A tale of two approaches to heteroclinic solutions for Φ-Laplacian systems

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

Existence of heteroclinic solutions for a class of problems involving the fractional Laplacian

In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type [Formula: see text] where [Formula: see text], [Formula: see text] are continuous functions verifying some technical conditions. For example [Formula: see text] can be asymptotically periodic and potential [Formula: see text] can be the Ginzburg–Landau potential, that is, [Formula: see text].

Homoclinic and heteroclinic solutions to a hepatitis C evolution model

Homoclinic and heteroclinic solutions to a standard hepatitis C virus (HCV) evolution model described by T. C. Reluga, H. Dahari and A. S. Perelson, (SIAM J. Appl. Math., 69 (2009), pp. 999–1023) are considered in this paper. Inverse balancing and generalized differential techniques enable derivation of necessary and sufficient existence conditions for homoclinic/heteroclinic solutions in the considered system. It is shown that homoclinic/heteroclinic solutions do appear when the considered system describes biologically significant evolution. Furthermore, it is demonstrated that the hepatitis C virus evolution model is structurally stable in the topological sense and does maintain homoclinic/heteroclinic solutions as diffusive coupling coefficients tend to zero. Computational experiments are used to illustrate the dynamics of such solutions in the hepatitis C evolution model.