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Black holes and solitons of the quantized dispersionless NLS and DNLS equations

Published online by Cambridge University Press:  17 February 2009

Oktay K. Pashaev  and
Jyh-Hao Lee
Oktay K. Pashaev
Affiliation:
Department of Mathematics, Izmir Institute of Technology, Liyla-Izmir, 35437Turkey; e-mail: [email protected]. Joint Institute for Nuclear Research, Dubna, 141980, Russian Federation. Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, ROC; e-mail: [email protected].
Jyh-Hao Lee
Affiliation:
Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, ROC; e-mail: [email protected].
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Abstract

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The classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 73 - 81
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Auberson, G. and Sabatier, P. C., “On a class of homogeneous nonlinear Schrödinger equations”, J. Math. Phys. 35 (8) (1994) 4028–4040.CrossRefGoogle Scholar
[2]Clarkson, P. A. and Cosgrove, C. M., “Painlevé analysis of the nonlinear Scrödinger family of equations”, J. Phys. A 20 (1987) 2003–24.Google Scholar
[3]Jin, S., Levermore, C. D. and McLaughlin, D. W., “The semiclassical limit of the defocusing NLS hierarchy”, Comm. Pure Appl. Math. 52 (5) (1999) 613–654.Google Scholar
[4]Kaup, D. J. and Newell, A. C., “An exact solution for a derivative nonlinear Scrödinger equation”, J. Math. Phys. 19 (1978) 789–801.Google Scholar
[5]Kruskal, M. D., “Maximal extension of Schwarzschild metric”, Phys. Rev. (2) 119 (1960) 1743–1745.CrossRefGoogle Scholar
[6]Lee, J.-H., Lin, C.-K. and Pashaev, O. K., “Equivalence relation and bilinear representation for derivative nonlinear Scrödinger type equations”, in Proceeding of Workshop on Nonlinearity, Integrability and all that: Twenty years after NEEDS' 79 (eds. Boiti, M., Martina, L., Pempinelli, F., Prinari, B. and Soliani, G.), (World Scientific, Singapore, 2000) 175–181.CrossRefGoogle Scholar
[7]Martina, L., Pashaev, O. K. and Soliani, G., “Integrable dissipative structures in the gauge theory of gravity”, Classical Quantum Gravity 14 (1997) 3179–3186.Google Scholar
[8]Martina, L., Pashaev, O. K. and Soliani, G., “Bright solitons as black holes”, Phys. Rev. D (3) 58 (1998) 084025.Google Scholar
[9]Nakamura, A. and Chen, H. H., “Multisoliton solutions of a derivative nonlinear Scrödinger equation”, J. Phys. Soc. Japan 49 (1980) 813–816.Google Scholar
[10]Pashaev, O. K., “Integrable models as constrained topological gauge theory. Constrained dynamics and quantum theory”, Nucl. Phys. B Proc. Suppl. 57 (1997) 338–341.Google Scholar
[11]Pashaev, O. K. and Lee, J.-H., “Soliton resonances, black holes and Madelung fluid”, J. Nonl. Math. Phys. Supplement Vol. 8 (2001) 230–234.CrossRefGoogle Scholar
[12]Pashaev, O. K. and Lee, J.-H., “Resonance NLS solitons as black holes in Madelung fluid”, hepth/9810139.Google Scholar
[13]Zakharov, V. E. and Shabat, A. B., “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Soviet Physics JETP 34 (1) (1972) 62–69.Google Scholar