几类非线性色散偏微分方程的研究
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摘要
非线性现象是自然界中普遍存在的一种重要现象。许多实际的非线性问题最终都可归结为非线性系统来描述。最近几十年来,物理、力学、化学、生物、工程、航空航天、医学、经济和金融等领域中诞生了许多非线性偏微分方程,但是由于方程的非线性以及本身的复杂性,使得对这些方程的研究具有很大的挑战性。本文研究了几类有着深刻物理背景的非线性色散偏微分方程,即广义Camassa-Holm方程,变形Camassa-Holm方程,广义Degasperis-Procesi方程,Fornberg-Whitham方程,和浸入K(2,2)方程。
对于广义Camassa-Holm方程(3.18),研究了它的Cauchy问题,得到了尖峰孤立波解是方程(3.18)的Cauchy问题整体弱解的结论。并指出尖峰孤立波是轨道稳定的。研究了方程(3.18)的初边值问题(3.36),利用Kato定理证明了初边值问题(3.36)在适当函数空间上是局部适定性,结合守恒律得到了初边值问题(3.18)的两个blow up结果。还发现了方程(3.18)的一个隐性线性结构,并由此得到了它的一种多重解的叠加解,对于变形Camassa-Holm方程(3.17),研究了其行波解,数值模拟表明,该方程具有一类定义在半实轴上的精确行波解。
对于广义Degasperis-Procesi方程(4.7),研究了它的一类初边值问题(4.11),同样得到了初边值问题(4.11)在适当函数空间上是局部适定性,利用微分不等式得到了初边值问题(4.11)的解的blow up结果。利用激波ansatz首先将方程(4.7)约化为一个常微分方程,然后通过求解此常微分方程得到了方程(4.7)由尖峰孤立波和反尖峰孤立波相互碰撞形成的一类特殊的波—激波的表达式。利用平面动力系统分叉方法结合数值模拟得到方程(4.7)的类扭结解和类反扭结解。同时还得到了方程(4.7)的峰状和谷状光滑孤立波解、尖峰孤立波解和周期波解,并指出尖峰孤立波可看作是光滑孤立波和周期波的极限。最后还发现了方程(4.7)的一个隐性线性结构,并由此得到了它的一种多重解的叠加解。
对于Fornberg-Whitham方程(5.1),研究了其Cauchy问题,得到了其Cauchy问题在H~s(R)(s＞3/2)空间中是局部适定的。利用分叉方法的得到了其光滑孤立波解、尖峰孤立波解和周期尖角子解,并指出尖峰孤立波可看作是光滑孤立波和周期波的极限。同时结合数值模拟得到了其类扭结解和类反扭结解。最后利用椭圆积分得到了方程(5.1)的反向环状孤立波解、峰状光滑孤立波解,以及其它各种周期解。
对于浸入K(2,2)方程(6.7),利用平面动力系统分叉方法结合数值模拟得到了其类扭结解和类反扭结解。最后还得到了其峰状和谷状光滑孤立波解。
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 u_0∈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.
对于广义Camassa-Holm方程(3.18),研究了它的Cauchy问题,得到了尖峰孤立波解是方程(3.18)的Cauchy问题整体弱解的结论。并指出尖峰孤立波是轨道稳定的。研究了方程(3.18)的初边值问题(3.36),利用Kato定理证明了初边值问题(3.36)在适当函数空间上是局部适定性,结合守恒律得到了初边值问题(3.18)的两个blow up结果。还发现了方程(3.18)的一个隐性线性结构,并由此得到了它的一种多重解的叠加解,对于变形Camassa-Holm方程(3.17),研究了其行波解,数值模拟表明,该方程具有一类定义在半实轴上的精确行波解。
对于广义Degasperis-Procesi方程(4.7),研究了它的一类初边值问题(4.11),同样得到了初边值问题(4.11)在适当函数空间上是局部适定性,利用微分不等式得到了初边值问题(4.11)的解的blow up结果。利用激波ansatz首先将方程(4.7)约化为一个常微分方程,然后通过求解此常微分方程得到了方程(4.7)由尖峰孤立波和反尖峰孤立波相互碰撞形成的一类特殊的波—激波的表达式。利用平面动力系统分叉方法结合数值模拟得到方程(4.7)的类扭结解和类反扭结解。同时还得到了方程(4.7)的峰状和谷状光滑孤立波解、尖峰孤立波解和周期波解,并指出尖峰孤立波可看作是光滑孤立波和周期波的极限。最后还发现了方程(4.7)的一个隐性线性结构,并由此得到了它的一种多重解的叠加解。
对于Fornberg-Whitham方程(5.1),研究了其Cauchy问题,得到了其Cauchy问题在H~s(R)(s＞3/2)空间中是局部适定的。利用分叉方法的得到了其光滑孤立波解、尖峰孤立波解和周期尖角子解,并指出尖峰孤立波可看作是光滑孤立波和周期波的极限。同时结合数值模拟得到了其类扭结解和类反扭结解。最后利用椭圆积分得到了方程(5.1)的反向环状孤立波解、峰状光滑孤立波解,以及其它各种周期解。
对于浸入K(2,2)方程(6.7),利用平面动力系统分叉方法结合数值模拟得到了其类扭结解和类反扭结解。最后还得到了其峰状和谷状光滑孤立波解。
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 u_0∈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.
引文
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