Scattering of Solitons in Derivative Nonlinear Schrödinger Model

We show that the chiral soliton model recently introduced by Aglietti et al. can be made integrable by adding an attractive potential with a fixed coefficient. The modified model is equivalent to the derivative nonlinear Schrödinger model which does not possess parity and Galilean invariance. We obtain explicit one and two classical soliton solutions and show that in the weak coupling limit, they correctly reproduce the bound state energy as well as the time delay of two-body quantum mechanics of the model. 1 E-mail address; [email protected] It has been known for sometime that soliton solutions to certain nonlinear field equations can be associated with elementary particles in quantum field theory. In particular, the nonlinear Schrödinger system shows that its classical solitons behave as quantized particles of the same theory. This may be compared with the sine-Gordon solitons which are associated with quantized elementary particles of the massive Thirring model. Recently, this correspondence has been extended to a new type of 1+1-dim nonlinear field equation[1, 2], which arose in the study of the dimensionally reduced 2+1-dim nonlinear Schrödinger model coupled to Chern-Simons gauge theory[3], breaking the Galilean invariance. It has been noted that this theory supports a chiral soliton solution, whose mass formula justifies the interpretation of a soliton as a bound state of elementary particles of the quantized theory in the weak coupling limit[1]. However, one serious drawback of the theory was its lacking of integrability structure which made impossible of finding multi-soliton solutions and their subsequent scattering behaviors. In this Letter, we show that this problem can be cured nicely when we add an attractive potential term to the theory with a fixed coefficient. This allows us to find exact one and two soliton solutions and address their quantum mechnical particle-like behavior. We find that one soliton solution reproduces the bound state energy of two identical particles, and reduces to the one soliton solution of Ref.[1] in the weak coupling limit. Two soliton solution describes the soliton-soliton scattering from which we obtain the time delay of each solitons during the scattering process. It is shown that these time delays agree with those of quantum particle scattering thereby confirming the quantum particle interpretation of solitons of the present model. The model we consider is given by the Lagrangian,