Model equation of the theory of solitons
We consider the hierarchy of integrable (1+2)-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain n arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable (1+1)-dimensional equations.
Keywordshierarchy of commuting vector fields Riemann invariant Dubrovin equations
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- 1.L. Martínez Alonso and A. B. Shabat, Phys. Lett. A, 300, 58–64 (2002); J. Nonlinear Math. Phys., 10, 229–242 (2003); Theor. Math. Phys., 140, 1073–1085 (2004); A. B. Shabat and L. Martínez Alonso, “On the prolongation of a hierarchy of hydrodynamic chains,” in: New Trends in Integrability and Partial Solvability (NATO Sci. Ser. II, Math. Phys. Chem., Vol. 132, A. B. Shabat et al., eds.), Kluwer, Dordrecht (2004), pp. 263–280.zbMATHCrossRefMathSciNetGoogle Scholar
- 6.V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov Theory of Solitons: The Inverse Scattering Method, Plenum, New York (1984).zbMATHGoogle Scholar