Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2\pi$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.Comment: 25 page

Symmetry of periodic waves for nonlocal dispersive equations

Of concern is the $\textit{a priori}$ symmetry of traveling wave solutions for a general class of nonlocal dispersive equations $$u_t + (u^2 + Lu)_x = 0,$$ where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols m guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formulate a $\textit{reflection criterion}$, which allows to affirm the symmetry of periodic traveling waves. The reflection criterion weakens the assumption of monotonicity between trough and crest and enables to treat $\textit{a priori}$ solutions with multiple crests of different sizes per period. Moreover, our result not only applies to smooth solutions, but also to traveling waves with a non-smooth structure such as peaks or cusps at a crest. The proof relies on a so-called $\textit{touching lemma}$, which is related to a strong maximum principle for elliptic operators, and a weak form of the celebrated $\textit{method of moving planes}$

Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations

We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves

On a thin film model with insoluble surfactant

This paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational, capillary, and van der Waals forces. We prove the existence of global weak solutions for medium sized initial data in large function spaces. Moreover, exponential decay towards the flat equilibrium state is established, where an estimate on the decay rate can be computed explicitly

On the thin film Muskat and the thin film Stokes equations

The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by explicit constants depending on the physical parameters of the system. Eventually, it is shown that if the initial datum satisfies additional (low order) Sobolev regularity, we can propagate Sobolev regularity for the corresponding solution. The proofs are based on a priori energy estimates in Wiener and Sobolev spaces

Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order -r, where r > 1. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol m(k) = k$^{-2}$. Thereby we recover its unique highest 2π-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest

Traveling waves for a quasilinear wave equation

We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions V and Γ. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coefficient Γ we distinguish between two cases: (a) Γ ∈ L∞ being regular and (b) Γ = γδ0 being a multiple of the delta potential at zero. For both cases we use bifurcation theory to prove the existence of nontrivial smallamplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidized by explicit examples for the coefficients V in front of the (linear) second time derivative for which our results hold