Special Session 56: Dynamical properties of nonlinear partial differential equations

Steady pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the B\`enard--Marangoni problem

Stefano Boehmer Lund University Sweden Co-Author(s):    Bastian Hilder, Jonas Jansen

Abstract: It is experimentally known that thin films of viscous fluids on heated plates develop polygonal, spatially periodic patterns. This is due to a self-sustaining thermocapillary effect causing an instability of the trivial constant state. Building upon a previous work on the one-dimensional case, we consider a two-dimensional thin-film equation. It can be formally derived from the B\`enard--Marangoni problem via a long-wave approximation. We consider the stationary problem, which we are able to reduce to a second-order equation amenable to analytic bifurcation theory. The constant solution destabilizes via a (conserved) long-wave instability and we prove existence of a global bifurcation branch of stationary solutions of fixed mass, which are symmetric and periodic with respect to a fixed square or hexagonal lattice. We finally analyse qualitative aspects of the solutions on the branch both analytically and numerically. Most importantly, we prove that solutions exhibit film rupture, that is, their minimal height tends to zero, under the condition that the Marangoni number is uniformly bounded on the bifurcation branch. This conditional result is substantiated by numerical experiments. This is joint work with Bastian Hilder and Jonas Jansen.

A plethora of fully localised solitary waves for the full-dispersion Kadomtsev-Petviashvili equation

Mats Ehrnstrom NTNU Norwegian University of Science and Technology Norway Co-Author(s):    M. D. Groves, Saarland University

Abstract: The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number \(\beta > 1/3\) also called strong surface tension.This equation has recently been shown to have a family of nondegenerate, symmetric fully localised solitary waves which decay to zero in all spatial directions.The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper we show that the FDKP-I equation also has a family of symmetric fullly localised solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

Convective Stability of Front Superpositions: The Role of Unstable Connecting States

Bastian Hilder Technical University of Munich Germany Co-Author(s):    Louis Gar\\`{e}naux

Abstract: In pattern-forming systems, an unstable `trivial' state is often first invaded by stripe patterns, which are themselves unstable to hexagonal patterns that emerge through a secondary (slower) invasion front. Such cascading invasions naturally give rise to a non-steady (i.e., the fronts travel at different speeds) two-front superposition in which the connecting state is unstable. In this talk, I present recent results on the linear convective stability of such a superposition of two travelling waves with unstable connecting state in a reaction-diffusion system. We find that convective stability holds for a strictly smaller range of propagation speeds than for the corresponding single-front waves, reflecting the long-range influence the fronts exert on one another. Since the superposition is time-dependent, classical techniques are not applicable to analyse this phenomenon. We therefore rely on numerical range estimates that imply time-uniform resolvent bounds. I will also discuss open challenges in the nonlinear analysis. This is joint work with Louis Gar\`{e}naux (INRIA Saclay).

Full Benjamin-Feir instability of capillary-gravity Stokes waves

Ting-Yang Hsiao SISSA Taiwan Co-Author(s):    Alberto Maspero

Abstract: We study the modulational (Benjamin-Feir) instability of small-amplitude periodic Stokes waves for the two-dimensional gravity-capillary water waves equations. We analyze the spectral stability of such waves under long-wave longitudinal perturbations. The linearized operator exhibits a defective zero eigenvalue of multiplicity four due to the symmetries of the system. Using Bloch-Floquet theory, we investigate the associated family of periodic spectral problems and obtain a complete description of the eigenvalues near the origin in the small-amplitude regime. We prove that the four eigenvalues undergo a full splitting and rigorously characterize the transition between stability and instability. In the unstable regime, the spectrum near the origin forms the characteristic Figure 8 pattern associated with Benjamin-Feir instability, while in the stable regime it remains purely imaginary.

Capillary drops with constant vorticity

Giuseppe La Scala Scuola Superiore Meridionale Italy Co-Author(s):    Pietro Baldi, Domenico Angelo La Manna

Abstract: We consider the problem of capillary liquid 2D and 3D capillary drops with the presence of constant vorticity. In the 2D case, we show that the problem is well-posed, the rotating circle solution is energetically stable and about it, small oscillations like rotating waves are produced. In the 3D case, we show that if capillary effects are stronger than vorticity ones and the equatorial section of the drop is strictly convex, then the drop is a surface of revolution.

The Fourier spectral approach to the spatial discretization of quasilinear hyperbolic systems

JOHANNA MARSTRANDER NTNU - Norwegian University of Science and Technology Norway Co-Author(s):    Vincent Duch\^{e}ne

Abstract: We will discuss the rigorous justification of the numerical spatial discretization by means of Fourier spectral methods of equations modelling the dynamical evolution of surface gravity waves. The Fourier spectral method is especially indicated for systems such as the Boussinesq and Whitham-Boussinesq systems since the nonlinear contributions are quadratic and dispersive contributions take the form of Fourier multipliers. The spatial discretization amounts to considering the modified PDE with low-pass filters for wave numbers $\left|k \right| \leq N$ applied to the initial data and nonlinear terms. Earlier convergence results for the discretization of Boussinesq systems have the shortcoming that they lack uniformity in the non-dispersive limit. Overcoming this requires a good understanding of the underlying (non-dispersive) quasilinear system, which will be the focus of the talk. Using energy estimates, we investigate the stability and convergence of the approximate solution as $N\to \infty$. The results depend on the regularity of the initial data and the structure of the system. We consider both sharp- and smooth low-pass filters and argue that - at least theoretically - smooth low-pass filters are operable on a larger class of systems.

