Domain Walls and Vector Solitons in the Coupled Nonlinear Schrodinger Equation

Abstract

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrodinger (CNLS) equation with general coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z (N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian-Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor (polarization). We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two polarized DB solitons typically feature a mass exchange that changes the parameters of the two bright components and the two soliton velocities.

Figures (5)

• Figure 1: Time evolution in Eq. (\ref{['Eq:GCNLS']}) of two asymmetric DWs between $ZZ$ and $NN$ assembled back-to-back plotted in the comoving frame. The CNLS parameters are $d_1=d_2=-5$, $g_1=-1$, $g_2=1$, $g_3=4$, and $g_4=-5$, and the travelling wave parameters are $(C_g,\omega_A,\omega_B)=(5.408,-0.5,5.5)$.
• Figure 2: Time evolution in Eq. (\ref{['Eq:GCNLS']}) of a Turing soliton plotted in the comoving frame. The CNLS parameters are $d_1=-2$, $d_2=-6$, $g_1=-7$, $g_2=-3$, $g_3=1$, and $g_4=6$, and the traveling wave parameters are $(C_g,\omega_A,\omega_B)=(9,-9.03,-4)$.
• Figure 3: Network of pitchfork bifurcations in Eq. (\ref{['eq:ODE-0_R_real']}) with the nodes representing the four equilibria and the edges representing the bifurcation points. The two Maxwell points respectively for asymmetric and symmetric DWs would occupy the two diagonal edges, but they are not shown explicitly.
• Figure 4: (Color online) Bifurcation diagrams and the profiles of the accompanying solutions for numerical continuations in $\omega_A$ shown in panel (a), followed by $C_g$ shown in panel (b). The solution profiles are shown in terms of their real (blue-solid) and imaginary (red-dashed) parts. The CNLS parameters are $d_1=-1$, $d_2=-3$, $g_1=2$, $g_2=4$, $g_3=5$, and $g_4=8$, and the traveling wave parameters at the starting point $A_1$ in panel (a) is $(C_g,\omega_A,\omega_B)=(5.093,-9.9,-9.8)$. (a) The profile at $A_1$ is shown as $\phi_A$ and $\psi_A$, and the profile at $B_1$ is shown as $\phi_B$ and $\psi_B$. (b) The profile at $A_2$ is shown as $\phi_A$ and $\psi_A$, and the profile at $B_2$ is shown as $\phi_B$ and $\psi_B$.
• Figure 5: (Color online) Collisions between two complex DB solitons with different polarizations. The CNLS parameters are $d_1=d_2=-1$, $g_1=g_2=g_3=2$, and $g_4=3$. The traveling wave parameters $(\omega_A, \omega_B, k_B)=(-9.7,-10,-2.697)$ are the same for both solitons, the velocities of the left and right solitons are respectively $(C_g^{(1)},C_g^{(2)})=(5.789,5)$, and the polarization of the right soliton is $\rho_R=0$. (a, b) Space-time plots in the frame comoving with velocity $C_g^{(0)}=5.395$. The polarization of the left soliton is $\rho_L=0$ in panel (a) and $\rho_L=\pi/2$ in panel (b). (c) Soliton profiles after the unpolarized collision in panel (a) (red), after the polarized collision in panel (b) (blue), and before collisions (black-dashed). The left soliton is always shown in the left half-plane, $x<0$, and the right soliton is always shown in the right half-plane, $x>0$.