  We study the following coupled Schr\"{o}dinger equations which have appeared
as several models from mathematical physics: {displaymath} {cases}-\Delta u_1
+\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega, -\Delta u_2 +\la_2
u_2 =\mu_2 u_2^3+\beta u_1^2 u_2, \quad x\in \Om, u_1=u_2=0 \,\,\,\hbox{on
\,$\partial\Om$}.{cases}{displaymath} Here $\Om$ is a smooth bounded domain in
$\R^N (N=2, 3)$ or $\Om=\RN$, $\la_1,\, \la_2$, $\mu_1,\,\mu_2$ are all
positive constants and the coupling constant $\bb<0$. We show that this system
has infinitely many sign-changing solutions. We also obtain infinitely many
semi-nodal solutions in the following sense: one component changes sign and the
other one is positive. The crucial idea of our proof, which has never been used
for this system before, is turning to study a new problem with two constraints.
Finally, when $\Om$ is a bounded domain, we show that this system has a least
energy sign-changing solution, both two components of which have exactly two
nodal domains, and we also study the asymptotic behavior of solutions as
$\beta\to -\infty$ and phase separation is expected.

Analysis of PDEs