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 tends to infinity. 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 tophat, 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 tophat 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 freesurface flow over localised topography', J. Fluid Mech., 832, 7396) is extended to describe the behaviour on any given solution branch using a method for glueing homoclinic connections.
Original language  English 

Pages (fromto)  532–561 
Number of pages  30 
Journal  Nonlinearity 
Volume  34 
DOIs  
Publication status  Published  21 Jan 2021 
Keywords
 Branch termination
 Homoclinic glueing
 Nonlinear ordinary differential equation
Profiles

Mark Blyth
 School of Mathematics  Reader
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research