非线性演化方程的奇异行波解研究
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摘要
在物理学,力学、生物学与大气动力学等众多自然科学领域的研究中都发现反映众多因子之间相互制约和相互依存的关系方程都是非线性方程,也就是一般被称为得非线性演化方程,而其中恰有为数不少的一部分是具有某种奇异性质的可积的偏微分方程也就是说具有Hamilton结构。
在1993年,Camassa与Holm利用Hamilton原理获得一类新的完全可积色散的浅水波方程(Camassa-Holm方程)并证明该方程在k=0时,具有一个在它的波峰处具有不连续的一阶导数的孤立波解(孤立尖波)。同年,Rosenau与Hyman为了理解非线性色散在液体滴落模式中的作用,引入并研究一类完全非线性的Korteweg-de Vries(KdV)方程K(m,n),得到了K(2,2)方程的一个特别的具有孤立子形状的解被他们命名为紧孤立子(紧孤立波)。自此之后,这些由光滑的初值条件经过传播后形成的非光滑解的出现的奇异现象就一直被广泛关注,近来也得到了一定的发展。
我们关心的是偏微分方程的行波解,在行波变换下非线性偏微分方程就转化为一个以波速为参数的常微分方程,而具有Hamilton结构的偏微分方程往往就化为可积的常微分方程,从而我们可以借助动力系统分支理论,从微分动力系统理论的角度,以解析的方法来说明在该方程中这类奇异解的出现的真正原因,并求出方程的行波解。
我们研究重点是研究这些奇异行波解并力图解释一些类型的非线性发展方程的行波解的某些动力学行为,研究方程的光滑的周期波、孤立子、扭波等以及非光滑的奇异行波解出现的参数条件,例如紧孤子解、周期尖波、尖孤立子、紧扭波等的存在性以及其在某些参数条件下的解析表达式。
在本文中,我们研究了几类经典的非线性波方程,广义Camassa-Holm方程,Degasperis-Procesi方程,一类非常广泛的方程的具有非线性色散项的广义B(m,n)方程,广义非线性Schr(?)dinger方程,非线性色散的K(m,n,k)方程以及广义非线性Klein-Gordon模型方程的行波解的分支,特别是其奇异行波解,特别地,在广义非线性Klein-Gordon模型方程的研究中首次提出了两类不同于以前已知的奇异解,并给出了其积分表达式。由此可见,利用微分动力系统的动力学分支理论,通过形变的技巧,不仅可以清楚地解释了已知的奇异行波解产生的动力学原因,而且可以了解其他的一些奇异解。
The mathematical modeling of enormously important phenomena arising in physics, mechanics, biology and other research fields often leads to nonlinear equations which are usually named nonlinear evolution equations. It is remarkable that many of these nonlinear equations possess a regular behavior, typical of integrable partial differential equations, that is there possess Hamilton structures.
In 1993, Camassa and Holm derived a completely integrable dispersive shallow water equation by using Hamiltonian methods and showed a soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak. In the same year, to understand the role of nonlinear dispersion in pattern formation, Rosenau and Hyman introuduce and study Korteweg-de Vries-like equation K(m,n) with nonlinear dispersion and presented their solitary wave solutions has compact support (which was named compacton), that is they vanish identically outside a finite core region. The appearences of these non-smooth travelling waves by propagation of the smooth initial conditions attracted many research attentions. Thse topics have seen significant advances and research is also very active.
We consider the travelling wave solution of these partial differential equations which are reduced into ordernary differential equations with travelling wave speed c as a parameter and a PDE with Hamiltonian structures into a integrable ODE which can be studied by using of the bifurcation theores of the dynamical system and thus understand their dynamics for some classes of the classical nonlinear evolution equations.
The aim of this dissertation is to study and understand the dynamics.of the traveling wave soltions of nonlinear wave equations, such as smooth periodic wave solutions, solitarys, kink waves and other non-smooth wave solutions such as compacton, cuspon and periodic cusp waves, et al and with a special emphasis on singular wave.
In this dissertation, we emplied the bifurcation theory of the dynamical systems and the transformation techniques to study the traveling wave soltions of several classes of extensively used classical nonlinear wave equations including the generalized Camassa-Holm equation, Degasperis-Procesi equation, generalized B(m,n) equation with nonlinear dispersion, generalized Schrodinger eqution, generalized K(m,n,k) equation and generalized Klein -Gordon model equation and their bifurcations with an emphasis on singular waves. It is worthy to point out that we present two types of singular waves which are different to the common known ones. One will see from this dissertation that the bifurcation theory of the dynamical systems and the transformation techniques can be used not only to clarify the reason of the existences of these common known singular waves but also to find some other typies ones.
