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

In this paper we consider the Allen-Cahn equation $$ -Δu = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ 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. $\| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$ 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.

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