We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of 2π-periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x| 1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the Hölder and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities.
We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves.
We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on R with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions. variety of applications including the continuum limit of discrete models with long range interaction [39], dislocation dynamics in crystals [12], mathematical biology [44], water wave dynamics [33], and financial mathematics [14]; see also [10] for a recent discussion on applications. For such α, the nonlocal fNLS (1.1) has been introduced by Laskin [40] in the context of fractional quantum mechanics, in which one generalizes the standard Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. A rigorous derivation in the context of charge transport in bio polymers (like DNA), can be found in [39]. Finally, we point out the case α = 1 may be thought to describe the relativistic dispersion relation ω(ξ) = |ξ| 2 + m 2 , an observation recently utilized in the mathematical description of Boson-stars; see [19].Throughout our analysis, we will be concerned with solutions of the formwhere ω, c ∈ R are parameters and φ is a bounded solution to the (generally) nonlocal profile equationwhere denotes differentiation with respect to the spatial variable. When c = 0, the focusing fNLS is well-known to admit standing solitary waves that are asymptotic to zero at spatial infinity. Among such solitary wave solutions, specific attention is often paid to the positive, radially symmetric solutions typically referred to as "ground states". The stability of such ground states dates back to the work of Cazenave and Lions [13] and Weinstein [51, 52] on the classical case α = 2, where the authors use the method of concentration compactness along with the construction of appropriate Lyapunov functionals. For α ∈ (1, 2], such ground states are known to be orbitally stable provided the nonlinearity is energy sub-critical, i.e. if 0 < σ < α; see [27,52], for example. While no nontrivial localized solutions exist in the defocusing case γ = −1, it is known in the classical case α = 2 to admit so-called black solitons of the form (1.2) corresponding to monotone front-like solutions asymptotic to constants as x → ±∞. The dynamics and stability of black solitons has been studied in numerous works; see, for example, [4,23,24]. For the fractional case, the authors plan to report on the existence, nondegeneracy, and stability of black solitons in the defocusing case in a future work. The stabilit...
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