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**Algebraic-geometric \(n\)-orthogonal curvilinear coordinate systems and solutions of the associativity equations.**
*(English.
Russian original)*
Zbl 1004.37052

Funct. Anal. Appl. 31, No. 1, 25-39 (1997); translation from Funkts. Anal. Prilozh. 31, No. 1, 32-50 (1997).

From the introduction: The problem of constructing \(n\)-orthogonal curvilinear coordinate systems, or flat diagonal metrics
\[
ds^2=\sum^n_{i=1} H^2_i(u) (du^i)^2, \quad u=(u^1,\dots, u^n),\tag{1}
\]
for more than a century since the famous work of Dupin and Binet published in 1810 was one of the most important problems of differential geometry. Treated as a classification problem, it was mainly solved in the beginning of the 20th century. The crucial contribution here was due to G. Darboux.

In the beginning of the 1980s, it was found that this classical problem has deep connections with the modern theory of integrable quasilinear hydrodynamic type systems in \((1+1)\)-dimensions. This theory was proposed by B. Dubrovin and S. Novikov as a Hamiltonian theory of the averaged (Whitham) equations for periodic solutions of integrable soliton equations in \((1+1)\)-dimensions. Later, it was noticed that the classification of Egorov metrics, i.e., flat diagonal metrics such that \[ \partial_j H^2_j= \partial_iH^2_j, \quad\partial_i =\partial/ \partial u^i,\tag{2} \] is equivalent to the classification problem for massive topological field theories. It was shown that locally the general solution of the Lamé equations \[ \partial_k \beta_{ij}= \beta_{ik}\beta_{kj}, \quad i\neq j\neq k, \tag{3} \]

\[ \partial_i \beta_{ij}+ \partial_j \beta_{ji}+ \sum_{m\neq i,j} \beta_{mi} \beta_{mj}= 0, \quad i\neq j,\tag{4} \] for the rotation coefficients \[ \beta_{ij}= \partial_i H_j/H_i,\quad i\neq j,\tag{5} \] depends on \(n(n-1)/2\) arbitrary functions of two variables. The system (1.3), (1.4) is equivalent to the vanishing of all a priori nontrivial components of the curvature tensor.

Quite recently, solutions of (3) and (4) have been constructed by V. Zakharov with the help of the “dressing procedure” within the framework of the inverse problem method. The main goal of this paper is not merely constructing finite-gap or algebraic-geometric solutions of the Lamé equations (3), (4) but proposing a scheme that simultaneously solves the complete system i.e., gives both the Lamé coefficients \(H_i\) and the flat coordinates \(x^i(u)\).

We use the method of Baker-Akhiezer functions, for constructing algebro-geometrical solutions to the Lamé equations. These solutions are explicitly expressed in terms of the theta functions of auxiliary algebraic curves.

In the beginning of the 1980s, it was found that this classical problem has deep connections with the modern theory of integrable quasilinear hydrodynamic type systems in \((1+1)\)-dimensions. This theory was proposed by B. Dubrovin and S. Novikov as a Hamiltonian theory of the averaged (Whitham) equations for periodic solutions of integrable soliton equations in \((1+1)\)-dimensions. Later, it was noticed that the classification of Egorov metrics, i.e., flat diagonal metrics such that \[ \partial_j H^2_j= \partial_iH^2_j, \quad\partial_i =\partial/ \partial u^i,\tag{2} \] is equivalent to the classification problem for massive topological field theories. It was shown that locally the general solution of the Lamé equations \[ \partial_k \beta_{ij}= \beta_{ik}\beta_{kj}, \quad i\neq j\neq k, \tag{3} \]

\[ \partial_i \beta_{ij}+ \partial_j \beta_{ji}+ \sum_{m\neq i,j} \beta_{mi} \beta_{mj}= 0, \quad i\neq j,\tag{4} \] for the rotation coefficients \[ \beta_{ij}= \partial_i H_j/H_i,\quad i\neq j,\tag{5} \] depends on \(n(n-1)/2\) arbitrary functions of two variables. The system (1.3), (1.4) is equivalent to the vanishing of all a priori nontrivial components of the curvature tensor.

