Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Dec 2009
Heteroclinic Solutions To An Asymptotically Autonomous Second-Order Equation, Gregory S. Spradlin
Gregory S. Spradlin
We study the differential equation ¨x(t) = a(t)V' (x(t)), where V is a double-well potential with minima at x = ±1 and a(t) → l > 0 as |t| → ∞. It is proven that under certain additional assumptions on a, there exists a heteroclinic solution x to the differential equation with x(t) → −1 as t → −∞ and x(t) → 1 as t → ∞. The assumptions allow l − a(t) to change sign for arbitrarily large values of |t|, and do not restrict the decay rate of |l −a(t)| as |t| → ∞.
Interacting Near-Solutions To A Hamiltonian System, Gregory S. Spradlin
Mar 2004
Interacting Near-Solutions To A Hamiltonian System, Gregory S. Spradlin
Gregory S. Spradlin
A Hamiltonian system with a superquadratic potential is examined. The system is asymptotic to an autonomous system. The difference between the Hamiltonian system and the “problem at infinity,” the autonomous system, may be large, but decays exponientially. The existence of a nontrivial solution homoclinic to zero is proven. Many results of this type rely on a monotonicity condition on the nonlinearity, not assumed here, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. The proof employs variational, minimax arguments. In some similar results requiring the monotonicity condition, solutions inhabit a manifold homeomorphic to the …
Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin
Dec 2003
Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin
Gregory S. Spradlin
We study a Hamiltonian system that has a superquadratic potential and is asymptotic to an autonomous system. In particular, we show the existence of a nontrivial solution homoclinic to zero. Many results of this type rely on a convexity condition on the nonlinearity, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. This paper replaces that condition with a different condition, which is automatically satisfied when the autonomous system is radially symmetric. Our proof employs variational and mountain-pass arguments. In some similar results requiring the convexity condition, solutions inhabit a submanifold homeomorphic to the …
A Perturbation Of A Periodic Hamiltonian System, Gregory S. Spradlin
Nov 1999
A Perturbation Of A Periodic Hamiltonian System, Gregory S. Spradlin
Gregory S. Spradlin
No abstract provided.
A Hamiltonian System With An Even Term, Gregory S. Spradlin
Aug 1997
A Hamiltonian System With An Even Term, Gregory S. Spradlin
Gregory S. Spradlin
No abstract provided.