Old and New Reductions of Dispersionless Toda Hierarchy

Symmetry Integrability and Geometry-Methods and Applications

article

Kanehisa Takasaki¹ 

[1] Graduate School of Human and Environmental Studies, Kyoto University

关键词: dispersionless Toda hierarchy;    finite-variable reduction;    waterbag model;    Landau–Ginzburg potential;    L¨owner equations;    Gibbons–Tsarev equations;    hodograph solution;    Darboux equations;    Egorov metric;    Combescure transformation;    Frobenius manifold;    flat coordinates;   

DOI  :  10.3842/SIGMA.2012.102

来源: National Academy of Science of Ukraine

【 摘 要 】

This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of ''Landau-Ginzburg potentials'' that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields a radial version of the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Flat coordinates of the underlying Egorov metrics are presented.

【 授权许可】

Unknown