Bright, dark, and mixed vector soliton solutions of the general coupled nonlinear Schrödinger equations

The reduction procedure for the general coupled nonlinear Schrödinger (GCNLS) equations with four-wave mixing terms is proposed. It is shown that the GCNLS system is equivalent to the well known integrable families of the Manakov and Makhankov $U(n,m)$-vector models. This equivalence allows us to construct bright-bright and dark-dark solitons and a quasibreather-dark solution with unconventional dynamics: the density of the first component oscillates in space and time, whereas the density of the second component does not. The collision properties of solitons are also studied.

Figure 1

The dependence of amplitudes of solution (7) on $b_{\text{Am}}$ (left) and $b_{\text{Ph}}$ (right). The red (solid) and blue (dashed) curves correspond to $|A_1|$ and $|A_2|$, respectively. In the left plot $b_{\text{Ph}}$ is fixed to $-\pi /3$ and in the right plot $b_{\text{Am}}$ is fixed to $1.51$. The other parameters are chosen as in the text.

Figure 2

The collisional dynamics of the BB two-soliton solution (8) on the $(x,t)$ plane. (a) Transmissional scenario of soliton collision. Parameters: $a=1,c=4,b=-1.9-0.1i,k_1=-2+i,k_2=3-i,{\alpha }_{1,1}=2-i,{\alpha }_{1,2}=1,{\alpha }_{2,1}=1.25+i,{\alpha }_{2,2}=-1+i$. (b) Reflectional scenario of soliton collision. Parameters: $a=-2,c=10.5,b=1.8-i,k_1=4-0.8i,k_2=-4+i,{\alpha }_{1,1}=2.2+i,{\alpha }_{1,2}=-5.93+0.82i,{\alpha }_{2,1}=-3+2i,{\alpha }_{2,2}=1.5-i$.

Figure 3

(a) The DD soliton (10) on the $(x,t)$ plane. (b) Snapshots of the density profile for $t=-7$. Parameters: $a=1,c=1,k=4,s_1=1,s_2=1.2,b=3.54-0.5i,\omega =1,{\rho }_1=2,{\rho }_2=0.927,{\xi }_{0,1}=4,{\xi }_{0,2}=3.1,{\theta }_0=1$.

Figure 4

The dependence of the grayness of solution (12) on $b_{\text{Am}}$. The left and right plots correspond to the density profile of component $q_1$ and $q_2$, respectively. The dashed curves correspond to the black soliton. The parameter $b_{\text{Ph}}$ is fixed to $\pi /3$. Other parameters are chosen as in the text.

Figure 5

(a) The QBD soliton solution (15) on the $(x,t)$ plane. (b) Snapshots of the density profile for $t=0.3$. Parameters: $a=3,c=1.5,k=1+6i,s=1.4,\rho =0.5,\alpha =2-2i,{\xi }_0=1,b=1-1.3i.$

Figure 6

(a) The collisional dynamics of the QBD two-soliton solutions (16) on the $(x,t)$ plane. (b),(c) Snapshots of density profile, before $\left(t=-0.34\right)$ and after $\left(t=0.5\right)$ collision. Parameters are chosen as in the text.