Global Bifurcation for the Whitham Equation

Abstract

Abstract. We prove the existence of a global bifurcation branch of 2π-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a 'highest', cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.

On Whitham and Related Equations

“…We will restrict in this paper to the Whitham equation posed on the real line. Many interesting aspects of the periodic problem, in particular, concerning periodic solitary waves, can be found in , , , , and . However, our results concerning the Cauchy problem and its link with the KdV one are straightforwardly valid in the periodic case.…”

“…Assume that s > 21 16 . Then, for every u 0 ∈ H s (R), there exists a positive time T = T ( u 0 H s ) = 0(1/ √ ) (which can be chosen as a nonincreasing function of its argument), and a unique solution u to (5) satisfying u(·, 0) = u 0 such that…”

“…We will focus here on nonperiodic solitary wave solutions of the Whitham equation (1), which are solutions of the form u(x − ct). We refer to [44], [5], [45], [6], and [8] for interesting (theoretical and numerical) studies on periodic traveling waves. In particular, the existence of a global bifurcation branch of 2π -periodic smooth traveling-wave solutions is established in [45].…”

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

“…Moreover, note that because φ is even, so is Kφ, where again K is defined as in (3) . It follows we can approximate Kφ(x) similarly by truncating cosine series expansion and again approximating coefficients with (14):…”

“…In fact, we believe the profiles along the global bifurcation branches will be smooth for arbitrarily large waveheight. To understand this, observe that differentiating the profile equation (37) and rearranging yields ψ = c 2 (1 + ψ) 3 Kψ . Figure 15.…”