Quantum expansion of soliton solutions

Giacomo Mauro D’Ariano, Mario Rasetti

Leonid A. Nesterov, Nikolay A. Veretenov, Nikolay N. Rosanov

Quantum fluctuations of two-dimensional dark dissipative solitons in an interferometer filled with a Kerr nonlinear medium are analyzed theoretically. The boundaries of the stability region of classical solitons excited in the considered interferometer by an external radiation are refined. The solution of the quantum problem is studied in the linear approximation. For this purpose, the formalism of expansion of the sought solution in discrete spectral functions is extended to the two-dimensional case. Using this expansion made it possible to construct a solution that describes fluctuations of collective variables of a two-dimensional soliton, namely, the coordinate of the center and momentum. Mean squares of quantum fluctuations of these quantities are calculated. Parameter domains in which there exist squeezed states of solitons with respect to each of the collective variables are determined. Comparative analysis of squeezing with respect to these variables for light and dark solitons is performed.

Andreas Sizmann

Leonid A. Nesterov, Nikolay A. Veretenov, Nikolay N. Rosanov

Quantum fluctuations of one-dimensional dark dissipative solitons sustained by an external radiation in an interferometer with a Kerr nonlinearity are analyzed theoretically. The stability region of classical solitons in this interferometer is studied. The boundaries of this region are determined, and types of excited solitons are classified. Quantum fluctuations of solitons are analyzed in an approximation linear in fluctuations. This problem was solved by linearizing the quantum Langevin equation in a neighborhood of a classical solution for the main type of a soliton from the obtained stability region. The main attention has been paid to studying quantum fluctuations of collective variables of dissipative solitons, namely, the coordinate of the center and momentum of the soliton. Based on the expansion of solutions of the linearized equation in eigenfunctions of the discrete spectrum of this equation, a solution describing quantum fluctuations of these variables is constructed. Using this expansion scheme made it possible to give a rigorous definition of the dissipative soliton position fluctuation operator. The study performed based on this scheme has made it also possible to construct a solution for a one-dimensional dark relaxing dissipative soliton. This soliton generalizes the stationary soliton with allowance for the shift of its center and deformation of its profile followed by the recovery of its initial shape. Average squares of quantum fluctuations of collective variables are calculated. A domain of parameters in which there exist quantum states of solitons with an initially high degree of squeezing with respect to the momentum is found. It is shown that such states are in correspondence with significantly higher velocities of soliton center drift. An experiment that could detect the relative squeezing with respect to the momentum due to the soliton center drift is discussed.

F. Kh. Abdullaev, Sergei Darmanyan, Pulat K. Khabibullaev

Dorota Jakubczyk

This paper addresses the singular 1-soliton solution of the K(m, n) equation that is being considered with generalized evolution.The ansatz method will be used to extract the singular 1-soliton solution of this equation.A couple of constraint conditions will fall out in order for the singular soliton solutions to exist.Subsequently, the special cases of this equation will be studied.They are the KdV and the mKdV equations where the extended (G /G)-expansion method will be employed to extract a few nonlinear wave solutions.