 Solution

The Euler–Lagrange equation is $\kappa {\varphi }^{\prime \prime }=a\varphi +b{\varphi }^3$. Multiplication by ${\varphi }^{\prime }$ and integration gives the first integralWith $\varphi ={\varphi }_{B}g$, $u=x/{\xi }_0$, and ${\xi }_0^2=-2\kappa /a$, this becomesThe monotone heteroclinic solutions areTranslation invariance leaves $x_0$ undetermined in an infinite system. A fixed global composition, boundary condition, or pinning field fixes this midpoint position.