space is cut into three parts clearly by the complete discrimination system for the third order polynomial to show the stable and unstable domains.By adjusting the parameters,we can control the patterns.In summary,we study a series of wave motion models arising from petroleum engineering field,and give the representations of solutions of each model under parameters conditions,and then understand the propagation of waves better.The obtained results are more complete than those known results and provide the theoretical support for the applications of these models.The proposed new methods are more simple and powerful than other related methods,and can also be applied to solve the similar problems.