New entire positive solution for the nonlinear Schrödinger equation: Coexistence of fronts and bumps

In this paper we construct new kinds of positive solutions of $$ \Delta u-u+u^{p}=0\quad{\rm on}\ {\Bbb R}^2 $$ when $p> 2$. These solutions have the following asymptotic behavior $$ u(x,z)\sim\omega(x-f(z))+\sum_{i=1}^{\infty}\omega_0((x, z)-\xi_i\vec{e}_1) $$ as $L\rightarrow +\infty$ where $\omega$ is a unique positive homoclinic solution of $\omega''-\omega+\omega^{p}=0$ in $\Bbb{R}$; $\omega_{0}$ is the two dimensional positive solution and $\vec{e}_1 = (1,0)$ and $\xi_j$ are points such that $\xi_j = jL + {\cal O} (1)$ for all $j\geq 1$. This represents a first result on the {\it coexistence} of fronts and bumps. Geometrically, our new solutions correspond to {\it triunduloid} in the theory of CMC surface.