## 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

### Keywords:

D’Alembert equation; formal integrability
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### References:

 [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 [7] Д. В. Трещёв 2005 Тр. МИАН 250 226-261 [8] English transl. D. V. Treschev 2005 Proc. Steklov Inst. Math.250 211-244 [9] W. Miller, Jr., S. Post, and P. Winternitz 2013 Classical and quantum superintegrability with applications 1309.2694v1 124 pp. [10] В. В. Козлов, Д. В. Трещев 2004 ТМФ140 3 460-479 [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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