Coulomb Gas Representation for Rational Solutions of the Painlevé Equations
- 36 Downloads
We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II–VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations.
KeywordsDynamic System Stationary Equation Hamiltonian Formalism Rational Solution Spin Representation
Unable to display preview. Download preview PDF.
- 1.E. L. Ince, Ordinary Differential Equations [in Russian], DNTVU, Khar'kov (1938); English transl., Dover, New York (1956).Google Scholar
- 2.M. Boiti and F. Pempinelli, Nuovo Cimento B, 59, 40-58 (1980).Google Scholar
- 3.V. G. Marikhin, A. B. Shabat, M. Boiti, and F. Pempinelli, JETP, 90, 553-561 (2000).Google Scholar
- 4.M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).Google Scholar
- 5.V. G. Marikhin and A. B. Shabat, Theor. Math. Phys., 118, 173-182 (1999).Google Scholar
- 6.V. é. Adler, A. B. Shabat, and R. I. Yamilov, Theor. Math. Phys., 125, 1603-1661 (2000).Google Scholar
- 7.V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1973); English transl.: by S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Plenum, New York (1984).Google Scholar
- 8.K. Okamoto, Proc. Japan Acad. Ser. A, 56, No. 6, 264-268 (1980).Google Scholar
- 9.V. E. Adler, Physica D, 73, 335-351 (1994).Google Scholar