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Linear systems with quadratic integral and complete integrability of the Schrödinger equation. (English. Russian original) Zbl 1442.37079

Russ. Math. Surv. 74, No. 5, 959-961 (2019); translation from Usp. Mat. Nauk 74, No. 5, 189-190 (2019).
The paper addresses a multidimensional D’Alembert (wave) equation in the \(n\)-dimensional space: \[ u_{tt}=a^2\Delta u. \] The integrability of this equation is a known fact. In the paper, the author extends this aspect of the equation, by explicitly constructing an infinite series of dynamical invariants, the lowest one being the standard Hamiltonian, \[ H=\frac{1}{2}\int d^nx\left[u_t^2+a^2\sum_n\left(u_{x_n}^2\right)\right]. \] Higher-order dynamical invariants are identified as integral expressions with the density expressed as quadratic expressions with respect the \(t\)- and \(x\)-derivatives of field \(u\). In particular, the next dynamical invariant in the series is \[ \frac{a^2}{2}\int d^nx\left[\sum_n\left(u_{tx_n}\right)^2+a^2\left(\Delta u\right)^2\right]. \]

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
35L05 Wave equation
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[1] В. В. Козлов 1992 ПММ56 6 900-906
[2] English transl. V. V. Kozlov 1992 J. Appl. Math. Mech.56 6 803-809 · Zbl 0792.70014
[3] V. V. Kozlov 2018 Regul. Chaotic Dyn.23 1 26-46 · Zbl 1400.37061
[4] Д. В. Трещёв, А. А. Шкаликов 2017 Матем. заметки101 6 911-918
[5] English transl. D. V. Treshchev and A. A. Shkalikov 2017 Math. Notes101 6 1033-1039 · Zbl 06769031
[6] L. D. Faddeev 2007 Amer. Math. Soc. Transl. Ser. 2 220 83-90
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[9] W. Miller, Jr., S. Post, and P. Winternitz 2013 Classical and quantum superintegrability with applications 1309.2694v1 124 pp.
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[11] English transl. V. V. Kozlov and D. V. Treschev 2004 Theoret. and Math. Phys.140 3 1283-1298 · Zbl 1178.81158
[12] В. В. Козлов 2005 Докл. РАН401 5 603-606
[13] English transl. V. V. Kozlov 2005 Dokl. Math.71 2 300-302
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