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Scattering theory for the Korteweg-de Vries (KdV) equation and its Hamiltonian interpretation. (English) Zbl 0618.35100

The paper deals with the Korteweg-de Vries equation of the form \(u_ t- 6uu_ x+u_{xxx}=0\) and shows that the scattering transform for it is not canonical in the naive sense as it produces ”paradoxes”, one of which is also presented. An explanation of this phenomenon is found, a correct Hamiltonian formulation of the scattering theory is proposed.
It is also shown that a necessary condition is the presence of a boundary in the continuous spectrum of the corresponding auxiliary linear problem: the continuous spectrum should not fill the whole real axis (for KdV equation spectrum fills the semi-axis \(\lambda =k^ 2\geq 0\), for the nonlinear Schrödinger equation it fills the real axis with a gap \(| \lambda | <\omega\), for the Toda chain - the interval \(-2\leq \lambda =2\cos k\leq 2)\).
Reviewer: J.Kostarcuk

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
35P25 Scattering theory for PDEs
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[1] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Phys. rev. lett., 19, 1095, (1967)
[2] Faddeev, L.D.; Takhtajan, L.A., Lett. math. phys., 10, (1985), to be published
[3] Zakharov, V.E.; Faddeev, L.D., Funkcional anal. i prilozen., Functional anal. appl., 5, 18, (1971), (in Russian); English transl. in
[4] Gardner, C.S., J. math. phys., 12, 1548, (1971)
[5] Faddeev, L.D., The inverse problem of quantum scattering theory. II, (), J. soviet math., 5, (1976), (in Russian); English transl. in · Zbl 0373.35014
[6] Shabat, A.B., Doklady AN SSSR, 211, 1310, (1973), (in Russian)
[7] Zakharov, V.E.; Manakov, S.V., Jetp, Sov. phys. JETP, 44, 203, (1976), (in Russian); English transl. in
[8] Ablowitz, M.J.; Segur, H., Stud. appl. math., 57, 13, (1977)
[9] Buslaev, V.S.; Sukhanov, V.V., (), 35, (in Russian)
[10] Buslaev, V.S.; Sukhanov, V.V., Problems of mathematical physics leningrad, 11, 78, (1985), (in Russian) · Zbl 0606.35078
[11] Zakharov, V.E.; Manakov, S.V.; Novikov, S.P.; Pitaevsky, L.P., Theory of solitons, (1980), Nauka Moscow, (in Russian) · Zbl 0598.35003
[12] Kulish, P.P.; Manakov, S.V.; Faddeev, L.D., Theor. math. phys., Theoret. math. phys., 28, 38, (1977), (in Russian); English transl. in
[13] Faddeev, L.D.; Korepin, V.E., Phys. rep., 42C, (1978)
[14] Faddeev, L.D.; Takhtajan, L.A., Hamiltonian approach in soliton theory, (), 86 · Zbl 1327.39013
[15] Kulish, P.P.; Reiman, A.G., (), 134, (in Russian)
[16] Magri, F., J. math. phys., 19, 1156, (1978)
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