Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations

A different kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable $N$-coupled nonlinear Schrödinger (CNLS) equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients. The corresponding soliton solutions for the $N=2$ case are obtained by using Hirota’s bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions for which nonsingular soliton propagation can occur. The multisoliton solutions and a generalization of the results to the multicomponent case with arbitrary $N$ are also presented. An appealing feature of soliton collision in the present case is that all the components of a soliton can simultaneously enhance their amplitudes, which can lead to a different kind of amplification process without induced noise.

Figure 1

Intensity plots of singular one soliton solution of Eq. (1) for $N=2$: (a) for the case $\mid {\alpha }_1^{\left(1\right)}\mid =\mid {\alpha }_1^{\left(2\right)}\mid$; (b) for the case $\mid {\alpha }_1^{\left(1\right)}\mid <\mid {\alpha }_1^{\left(2\right)}\mid$.

Figure 2

Intensity plots of regular one soliton solution of Eq. (1) for the $N=2$ case.

Figure 3

Stationary singular two soliton solution for the $N=2$ case.

Figure 4

Shape changing (intensity redistribution) collision of two solitons in the mixed CNLS system for the $N=2$ case.

Figure 5

Elastic collision of two solitons in the mixed CNLS system for the $N=2$ case.

Figure 6

Shape changing (intensity redistribution) collision of three solitons in the mixed CNLS system for the $N=2$ case.

Figure 7

Elastic collision of three solitons in the mixed CNLS system for the $N=2$ case.

Figure 8

Stationary singular three soliton solution for the $N=3$ case.

Figure 9

Shape changing (intensity redistribution) collision of two solitons in the mixed CNLS system, for the $N=3$ case, exhibiting same kind of shape changes for a given soliton in all the three components.

Figure 10

Shape changing (intensity redistribution) collision of two solitons in the mixed CNLS system, for the $N=3$ case, exhibiting same kind of shape changes for a given soliton in the $q_1$ and $q_3$ components and an exactly opposite collision scenario in the $q_2$ component.