Two-end solutions to the Allen-Cahn equation in ${\mathbb{R}}^3$

In this paper we consider the Allen-Cahn equation $$-\mathrm{\Delta }u=u-u^{3}in{\mathbb{R}}^3$$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$\mathrm{\parallel }u-H(\cdot -k\ln r+c_k){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.