Schedule

On two new constructions of solitary waves of the nonlinear and nonlocally dispersive Whitham equation

Some of you have already heard me talk about solitary waves in the Whitham equation. In this talk I give some details, in particular about the Orlicz-space construction. Full abstract below.

Solitary waves in dispersive and water wave equations are often constructed using either constrained minimisation or pertubative techniques around a trivial flow. In both cases, the resulting waves are typically small, because of nonlinear control. We present here two new proofs for existence of solitary waves in the nonlinear and nonlocal evolution equation

\[ 

u_t + L u_x + u u_x = 0, \qquad \mathcal{F} (Lu)(\xi) = \left( \frac{\tanh(\xi)}{\xi} \right)^{1/2} \mathcal{F} u(\xi),

\]

also called the Whitham equation. The first proof is based on a priori estimates of periodic waves of all heights, and uses a limiting argument in the periodic to obtain a family of solitary waves up to the highest wave. The second uses a maximisation technique perhaps not earlier used in the water wave setting, where the dispersive part of the energy functional is maxmimised whereas remaining terms are held as a constraint in an Orlicz space constructed directly for this purpose. That is in many respects an L^p-based maximisation technique. We find in the second work small and intermediate-sized waves, although not necessarily a highest solitary wave.

The first work is joint with K. Nik and C. Walker; the second with A. Stefanov and M. N. Arnesen.

Lump solutions for the fKP-I equation

 In this talk, I will present a joint result with H. Borluk (Istanbul) and D. Nilsson (Lund) on the existence and decay of fully localised traveling waves for the fractional KP-I equation.

Existence is established via variational methods in the energy subcritical regime. The decay is studied in the spirit of the classical paper by Bona & Li from 1997, where the steady equation is rewritten as a convolution equation so that the decay of the solution can be deduced from the decay properties of the convolution kernel. We show that irrespective of the strength of dispersion in the fractional KP-I equation, the decay is quadratic.

Existence of solitary waves in equations with nonlocal nonlinearities

This talk concerns the existence of solitary-wave solutions to a class of nonlinear, dispersive evolution equations with Coifman-Meyer nonlinearities. These equations generalize a class of unidirectional, nonlinear wave equations for which the existence of solitary waves has been extensively studied, but where the nonlinearity is a local function. Inspired by several models where nonlinear frequency interaction appears, we extend the theory to allow for nonlocal nonlinearities in the form of bilinear Fourier multipliers. We establish the existence of smooth solitary waves when the linear multiplier is of positive and slightly higher order than the operator on the nonlinear term. The proof is based on Weinstein’s argument for $L^2$-constrained minimization using Lions’ method of concentration-compactness.

Transverse instability of line periodic waves to the KP-I equation

 The passage from linear instability to nonlinear instability has been shown for 1D solitary waves under 2D perturbations. Although transverse instability of periodic waves to the KdV equation under the KP-I flow has been expected to be true from spectral instability for a long time, it has not been clear how to adapt the general instability theory for solitary waves to periodic waves until now. In this talk, we present how such an adaptation works with the aid of exponential trichotomies and multivariable Puiseux series.

Joint work with E. Wahlén.

Deep- and shallow-water limits of statistical equilibria for the intermediate long wave equation

The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between the deep-water regime (= the Benjamin-Ono (BO) regime) and the shallow-water regime (= the KdV regime).

Exploiting the complete integrability of ILW, I will discuss the statistical convergence of ILW to both BO and KdV, namely, the convergence of the higher order conservation laws for ILW and their associated invariant measures. In particular, as KdV possesses only half as many conservation laws as ILW and BO, we observe a novel 2-to-1 collapse of ILW conservation laws to those of KdV, which yields alternating modes of convergence for the associated measures in the shallow-water regime.

This talk is based on joint work with Guopeng Li (Univ. Edinburgh) and Tadahiro Oh (Univ. Edinburgh).

Vortex-carrying solitary gravity waves of large amplitude

 In this talk, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family C of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on C – including those that are large amplitude or even overhanging – the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air.

Invariant measures for mKdV and KdV on the line

I will discuss a recent result on the invariance of the Gibbs measure on the line for the real-valued defocusing modified Korteweg de-Vries equation (mKdV). Applying the Miura transform, we obtain a new invariant, non-Gaussian probability measure for KdV on the line at the same local regularity as the white noise. This talk is based on joint work with R. Killip and M. Visan (UCLA).

