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Position dependent non-linear Schrödinger hierarchies: involutivity, commutation relations, renormalisation and classical invariants. (English) Zbl 1207.35257

Summary: We consider a family of explicitly position dependent hierarchies \((I_n)_0^\infty\), containing the NLS (nonlinear Schrödinger) hierarchy. All \((I_n)_0^\infty\) are involutive and fulfill \({\mathcal D}I_n=nI_{n-1}\), where \({\mathcal D}=D^{-1}V_0\), \(V_0\) being the Hamiltonian vector field \(v\frac{\delta}{\delta v}- u\frac{\delta}{\delta u}\) afforded by the common ground state \(I_0=uv\). The construction requires renormalisation of certain function parameters.
It is shown that the ‘quantum space’ \(\mathbb C[I_0,I_1,\dots]\) projects down to its classical counterpart \(\mathbb C[p]\), with \(p=I_1/I_0\), the momentum density. The quotient is the kernel of \({\mathcal D}\). It is identified with classical semi-invariants for forms in two variables.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
17B63 Poisson algebras
17B80 Applications of Lie algebras and superalgebras to integrable systems
34C14 Symmetries, invariants of ordinary differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70S10 Symmetries and conservation laws in mechanics of particles and systems
76F30 Renormalization and other field-theoretical methods for turbulence
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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[1] Arnold, V. I., Mathematical Methods in Classical Mechanics (1978), Springer: Springer Berlin · Zbl 0386.70001
[2] Brandão, A., Symplectic structure for Gaussian diffusions, J. Math. Phys., 39, 9, 4257-4283 (1998) · Zbl 0976.81021
[3] Brandão, A.; Kolsrud, T., Phase space transformations of Gaussian diffusions, Potential Anal., 10, 2, 119-132 (1999) · Zbl 0924.58112
[4] Brandão, A.; Kolsrud, T., Time-dependent conservation laws and symmetries for classical mechanics and heat equations, (Harmonic Morphisms, Harmonic Maps, and Related Topics. Harmonic Morphisms, Harmonic Maps, and Related Topics, Brest, 1997. Harmonic Morphisms, Harmonic Maps, and Related Topics. Harmonic Morphisms, Harmonic Maps, and Related Topics, Brest, 1997, Chapman & Hall/CRC Res. Notes Math., vol. 413 (2000), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL), 113-125 · Zbl 0947.58033
[5] Dickey, L., Soliton Equations and Hamiltonian Systems (1991), World Scientific: World Scientific Singapore · Zbl 0753.35075
[6] Djehiche, B.; Kolsrud, T., Canonical transformations for diffusions, C. R. Acad. Sci. Paris I, 321, 339-344 (1995) · Zbl 0836.60084
[7] Faddeev, L. D.; Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (1986), Nauka: Nauka Moscow, (English translation by Springer-Verlag, Berlin, Heidelberg, 1987)
[8] Goldstein, H., Classical Mechanics (1980), Addison-Wesley: Addison-Wesley New York · Zbl 0491.70001
[9] Gurevich, G. B., Foundations of the Theory of Algebraic Invariants (1964), Noordhoff: Noordhoff Groningen, (English translation from the Russian) · Zbl 0128.24601
[10] Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations (1999), Wiley: Wiley Chichester · Zbl 1047.34001
[11] Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics (1983), Nauka: Nauka Moscow, (English translation by D. Reidel, Dordrecht, 1985)
[12] (Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1, Symmetries, Exact Solutions and Conservation Laws (1993), CRC Press: CRC Press Boca Raton, FL)
[13] Ibragimov, N. H.; Kolsrud, T., Lagrangian approach to evolution equations: symmetries and conservation laws, Nonlinear Dynam., 36, 1, 29-40 (2004) · Zbl 1106.70012
