Weierstrass Traveling Wave Solutions For Dissipative Benjamin, Bona, And Mahony (Bbm) Equation, Stefan C. Mancas, Greg Spradlin, Harihar Khanal
Jul 2013
Weierstrass Traveling Wave Solutions For Dissipative Benjamin, Bona, And Mahony (Bbm) Equation, Stefan C. Mancas, Greg Spradlin, Harihar Khanal
Gregory S. Spradlin
In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified Benjamin, Bona, and Mahony (BBM) equation by viscosity. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant makes the equation integrable in terms of Weierstrass ℘ functions. We will use a general formalism based on Ince's transformation to write …
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| → ∞.
Lanchester's Equations In Three Dimensions, Christina Spradlin, Greg Spradlin
Mar 2007
Lanchester's Equations In Three Dimensions, Christina Spradlin, Greg Spradlin
Gregory S. Spradlin
This paper generalizes Lanchester's equations of warfare to partial differential equations involving time and two spatial variables. Unlike in Lanchester's original ordinary differential equations, the distribution of armies over the battlefield must be considered. Four different modes of attack are introduced, generalizing Lanchester's equations for area fire and for direct fire. The effect of the distribution of forces and their movement on the outcome is considered, and numerical simulations given.
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 …
An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin
Dec 1999
An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin
Gregory S. Spradlin
We consider a singularly perturbed elliptic PDE that arises in the study of nonlinear Schrodinger equations. We seek solutions that are positive on the entirety of Euclidean space and that vanish at infinity. Under the assumption that the nonlinear term of the PDE satisfies super-linear and sub-critical growth conditions, we show that for small values of the epsilon parameter in the PDE, there solutions that concentrate near local minima of V (a coefficient function in the PDE) . The local minima may occur in unbounded components, as long as the Laplacian of V achieves a strict local minimum along such …
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.
