Termination points and homoclinic glueing for a class of inhomogeneous nonlinear ordinary differential equations

TL;DR

This paper investigates the solution structure of a class of inhomogeneous nonlinear ODEs with decaying forcing functions, revealing infinite solution branches, their termination behavior, and extending asymptotic analysis using homoclinic glueing techniques.

Contribution

It provides a detailed analysis of solution branches for inhomogeneous nonlinear ODEs with various decay forcings, including exact solutions, numerical computations, and asymptotic descriptions, especially for Lorentzian forcing.

Findings

Solution branches exist for different forcing types.

Branches terminate at discrete parameter values for top-hat and Gaussian.

Asymptotic analysis extended using homoclinic glueing methods.

Abstract

Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinary differential equations taking the form \[u'' + u^2 = \alpha f(x) \] for parameter $\alpha$ are studied. The problem is defined on the $x$ line with decay of both the solution $u(x)$ and the imposed forcing $f(x)$ as $|x| \to \infty $. The rate of decay of $f(x)$ is important and has a strong influence on the structure of the solution space. Three particular forcings are examined primarily: a rectilinear top-hat, a Gaussian, and a Lorentzian, the latter two exhibiting exponential and algebraic decay, respectively, for large $x$. The problem for the top hat can be solved exactly, but for the Gaussian and the Lorentzian it must be computed numerically in general. Calculations suggest that an infinite number of solution branches exist in each case. For the top-hat and the Gaussian the solution branches terminate at a discrete set of $\alpha$ values starting from zero. A general asymptotic description of the solutions near to a termination point is constructed that also provides information on the existence of local fold behaviour. The solution branches for the Lorentzian forcing do not terminate in general. For large $\alpha$ the asymptotic analysis of Keeler, Binder $\&$ Blyth (2018 "On the critical free-surface flow over localised topography", J. Fluid Mech., 832, 73-96) is extended to describe the behaviour on any given solution branch using a method for glueing homoclinic connections.