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

ANZIAM J. 44 (2002), 73-81

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

Oktay K. Pashaev
Department of Mathematics
Izmir Institute of Technology
Liyla-Izmir
35437 Turkey
pashaev@likya.iyte.edu.tr

and

Jyh-Hao Lee
Institute of Mathematics
Academia Sinica
Taipei 11529
Taiwan
ROC
leejh@math.sinica.edu.tw

Abstract

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.

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