Soliton equations and integrability
Goal: Understanding the main methods for studying solitons in partial differential/discrete equations, asimptotics and integrability
Supervisor: Stefan Carstea
Scheme:
A: Integrability through Lax pair; case of Korteweg de Vries and Toda lattice
B: Direct and inverse scattering transform, soliton solutionsC: Dressing method for finding solitons; Riemann-Hilbert approach, application to nonlinear Schrodinger equation
D. Direct methods, Hirota bilinear formalism
E. periodic solutions, Dubrovin-Novikov dispersion relations method
F. Asymptotic methods for obtaining amplitude and soliton equations
G. Supersymmetric and fermionic extension of soliton equations, interaction of super-solitons(advanced)
H. Multidimensional solitons, \dbar-problem (advanced)
References:
1.M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering transform, SIAM, 1981.
2. M. Toda, Theory of nonlinear lattices, Springer, 1981.
3. Manakov, Novikov, Pitaevski, Zakharov, Thoery of Solitons; the inverse scattering method, Consultants Bureau, 1984.
4. P. Drazin, R. Johnson, Solitons an Introduction, Cambridge Univ Press, 1989.
5. R. Hirota, Direct method in Soliton Theory, Cambridge Univ Press, 2004.
6. R. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Univ Press, 1997.
7. B. G. Konopelchenko, Introduction to Multidimensional Integrable equations, Plenum Press, 1992.
8. A. Jeffrey and T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory, Pitman, London, 1982.
9. B. de Witt, Supermanifolds, 1992.