Asymmetric Solutions to the Capillary-Gravity Whitham Equation

The Whitham equation has been a rich shallow water wave model for gravity waves. In the presence of surface tension, the natural generalisation of this water wave model is the capillary-gravity Whitham equation. Here we focus on periodic travelling waves that solve the capillary-gravity Whitham equation. These have been fully characterized in the case of small and even waves. This characterization is complemented by the work presented in this talk dealing with small asymmetric periodic travelling waves. Such asymmetric waves are far more scarce than the even ones and can only be constructed in certain cases for weak surface tension.

Convergence of Mechanical Balance laws for Water Waves: From KdV to Euler

The Korteweg-de Vries (KdV) equation is as an approximate model of long waves of small amplitude at the free surface with inviscid fluid. In this presentation we will demonstrate that the mechanical balance quantities (mass, momentum and energy), as defined by the solution of the KdV equation, rigorously approximate those in the Euler system within the $L^{\infty}$ space. Furthermore, these approximations are estimated in relation to the parameter  characterizing the long-wave behavior.

Strongly interacting solitary waves for the fractional modified Korteweg-de Vries equation

 The solitons (or solitary waves if the equation is non-integrable) are a non-linear phenomenon observed by John Scott Russell in 1834 in the context of shallow water. The solitons play a main role in the asymptotic time behavior of the solution of some non-linear dispersive equations. It was numerically observed in the 60s that any nice solution of the Fermi, Pasta, Tsingou, and Ulam problem decomposes in some of the soliton, called multi-soliton, plus a residual term. To understand this decomposition, the first step is the construction of solitary waves and multi-solitary waves. We will talk about the properties of the solitary waves and the construction of a dipole (two solitary waves moving at the same speed) for the fractional modified Korteweg-de Vries equation. We will explain the Martel-Merle method and how this method has been adapted in the non-local context for the construction of dipole. This talk is based on a joint work with F. Valet.  

Study of a generalized Dirichlet-Neumann operator with application to doubly periodic waves on Beltrami flows

A Beltrami flow is a flow in which the velocity field and the vorticity are collinear, that is $\curl \bfu =\alpha\bfu$ for some constant $\alpha$. Recently Groves and Horn (2020) showed that the equations governing Beltrami flows can be written in terms of a generalized Dirichlet-Neumann operator, which extends the classical Dirichlet-Neumann operator for irrotational flow. In this talk I will discuss some recent analytical results for this operator and describe how these can be used to prove existence of doubly periodic waves over a Beltrami flow.

This talk is based on a joint work with Mark Groves, Stefano Pasquali and Erik Wahlén. 

A Direct Construction of Solitary Waves for a Fractional Korteweg-De Vries Equation with an Inhomogeneous Symbol

 For the direct construction of solitary waves for a fractional Korteweg-de Vries equation, we parameterize the known periodic solution curves through the relative wave height and then use a priori estimates for these solutions. Such periodic waves converge locally uniformly to waves with negative tails, which are transformed into the desired branch of solutions. The obtained branch evolves from the zero solution and reaches the highest wave through a unique path in the wave speed–wave height space. The behavior of the highest wave near its crust depends on the value of s (s>0) and reflects that of the corresponding periodic waves.

Pattern Formation and Film Rupture in an Asymptotic Model of the Bénard--Marangoni Problem

Thin fluid films on heated planes exhibit the formation of spatially periodic structures. These can take the form of regular polygonal pattern, which was experimentally observed by Henri Bénard in 1900, or film rupture leading to dewetting phenomena. The emergence of these patterns is caused by the thermocapillary effect and the mathematical problem is known as the Bénard--Marangoni problem.

In this talk, I will derive a deformational asymptotic model for the Bénard–Marangoni problem in the thin-film limit. In this model, the state of constant film height destabilises via a (conserved) long-wave instability and periodic solutions bifurcate via a subcritical pitchfork bifurcation. I will demonstrate that the bifurcation curve can be extended to a global bifurcation branch. Furthermore, periodic film-rupture solutions can be constructed as limit points of the bifurcation branch.

The talk is based on joint work with Stefano Böhmer, Gabriele Brüll (both Lund) and Bastian Hilder (TU Munich).

Rigorous justification of the Benjamin-Ono equation as an internal wave model

The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the unidirectional propagation of weakly nonlinear, long internal waves in a two-layer fluid. The equation was introduced formally by Benjamin in the '60s and has been a source of active research since. For instance, the study of the long-time behavior of solutions, stability of traveling waves, and the low regularity well-posedness of the initial value problem. However, despite the rich theory for the BO equation, it was an open question whether its solutions are close to the ones of the original physical system.

In this talk, I will explain the main steps involved in the rigorous derivation of the BO equation.