The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type
The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type
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DOI：
10.1007/pl00005573
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发表时间：
2001-05
影响因子：
2.4
通讯作者：
M. Alber;R. Camassa;Yuri N. Fedorov;Darryl D. Holm;J. Marsden
中科院分区：
文献类型：
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作者：
M. Alber;R. Camassa;Yuri N. Fedorov;Darryl D. Holm;J. Marsden
推广了非线性可积偏微分方程的代数几何方法，得到了一类N分量非线性发展方程组的新的分段光滑弱解.这个类包括，除其他外，方程从Dym和浅水方程层次。本文的主要目的是给出这些非线性偏微分方程的分段光滑弱解的显式θ-泛函表达式，这些解与超椭圆Jacobian的非线性子簇有关。首先，我们展示了可积偏微分方程的一些特殊功能，承认分片光滑弱解，这使得他们不同于方程的解决方案是全球亚纯，如KdV方程。其次，我们融合了代数几何和弱解偏微分方程的技术，以获得进一步的洞察力，并明确的公式，分段光滑有限间隙solutions.的基本技术用于实现这些目标是相当不同的，从早期的文件处理峰值解决方案。首先，有限间隙分段光滑解的轮廓与某些有限维台球动力系统和椭圆台球有关。其次，在将某些有限维Hamilton系统在Riemann曲面上的解归结为非标准Jacobi反演问题的解之后，通过引入新的参数化来解决这一问题。在代数几何方法的其他自然结果中，我们发现有限维可积Hamilton动力系统描述了有限间隙中的峰的运动以及极限（孤子）情况，并精确地解决它们。峰的动力学也通过使用Jacobi反演问题获得。最后，我们将我们的方法与冲击波方法的弱解的波动方程，通过确定跳跃条件的峰值位置。
An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class ofN-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians.The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations.Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.