Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Abstract

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t = 0.We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our J Nonlinear Sci (2010) 20: 277-308 numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

Continuation of periodic solutions in the waveguide array mode-locked laser

“…Recently [26][27][28], Wilkening and Ambrose introduced an efficient method of computing timeperiodic solutions of nonlinear PDEs. We will refer to this method as the Adjoint Continuation Method (ACM).…”

“…Following the basic approach in [26][27][28], we define a nonlinear functional G of the initial conditions and supposed period that is zero if and only if the solution is z-periodic. As we wish to determine the dependence of G on the parameter g 0 in (2), we add to (1) the equation ∂g 0 /∂z = 0.…”

“…One of these variables (usually g 0 or Z) is taken as a bifurcation parameter in the continuation algorithm and removed from the list of variables that BFGS is allowed to vary in search of a minimum. Alternatively, a penalty function [26] can be used to enforce the value of the bifurcation parameter. In either approach, it is useful to be able to use any variable as a bifurcation 9…”

“…Store the solution, zderivative, and Hermite correction data [26,27] Evolve the adjoint equation until ζ = Z [34,35]. Use cubic Hermite interpolation to approximate X(z) [26,27] Compute G and ∇G from X(Z), X z (Z), X(Z) and X(0) [26,27] BFGS update step [38] Update initial condition as specified by BFGS [38] Has BFGS converged?…”

Cyclic steady states of nonlinear electro-mechanical devices excited at resonance

“…In [6], the authors consider the evolution of treaded rolling bodies. Other works on Couette turbulence [7], integrable equations [8], mode-locked lasers [9], corresponding linear isotropic permittivities R , V , and W . In such a case, and in the absence of any volume charge, one requires…”