  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.