[14] T. Kolsrud, Quantum constants of motion and the heat Lie algebra in a Riemannian manifold, Preprint TRITA-MAT, Stockholm, 1996; T. Kolsrud, Quantum constants of motion and the heat Lie algebra in a Riemannian manifold, Preprint TRITA-MAT, Stockholm, 1996
[15] T. Kolsrud, The hierarchy of the Euclidean non-linear Schrödinger equation is a harmonic oscillator containing KdV, Preprint TRITA-MAT, Stockholm, 2004; T. Kolsrud, The hierarchy of the Euclidean non-linear Schrödinger equation is a harmonic oscillator containing KdV, Preprint TRITA-MAT, Stockholm, 2004
[16] T. Kolsrud, Symmetries for the Euclidean non-linear Schrödinger equation and related free equations, in: Proc. of MOGRAN X, Cyprus, 2004. Preprint TRITA-MAT, Stockholm, 2005; T. Kolsrud, Symmetries for the Euclidean non-linear Schrödinger equation and related free equations, in: Proc. of MOGRAN X, Cyprus, 2004. Preprint TRITA-MAT, Stockholm, 2005
[17] Kolsrud, T.; Lobeau, E., Foliated manifolds and conformal heat morphisms, Ann. Global Anal. Geom., 21, 241-267 (2002) · Zbl 1007.58010
[18] Kolsrud, T.; Zambrini, J. C., The general mathematical framework of Euclidean quantum mechanics, (Stochastic Analysis and Applications. Stochastic Analysis and Applications, Lisbon, 1989 (1991), Birkhäuser), 123-143 · Zbl 0784.60106
[19] Kolsrud, T.; Zambrini, J. C., An introduction to the semiclassical limit of Euclidean quantum mechanics, J. Math. Phys., 33, 4, 1301-1334 (1992) · Zbl 0775.60008
[20] D. Laksov, personal communication, autumn 2005; D. Laksov, personal communication, autumn 2005
[21] Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics, vol. 1, Mechanics (1977), Pergamon Press: Pergamon Press Oxford
[22] Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics, vol. 3, Quantum Mechanics (1977), Pergamon Press: Pergamon Press Oxford
[23] Lie, S., Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen, Arch. Math., 6, 328-368 (1881) · JFM 13.0298.01
[24] Loubeau, E., Morphisms of the heat equation, Ann. Global Anal. Geom., 15, 6, 487-496 (1997) · Zbl 0910.58009
[25] McKean, H. P.; Vaninsky, K. L., Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math., 50, 6, 489-562 (1997) · Zbl 0990.35047
[26] McKean, H. P.; Vaninsky, K. L., Cubic Schrödinger: the petit canonical ensemble in action-angle variables, Comm. Pure Appl. Math., 50, 7, 593-622 (1997) · Zbl 0883.35032
[27] Mikhailov, A. V.; Shabat, A. B.; Yamilov, V. V., Extension of the module of invertible transformations. Classification of integrable systems, Comm. Math. Phys., 115, 1-19 (1988) · Zbl 0659.35091
[28] Morse, P. M.; Feshbach, H., Methods of Theoretical Physics, vols. I-II (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.40603
[29] Olver, P. J., Applications of Lie Groups to Differential Equations (1993), Springer: Springer Berlin, Heidelberg, New York · Zbl 0785.58003
[30] Olver, P. J., Equivalence, Invariants and Symmetry (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0837.58001
[31] Olver, P. J., Classical Invariant Theory (1999), London Math. Soc., Cambridge Univ. Press: London Math. Soc., Cambridge Univ. Press Cambridge · Zbl 0971.13004
[32] Simon, B., The \(P(\phi)_2\) Euclidean (Quantum) Field Theory (1974), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 1175.81146
[33] Thieullen, M.; Zambrini, J. C., Probability and quantum symmetries I. The theorem of Noether in Schrödinger’s euclidean quantum mechanics, Ann. Inst. Henri Poincaré, Phys. Théorique, 67, 3, 297-338 (1997) · Zbl 0897.60062
[34] Thieullen, M.; Zambrini, J. C., Symmetries in the stochastic calculus of variations, Probab. Theory Relat. Fields, 107, 401-427 (1997) · Zbl 0868.60064
[35] Treves, F., On the characterisation of the nonlinear Schrödinger hierarchy, Sel. Math., New Ser., 9, 601-656 (2003) · Zbl 1059.37058
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