An index theory with applications to homoclinic Orbits of Hamiltonian systems and Dirac equations

TL;DR

This paper introduces a new index pair for dual variational methods and demonstrates its application in establishing the existence and multiplicity of homoclinic orbits in Hamiltonian systems and solutions in Dirac equations.

Contribution

It defines a novel index pair and explores its relationship with existing indices, applying it to nonlinear Hamiltonian and Dirac systems.

Findings

Established the relationship between different index definitions.

Proved existence of homoclinic orbits in Hamiltonian systems.

Proved existence of solutions for nonlinear Dirac equations.

Abstract

In this paper, we will define the index pair $(i_A(B),\nu_A(B))$ by the dual variational method, and show the relationship between the indices defined by different methods. As applications, we apply the index $(i_A(B),\nu_A(B))$ to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations