Morse Index Theory For Heteroclinic Orbits Of Lagrangian Systems

The thesis consists of two main parts.At the first part,we mainly study the Morse index theory for heteroclinic orbits and half-clinic orbits of Lagrangian system.In order to establish this theory,we first provide the relation between Morse index and Maslov index of the heteroclinic orbits the sufficient and neces-sary condition on the Fredholm of Sturm-Liouville operator defined both on the line and on the half-line,thereby we provide the relation between Morse index and Maslov index of the heteroclinic orbits and the relation between Morse index and Maslov index of the half-clinic orbits for any self-adjoint boundary condition.In addition,we compute the difference of those two Morse indices of half-clinic orbits between any self-adjoint and Dirichlet boundary conditions.At the second part,we study the index theory for traveling waves in reaction diffusion systems with skew gradient structure.For this type of system,since the operator linearized along the traveling wave is not self-adjoint and lack of Hamiltonian structure,so the first part of the processing technique can't be ap-plied directly,and to overcome that we first give a suitable,symplecticspace in order to obtain the stable subspace and unstable subspace are both Lagrangian subspace with respect to this symplectic inner product,so a Maslov index of the traveling wave is well-defined,and we provide the relation between the unstable eigenvalues of the traveling wave and Maslov index.As applications,we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equation.