Quite recently, solutions of (3) and (4) have been constructed by V. Zakharov with the help of the “dressing procedure” within the framework of the inverse problem method. The main goal of this paper is not merely constructing finite-gap or algebraic-geometric solutions of the Lamé equations (3), (4) but proposing a scheme that simultaneously solves the complete system i.e., gives both the Lamé coefficients \(H_i\) and the flat coordinates \(x^i(u)\).

We use the method of Baker-Akhiezer functions, for constructing algebro-geometrical solutions to the Lamé equations. These solutions are explicitly expressed in terms of the theta functions of auxiliary algebraic curves.

### MSC:

37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |

58D29 | Moduli problems for topological structures |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

35Q58 | Other completely integrable PDE (MSC2000) |

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\textit{I. M. Krichever}, Funct. Anal. Appl. 31, No. 1, 25--39 (1997; Zbl 1004.37052); translation from Funkts. Anal. Prilozh. 31, No. 1, 32--50 (1997)

### References:

[1] | G. Darboux, Leçons sur le systèmes orthogonaux et les coordonnées curvilignes, Paris, 1910. · JFM 41.0674.04 |

[2] | B. A. Dubrovin and S. P. Novikov, ”The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR,27, 665–654 (1983). · Zbl 0553.35011 |

[3] | B. A. Dubrovin and S. P. Novikov, ”Hydrodynamics of weakly deformed soliton lattices: Differential geometry and Hamiltonian theory,” Usp. Mat. Nauk,44, 35–124 (1989). · Zbl 0712.58032 |

[4] | S. P. Tsarev, ”The geometry of Hamiltonian systems of hydrodynamic type. Generalized hodograph method,” Izv. Akad. Nauk SSSR, Ser. Mat.,54, No. 5, 1048–1068 (1990). |

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[7] | V. Zakharov, Description of then-orthogonal curvilinear coordinate systems and hamiltonian integrable systems of hydrodynamic type. Part 1, Integration of the Lamé equations, Preprint (to appear in Duke Math. J.). |

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[9] | E. Witten, ”The structure of the topological phase of two-dimensional gravity,” Nuclear Phys. B,340, 281–310 (1990). |

[10] | E. Verlinder and H. Verlinder, A solution of two-dimensional topological quantum gravity, Preprint IASSNS-HEP 90/40, PUPT-1176, 1990. |

[11] | I. M. Krichever, ”The algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976). · Zbl 0361.35007 |

[12] | I. M. Krichever, ”The integration of nonlinear equations with the help of algebraic-geometric methods,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977). · Zbl 0346.35028 |

[13] | A. R. Its and V. B. Matveev, ”On a class of solutions of the Korteweg-de Vries equation,” in: Problems of Mathematical Physics,8, Leningrad State University, 1976. |

[14] | I. Krichever, O. Babelon, E. Billey, and M. Talon, ”Spin generalization of the Calogero-Moser system and the matrix KP equation,” Am. Math. Soc. Transl. (2),170, 83–119 (1995). · Zbl 0843.58069 |

[15] | S. P. Novikov and A. P. Veselov, ”Finite-gap two-dimensional periodic Schrödinger operators: exact formulas and evolution equations,” Dokl. Akad. Nauk SSSR,279, No. 1, 20–24 (1984). · Zbl 0613.35020 |

[16] | I. M. Krichever, ”Algebraic-geometric two-dimensional operators with self-consistent potentials,” Funkts. Anal. Prilozhen.,28, No. 1, 26–40 (1994). · Zbl 0863.35109 |

[17] | B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”The Schrödinger equation in a periodic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR,229, No. 1, 15–18 (1976). · Zbl 0441.35021 |

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