Decay structure of an inviscid non-equilibrium radiation hydrodynamics system

Ramon G. Plaza Universidad Nacional Autonoma de Mexico Mexico Co-Author(s):    Corrado Lattanzio, Jose M. Valdovinos

Abstract: In this talk I will examine the diffusion approximation, non-equilibrium model of radiation hydrodynamics derived by Buet and Despr\`{e}s (J. Quant. Spectrosc. Radiat. Transf. 85, 2004). The latter describes a non-relativistic inviscid fluid subject to a radiative field under the non-equilibrium hypothesis, that is, when the temperature of the fluid is different from the radiation temperature. It is shown that local solutions exist for the general system in several space dimensions. It is also proved that only the one-dimensional model is genuinely coupled in the sense of Kawashima and Shizuta. A notion of entropy function for non-conservative parabolic balance laws is also introduced. It is shown that the entropy identified by Buet and Despr\`{e}s is an entropy function for the system in the latter sense. This entropy is used to recast the one-dimensional system in terms of a new set of perturbation variables and to symmetrize it. With the aid of genuine coupling and symmetrization, decay rates for solutions to the one dimensional linearized problem are obtained; these estimates, combined with the local existence result, yield the global existence and decay in time of perturbations of constant equilibrium states in one space dimension. This is joint work with Corrado Lattanzio (L`Aquila) and Jose M. Valdovinos (Toulouse).

Rediscovering shallow-water equations from experimental data

Douglas Svensson Seth Norwegian University of Science and Technology Sweden Co-Author(s):    Kjell Heinrich, Mats Ehrnstrom, Simen Ellingsen

Abstract: While data-driven methods have advanced the discovery of governing equations, extracting robust partial differential equations from real-world data remains challenging. Here we present a Fourier-based approach to rediscover a shallow water-equation akin to the Korteweg-De Vries (KdV) equation using only video footage of solitons. The Fourier-multiplier technique is also compared to another distinct method, weak-form sparse identification of nonlinear dynamics (WSINDy), which independently recover the same PDE, confirming its inherent structure in the data. We validate the discovered equation by solving it forward and comparing it to unseen experimental cases.

Three-dimensional doubly periodic gravity water waves on Beltrami flows

Erik Wahl\`en Lund University Sweden Co-Author(s):    Mark Groves, Dag Nilsson, Stefano Pasquali

Abstract: I will present an existence result for small-amplitude doubly periodic travelling gravity water waves on Beltrami flows, a class of rotational flows in which the velocity field is parallel to the vorticity. This extends the existence theory by Iooss and Plotnikov in the irrotational setting. The problem is technically challenging due to the appearance of small divisors. We overcome this by using a Nash--Moser theorem. I will highlight the key differences and new issues that appear in the Beltrami setting.

Mixed type problem and transonic flows

Dehua Wang University of Pittsburgh USA Co-Author(s):

Abstract: In this talk, we will consider the Euler equations of gas dynamics and applications in transonic flows. We will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles.

Mountain waves: Linear theory

Jörg Weber University of Vienna Austria Co-Author(s):

Abstract: In this talk, I will discuss the linear theory of mountain waves in two dimensions. After an introduction to the physical relevance and the typically observed wave patterns, we turn to the underlying boundary value problem for the compressible Euler equations coupled to the ideal gas law and the first law of thermodynamics. In particular, we are interested in their linearization at a background state, corresponding to an incoming horizontal wind profile. We show how the linearized equations can be reduced to a Helmholtz-like equation (the Scorer equation, well-known in the applied literature) for the vertical velocity on the upper-half plane. We then present a solution theory for the corresponding boundary value problem. Here, we have to pay special attention to a careful implementation of a physically correct radiation condition that is fundamentally different to typical radiation conditions \`a la Sommerfeld, which are relevant in the context of electromagnetic and acoustic waves, but physically incorrect for mountain waves. The talk is based on joint work with Adrian Constantin (U Vienna).

Statistics of long-crested crossing random waves over a varying bottom

Zibo Zheng Okinawa Institute of Science and Technology Japan Co-Author(s):    Zibo Zheng, Wooyoung Choi, Amin Chabchoub

Abstract: The evolution of a unidirectional wave train on varying bathymetry is well documented while the statistics of crossing seas over uneven bottoms remain poorly understood. This work investigates the spatial evolution of skewness and kurtosis for weakly nonlinear random waves propagating over a slowly varying bottom. To model this complex process, we develop a system of coupled nonlinear Schr\"odigner equations that accounts for the interaction between two obliquely propagating wave trains and variation of water depth. We perform comprehensive numerical simulations using split-step Fourier method and initialize the representative wave field using the JONSWAP spectrum. By examining how the incident angle and bathymetry alter wave statistics, we ultimately demonstrate their combined influence on freak wave generation.