Algebraic-Geometric $n$-Orthogonal Curvilinear Coordinate Systems and Solutions of the Associativity Equations
I. M. Krichever

This publication is cited in the following articles:

1. Polina A. Leonchik, Andrey E. Mironov, “Two-dimensional discrete operators and rational functions on algebraic curves”, São Paulo J. Math. Sci., 2024  
2. A. Senninger, “On discrete orthogonal coordinates corresponding to torsion-free sheaves over reducible spectral curves”, Siberian Math. J., 65:5 (2024), 1201–1207      
3. A. E. Mironov, A. Senninger, I. A. Taimanov, “Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves”, Math. Notes, 114:4 (2023), 573–582      
4. E. V. Glukhov, O. I. Mokhov, “Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type”, Izv. Math., 87:6 (2023), 1148–1160              
5. Shijie Zeng, Yaqing Liu, “The Whitham Modulation Solution of the Complex Modified KdV Equation”, Mathematics, 11:13 (2023), 2810  
6. E. V. Glukhov, “On an Algebro-Geometric Approach to Ribaucour Transformations and Bianchi Cubes”, Math. Notes, 111:5 (2022), 722–728        
7. Maltsev A.Ya. Novikov S.P., “Poisson Brackets of Hydrodynamic Type and Their Generalizations”, J. Exp. Theor. Phys., 132:4, SI (2021), 645–657    
8. R. C. Kulaev, A. B. Shabat, Trends in Mathematics, Operator Theory and Differential Equations, 2021, 79  
9. R. Ch. Kulaev, A. B. Shabat, “Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$”, Russian Math. (Iz. VUZ), 64:4 (2020), 35–43        
10. E. V. Glukhov, O. I. Mokhov, “On algebraic-geometry methods for constructing submanifolds with flat normal bundle and holonomic net of curvature lines”, Funct. Anal. Appl., 54:3 (2020), 169–178          
11. Bobenko I A., Schief W.K., Suris Yu.B., Techter J., “On a Discretization of Confocal Quadrics. a Geometric Approach to General Parametrizations”, Int. Math. Res. Notices, 2020:24 (2020), 10180–10230    
12. E. V. Glukhov, O. I. Mokhov, “On algebraic-geometry methods for constructing flat diagonal metrics of a special form”, Russian Math. Surveys, 74:4 (2019), 761–763                
13. Bobenko I A., Tsarev S., “The Curvature Line Parametrization From Circular Nets on a Surface”, J. Math. Phys., 59:9, SI (2018), 091410          
14. R. Ch. Kulaev, A. K. Pogrebkov, A. B. Shabat, “Darboux system as three-dimensional analog of Liouville equation”, Russian Mathematics, 62:12 (2018), 50–58      
15. R. Ch. Kulaev, A. K. Pogrebkov, A. B. Shabat, “Darboux system: Liouville reduction and an explicit solution”, Proc. Steklov Inst. Math., 302 (2018), 250–269            
16. Dunin-Barkowski P., Norbury P., Orantin N., Popolitov A., Shadrin S., “Primary Invariants of Hurwitz Frobenius Manifolds”, Topological Recursion and Its Influence in Analysis, Geometry, and Topology, Proceedings of Symposia in Pure Mathematics, 100, eds. Liu C., Mulase M., Amer Mathematical Soc, 2018, 297–331    
17. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937                
18. Dynnikov I.A. Glutsyuk A.A. Mironov A.E. Taimanov I.A. Vesnin A.Yu., “The Conference “Dynamics in Siberia”, Novosibirsk, February 26 - March 4, 2017”, Sib. Electron. Math. Rep., 14 (2017), A7–A30        
19. E. I. Shamaev, “O diskretizatsii parabolicheskikh koordinat”, Sib. elektron. matem. izv., 13 (2016), 1159–1169    
20. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175      
21. O. I. Mokhov, “On metrics of diagonal curvature”, J. Math. Sci., 248:6 (2020), 780–787    
22. O. A. Bogoyavlenskaya, “Ob odnom klasse konechnozonnykh krivolineinykh ortogonalnykh koordinat”, Sib. elektron. matem. izv., 12 (2015), 947–954    
23. Kodama Y. Konopelchenko B. Schief W.K., “Critical Points, Lauricella Functions and Whitham-Type Equations”, J. Phys. A-Math. Theor., 48:22 (2015), 225202            
24. Pavlov M.V. Sergyeyev A., “Oriented Associativity Equations and Symmetry Consistent Conjugate Curvilinear Coordinate Nets”, J. Geom. Phys., 85 (2014), 46–59            
25. Igonin S. Marvan M., “On Construction of Symmetries and Recursion Operators From Zero-Curvature Representations and the Darboux-Egoroff System”, J. Geom. Phys., 85 (2014), 106–123            
26. Shipman B.A., Shipman P.D., “A Lie-Algebraic Formulation For Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces”, J. Lie Theory, 24:2 (2014), 397–419      
