The Study On Several Nonlinear Dispersive Partial Differential Equations

Nonlinearity is universal and important phenomenon in nature. Most nonlinear problems can be described by nonlinear equations.In recent years,many nonlinear partial differential equations were derived from physics,mechanics,chemistry,biology,engineering,aeronautics, medicine,economy,finance and many other fields.Because of the non-linearity and complexity of themselves,it is a big challenge to deal with them.In the paper,we study several nonlinear partial differential dispersive equations,that is,a generalized Camassa-Holm equation,a modified Camassa-Holm equation,a generalized Degasperis-Procesi equation,the Fornberg-Whitham equation and the osmosis K(2,2) equation.Firstly,we study a generalized Camassa-Holm equation(3.18).We prove that the obtained peaked solitary wave solution of Eq.(3.18) is a global weak solution to the Cauchy problem of Eq.(3.18).We also point out that the peaked solitary wave solution is orbital stable.In addition, we study an initial boundary value problem of Eq.(3.18).With the aim of Kato's theorem,we prove the initial and boundary value problem (3.36) is local well-posed in some function space.Two blow-up results are established by combining conservation law.We also investigate a modified Camassa-Holm equation(3.17).With the aim of numerical simulations,we show Eq.(3.17) has a type of travelling wave solution that defined on some semifinal interval and possessing some properties of kink wave solution or antikink wave solution.Secondly,we study a generalized Degasperis-Procesi equation(4.7). We study an initial boundary value problem of Eq.(4.7) and obtain that the initial and boundary value problem(4.11) is local well-posed in some function space,and also obtain a blow-up result.By the shock wave ansatz,we convert Eq.(4.7) into a group of ordinary differential equations,then obtain a special solution of Eq.(4.7),that is shock wave solution.It can be regard as the result of the collision of peakon and antipeakon. By using the bifurcation method of planar dynamical systems and the numerical simulations,we obtain the kink-like and antikink-like wave solutions of Eq.(4.7).Meanwhile,the smooth solitary wave solutions of peak and valley form,the peaked solitary wave solutions and the period cusp wave solutions of Eq.(4.7) are also obtained.We point out that the peaked solitary wave solutions can be regarded as the limit of smooth solitary wave solutions and also the period cusp wave solutions. In addition,we find an implicit linear structure in Eq.(4.7).According to the linear structure,we give the superposition of multi-solutions of Eq.(4.7).This is an interesting result.Thirdly,we study the Fornberg-Whitham equation(5.1).By Kato's theorem,we prove that the Cauchy problem of Eq.(5.1) is local well- posed with the initial data u₀∈H^s(R)(s＞3/2).Employing the bifurcation method of planar dynamical systems we obtain the smooth solitary wave solutions of peak form,the peaked solitary wave solutions and the period cusp wave solutions of Eq.(5.1).We point out that the peaked solitary wave solutions can be regarded as the limit of smooth solitary wave solutions and also the period cusp wave solutions.Meanwhile,the kink-like and antikink-like wave solutions of Eq.(5.1) are obtained.We also make the numerical simulations of the reduced traveling wave system, and the numerical result showed that our theoretical results are correct.In addition,with the aim of elliptic integral,we obtain the inverted loop-like solitary wave solutions,the smooth solitary wave solutions of peak form,and many other period wave solutions of Eq.(5.1).Lastly,we study the osmosis K(2,2) equation Eq.(6.7).Employing the bifurcation method of planar dynamical systems and the numerical simulations,we obtain the kink-like and antikink-like wave solutions of Eq.(6.7).Meanwhile,the smooth solitary wave solutions of peak and valley form of Eq.(6.7) are also obtained.We point out that the peaked solitary wave solutions can be regarded as the limit of smooth solitary wave.