Geodesic
Nonlinear model of Schr\"odinger type: Conservation laws, Hamiltonian structure, and complete integrability
Teoretičeskaâ i matematičeskaâ fizika, Tome 65 (1985) no. 2, pp. 271-284

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A method is proposed for finding Lax type representations for nonlinear evolution (one-dimensional) equations of mathematical physics. It is shown that the Schrödinger type nonlinear model $\psi_t-i\psi_{xx}+2|\psi|^2\psi_x=0$ admits a Lax-type representation and is a Hamiltonian completely integrable dynamical system. Exact quasiperiodic (finite-gap, i.e  having only a finite number of stability bands in its spectrum) solutions of this system are obtained in terms of Riemann theta functions. 
@article{TMF_1985_65_2_a10,
     author = {N. N. Bogolyubov (Jr.) and A. K. Prikarpatskii and A. M. Kurbatov and V. G. Samoilenko},
     title = {Nonlinear model of {Schr\"odinger} type: {Conservation} laws, {Hamiltonian} structure, and complete integrability},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {271--284},
     publisher = {mathdoc},
     volume = {65},
     number = {2},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1985_65_2_a10/}
}
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N. N. Bogolyubov (Jr.); A. K. Prikarpatskii; A. M. Kurbatov; V. G. Samoilenko. Nonlinear model of Schr\"odinger type: Conservation laws, Hamiltonian structure, and complete integrability. Teoretičeskaâ i matematičeskaâ fizika, Tome 65 (1985) no. 2, pp. 271-284. http://geodesic.mathdoc.fr/item/TMF_1985_65_2_a10/