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Existence of heteroclinic orbits for a corner layer problem in anisotropic interfaces. (English) Zbl 1157.34047
The authors consider the problem of finding \(C^2\)-smooth heteroclinic solutions of the singularly perturbed system of second-order differential equations of gradient type:
\[ x''=g_x(x,y),\quad \varepsilon y''=g_y(x,y),\quad x,y\in\mathbb R^1,\tag{1} \]
withthe boundary conditions
\[ x(-\infty)=y(-\infty)=0, \quad x(+\infty)=x_1,\quad y(+\infty)=y_1\tag{2} \]
where the parameter \(\varepsilon\) is sufficiently small, and the function \(g(x,y)\) satisfies some salient conditions. In this paper the authors study problem (1)–(2) with the function \(g(x,y)=2(x^2+y^2)+\frac34\,y^4+x^4\). This special problem has a physical motivation: the multi-order-parameter phase field model (for anisotropy of interfaces in an ordered alloy), the description of crystalline interphase boundaries.
Solutions of the system (1) with sufficiently small parameter \(\varepsilon>0\) are approximated by a non-smooth connection for the limiting system obtained by setting \(\varepsilon=0\). The irregularity in the formal limit arises due to the branching nature of the set of solutions \((x,y)\) of the equation \(g_y(x,y)=0\), hence the critical manifold has a singularity. The mathematical interest of the problem stems from the fact that the smoothness and normal hyperbolicity of the critical manifold fail at certain points, thus the well-developed geometric singular perturbation theory does not apply. Using a functional analytic approach, the authors prove the existence of \(C^2\)-smooth heteroclinic solutions of the problem (1)–(2), and estimated their dependence on \(\varepsilon\).

34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations