Abstract
Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints, the dispersionless Veselov-Novikov equation is reduced to the (1+1)-dimensional hydrodynamic-type systems.
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References
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM (1981).
L. V. Bogdanov and B. G. Konopelchenko, Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type, Preprint arXiv:nlin.SI/0312013 (2003).
Y. Cheng and Y. S. Li, “The constraint of the Kadomtsev-Petviashvili equation and its special solutions,” Phys. Lett. A, 157, 22 (1991).
B. A. Dubrovin and S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices: differential geometry and Hamiltonian theory,” Russ. Math. Surv., 44, 35, (1989).
N. M. Ercolani et al., Eds., Singular Limits of Dispersive Waves, Nato Adv. Sci. Inst. Ser. B, Phys. 320, Plenum Press, New York (1994).
E. V. Ferapontov, “Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry,” Differ. Geom. Appl., 11, 117 (1999).
J. Gibbons and S. P. Tsarev, “Conformal maps and reductions of the Benney equations,” Phys. Lett. A, 258, 263 (1999).
Y. Kodama, “A method for solving the dispersionless KP equation and its exact solutions,” Phys. Lett. A, 129, 223 (1988).
Y. Kodama, “Solutions of the dispersionless Toda equation,” Phys. Lett. A, 147, 477 (1990).
Y. Kodama and J. Gibbons, “A method for solving the dispersionless KP hierarchy and its exact solutions, Phys. Lett. A, 135, No. 3, 167 (1989).
B. Konopelchenko and L. Martinez Alonso, “\(\overline \partial \)-equations, integrable deformations of quasi-conformal mappings, and Whitham hierarchy,” Phys. Lett. A, 286, 161 (2001).
B. G. Konopelchenko and L. Martinez Alonso, “Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations,” Stud. Appl. Math., 109, 313–336 (2002).
B. G. Konopelchenko, L. Martinez Alonso, and O. Ragnisco, “The \(\overline \partial \)-approach to the dispersionless KP hierarchy,” J. Phys. A: Math. Gen., 34, 10209 (2001).
B. G. Konopelchenko and A. Moro, “Geometrical optics in nonlinear media and integrable equations,” J.Phys. A: Math. Gen., 37, L105–L111 (2004).
B. G. Konopelchenko and A. Moro, “Integrable equations in nonlinear geometrical optics,” Stud. Appl. Math. (to appear); Preprint arXiv:nlin.SI/0403051 (2004).
B. G. Konopelchenko and U. Pinkall, “Integrable deformations of affine surfaces via Knizhnik-Veselov-Novikov equation,” Phys. Lett. A, 245, 239–245 (1998).
B. Konopelchenko, J. Sidorenko, and W. Strampp, “(1 + 1)-Dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems,” Phys. Lett. A, 157, 17 (1991).
I. M. Krichever, “Averaging method for two-dimensional integrable equations,” Funkts. Anal. Prilozh., 22, 37 (1988).
I. M. Krichever, “The τ-function of the universal Whitham hierarchy, matrix models, and topological field theories,” Commun. Pure Appl. Math., 47, 437 (1994).
A. Yu. Orlov, “Vertex operator, \(\overline \partial \)-problem, symmetries, variational identities, and Hamiltonian formalism for (2 + 1) integrable systems,” in: Nonlinear and Turbulent Processes in Physics (V. Baryakhtar, Ed.), World Scientific, Singapore (1988).
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
A. P. Veselov and S. P. Novikov, “Finite-zone two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations,” Dokl. Akad. Nauk SSSR, 279, 20 (1984).
V. E. Zakharov, “Benney equations and quasi-classical approximation in the inverse problem method,” Funkts. Anal. Prilozh., 14, 89 (1980).
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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.
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Bogdanov, L.V., Konopelchenko, B.G. & Moro, A. Symmetry constraints for real dispersionless Veselov-Novikov equation. J Math Sci 136, 4411–4418 (2006). https://doi.org/10.1007/s10958-006-0234-3
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DOI: https://doi.org/10.1007/s10958-006-0234-3