在1993年,Camassa与Holm利用Hamilton原理获得一类新的完全可积色散的浅水波方程(Camassa-Holm方程)并证明该方程在k=0时,具有一个在它的波峰处具有不连续的一阶导数的孤立波解(孤立尖波)。同年,Rosenau与Hyman为了理解非线性色散在液体滴落模式中的作用,引入并研究一类完全非线性的Korteweg-de Vries(KdV)方程K(m,n),得到了K(2,2)方程的一个特别的具有孤立子形状的解被他们命名为紧孤立子(紧孤立波)。自此之后,这些由光滑的初值条件经过传播后形成的非光滑解的出现的奇异现象就一直被广泛关注,近来也得到了一定的发展。
我们关心的是偏微分方程的行波解,在行波变换下非线性偏微分方程就转化为一个以波速为参数的常微分方程,而具有Hamilton结构的偏微分方程往往就化为可积的常微分方程,从而我们可以借助动力系统分支理论,从微分动力系统理论的角度,以解析的方法来说明在该方程中这类奇异解的出现的真正原因,并求出方程的行波解。
我们研究重点是研究这些奇异行波解并力图解释一些类型的非线性发展方程的行波解的某些动力学行为,研究方程的光滑的周期波、孤立子、扭波等以及非光滑的奇异行波解出现的参数条件,例如紧孤子解、周期尖波、尖孤立子、紧扭波等的存在性以及其在某些参数条件下的解析表达式。
在本文中,我们研究了几类经典的非线性波方程,广义Camassa-Holm方程,Degasperis-Procesi方程,一类非常广泛的方程的具有非线性色散项的广义B(m,n)方程,广义非线性Schr(?)dinger方程,非线性色散的K(m,n,k)方程以及广义非线性Klein-Gordon模型方程的行波解的分支,特别是其奇异行波解,特别地,在广义非线性Klein-Gordon模型方程的研究中首次提出了两类不同于以前已知的奇异解,并给出了其积分表达式。由此可见,利用微分动力系统的动力学分支理论,通过形变的技巧,不仅可以清楚地解释了已知的奇异行波解产生的动力学原因,而且可以了解其他的一些奇异解。
The mathematical modeling of enormously important phenomena arising in physics, mechanics, biology and other research fields often leads to nonlinear equations which are usually named nonlinear evolution equations. It is remarkable that many of these nonlinear equations possess a regular behavior, typical of integrable partial differential equations, that is there possess Hamilton structures.
In 1993, Camassa and Holm derived a completely integrable dispersive shallow water equation by using Hamiltonian methods and showed a soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak. In the same year, to understand the role of nonlinear dispersion in pattern formation, Rosenau and Hyman introuduce and study Korteweg-de Vries-like equation K(m,n) with nonlinear dispersion and presented their solitary wave solutions has compact support (which was named compacton), that is they vanish identically outside a finite core region. The appearences of these non-smooth travelling waves by propagation of the smooth initial conditions attracted many research attentions. Thse topics have seen significant advances and research is also very active.
We consider the travelling wave solution of these partial differential equations which are reduced into ordernary differential equations with travelling wave speed c as a parameter and a PDE with Hamiltonian structures into a integrable ODE which can be studied by using of the bifurcation theores of the dynamical system and thus understand their dynamics for some classes of the classical nonlinear evolution equations.
The aim of this dissertation is to study and understand the dynamics.of the traveling wave soltions of nonlinear wave equations, such as smooth periodic wave solutions, solitarys, kink waves and other non-smooth wave solutions such as compacton, cuspon and periodic cusp waves, et al and with a special emphasis on singular wave.
In this dissertation, we emplied the bifurcation theory of the dynamical systems and the transformation techniques to study the traveling wave soltions of several classes of extensively used classical nonlinear wave equations including the generalized Camassa-Holm equation, Degasperis-Procesi equation, generalized B(m,n) equation with nonlinear dispersion, generalized Schrodinger eqution, generalized K(m,n,k) equation and generalized Klein -Gordon model equation and their bifurcations with an emphasis on singular waves. It is worthy to point out that we present two types of singular waves which are different to the common known ones. One will see from this dissertation that the bifurcation theory of the dynamical systems and the transformation techniques can be used not only to clarify the reason of the existences of these common known singular waves but also to find some other typies ones.
引文
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