27. E. I. Shamaev, “O reshetkakh Darbu–Egorova v ${\mathbb R}^n$”, Sib. elektron. matem. izv., 10 (2013), 113–122  
28. Shemyakova E., “A Package to Work with Linear Partial Differential Operators”, Program. Comput. Softw., 39:4 (2013), 212–219            
29. Shemyakova E., “Proof of the Completeness of Darboux Wronskian Formulae for Order Two”, Can. J. Math.-J. Can. Math., 65:3 (2013), 655–674          
30. I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Russian Math. Surveys, 66:1 (2011), 107–144                  
31. I. P. Rybnikov, “Minimal Lagrangian submanifolds in $\mathbb C\mathrm P^n$ with diagonal metric”, Siberian Math. J., 52:1 (2011), 105–112        
32. D. A. Berdinskii, I. P. Rybnikov, “On orthogonal curvilinear coordinate systems in constant curvature spaces”, Siberian Math. J., 52:3 (2011), 394–401        
33. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., “Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation”, Xvith International Congress on Mathematical Physics, 2010, 23–44          
34. A. E. Mironov, “Spectral Data for Hamiltonian-Minimal Lagrangian Tori in $\mathbb C\mathrm P^2$”, Proc. Steklov Inst. Math., 263 (2008), 112–126              
35. Bazhanov, VV, “Quantum geometry of three-dimensional lattices”, Journal of Statistical Mechanics-Theory and Experiment, 2008, P07004        
36. A. I. Bobenko, Yu. B. Suris, “On organizing principles of discrete differential geometry. Geometry of spheres”, Russian Math. Surveys, 62:1 (2007), 1–43                  
37. M. V. Pavlov, “Integrability of the Egorov systems of hydrodynamic type”, Theoret. and Math. Phys., 150:2 (2007), 225–243                
38. A. E. Mironov, I. A. Taimanov, “Some algebraic examples of Frobenius manifolds”, Theoret. and Math. Phys., 151:2 (2007), 604–613                
39. Yu. A. Aminov, “Families of submanifolds of constant negative curvature of many-dimensional Euclidean space”, Sb. Math., 197:2 (2006), 139–152                
40. A. E. Mironov, I. A. Taimanov, “Orthogonal Curvilinear Coordinate Systems Corresponding to Singular Spectral Curves”, Proc. Steklov Inst. Math., 255 (2006), 169–184      
41. Burstall, FE, “The Ribaucour transformation in Lie sphere geometry”, Differential Geometry and Its Applications, 24:5 (2006), 503          
42. Bobenko, AI, “Discrete and smooth orthogonal systems: C-infinity-approximation”, International Mathematics Research Notices, 2003, no. 45, 2415        
43. Aratyn, H, “Solutions of the WDVV equations and integrable hierarchies of KP type”, Communications in Mathematical Physics, 239:1–2 (2003), 155            
44. Chekhov, L, “DV and WDVV”, Physics Letters B, 562:3–4 (2003), 323            
45. O. I. Mokhov, “Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics”, Theoret. and Math. Phys., 130:2 (2002), 198–212              
46. O. I. Mokhov, “Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability”, Funct. Anal. Appl., 36:3 (2002), 196–204              
47. van de Leur, J, “Twisted GL(n) loop group orbit and solutions of the WDVV equations”, International Mathematics Research Notices, 2001, no. 11, 551    
48. Adam Doliwa, Integrable Hierarchies and Modern Physical Theories, 2001, 355  
49. Manuel Mañas, Luis Martínez Alonso, Integrable Hierarchies and Modern Physical Theories, 2001, 391  
50. Manas, M, “Dressing methods for geometric nets: II. Orthogonal and Egorov nets”, Journal of Physics A-Mathematical and General, 33:40 (2000), 7181            
51. A. A. Akhmetshin, Yu. S. Vol'vovskii, I. M. Krichever, “Discrete Analogs of the Darboux–Egorov Metrics”, Proc. Steklov Inst. Math., 225 (1999), 16–39      
52. Krichever, I, “Periodic and almost-periodic potentials in inverse problems”, Inverse Problems, 15:6 (1999), R117            
53. van de Leur, JW, “The construction of Frobenius manifolds from KP tau-functions”, Communications in Mathematical Physics, 205:3 (1999), 587            
54. A. A. Akhmetshin, Yu. S. Vol'vovskii, I. M. Krichever, “A generating formula for solutions of the associativity equations”, Russian Math. Surveys, 54:2 (1999), 427–429              
55. Manas, M, “From Ramond fermions to Lame equations for orthogonal curvilinear coordinates”, Physics Letters B, 436:3–4 (1998), 316