Special Session 125: Models of Fluid Motion

The spectrum of steady, laser-driven convective flows

Benjamin Akers The Air Force Institute of Technology USA Co-Author(s):    Benjamin Akers, Jeremiah Lane, Wesley Coonradt

Abstract: It has recently been shown that continuous wave lasers can generate steady, parametrically-analytic, laser-driven convective fluid flows (Lane, Fluids 2021), (Lane Stud. App. Math. 2024). We investigate the stability of these flows, by calculating their spectral data. Steady internally forced convection is contrasted to classic Rayleigh-Bernard convection. The spectrum is shown to be parametrically analytic at simple eigenvalues. The spectral data of large amplitude flows are computed, and dynamics near unstable flows are presented.

From explicit formulas to Wave Kinetic theory for the Benjamin-Ono equation

Yvonne Alama Bronsard Massachusetts Institute of Technology USA Co-Author(s):

Abstract: In this talk, we begin by introducing the explicit solution formula for the Benjamin-Ono (BO) equation obtained by Patrick Gerard. Building on this representation, we present an overview of recent numerical schemes, which provide new tools for investigating the long-time dynamics of BO. In the setting of randomized initial data, we then explain how iterating this explicit formula, combined with non-commutative Khintchine's inequality, yields new insights into the Wave Kinetic theory for BO, up to the physically relevant kinetic time scale. The former results are joint work with Xi Chen (Basel), Matthieu Dolbeault (Nantes), Thierry Laurens (Wisconsin-Madison), and Ola Maehlen (Paris-Saclay), while the latter is a joint work with Gigliola Staffilani (MIT) and Felipe Hernandez (Penn State).

Analytical frameworks for spatially quasiperiodic or uniformly local free-surface flows

David M. Ambrose Drexel University USA Co-Author(s):

Abstract: We will present methods of existence theory for free-surface fluid flows in the case of uniformly local Sobolev data and in the case of spatially quasiperiodic initial data. These are both instances of non-decaying, non-periodic fluid flows. Such data could arise, for instance, when making a non-periodic perturbation of a periodic flow, or when wavetrains with different, non-commensurate periodicities interact. Among the topics to be discussed are a version of the Birkhoff-Rott integral and Hilbert transform which do not require decay or periodicity, and estimates for these in uniformly local Sobolev spaces. Another potential framework to be applied is the Cauchy-Kowalevski theorem, and some interesting consequences of this in the spatially quasiperiodic setting will be discussed.

Modeling Broadband Water-Wave Experiments with Generalizations of NLS

John Carter Seattle University USA Co-Author(s):    Joanna Van Liew and Diane Henderson

Abstract: Water waves generated by storms gradually disperse, forming swells that can travel thousands of kilometers across the ocean`s surface. The evolution of such swells on deep water can be modeled by the nonlinear Schrodinger (NLS) equation and its generalizations. However, most of these models assume a narrow range of wave frequencies and do not account for any form of energy loss. We compare predictions from new NLS generalizations that include dissipation and allow for broad ranges of frequencies with measurements from new laboratory experiments in which the frequency broadbandedness was varied.

Following Information Flow in the Majda-McLaughlin-Tabak Model

Christopher Curtis San Diego State University USA Co-Author(s):    Erik Bollt

Abstract: In this work, we quantify the timescales and magnitude of information flow associated with multiscale energy transfer in the Majda-McLaughlin-Tabak model. As we show, both forward and inverse cascade features are detected. However, our method allows for a more intricate picture to emerge which quantifies how wave mixing allows for widely disparate scales to be causally linked. This should be of use in a range of similar problems, and thus we present a method that, beyond canonical averaging techniques, provides a more detailed characterization of chaotic multiscale flows.

The Benjamin-Feir instability in KdV-like equations

Bernard Deconinck University of Washington USA Co-Author(s):    Bhavna Kaushik

Abstract: Nonlinear waves in dispersive media can be succeptible to modulational insta- bilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For small-amplitude traveling wave solutions, we provide a complete characterization of the spectrum near the origin of the linear operator obtained from linearizing about periodic traveling waves. We prove rig- orously that, when the modulational instability is present, the spectrum connected to the origin consists of curves that invariably form a closed figure-eight pattern.

Spectral Evidence of Integrability Breakdown in Shallow-Water Multisoliton Fission: A pIST Benchmark of Wave Models

Jose Galaz Pontificia Universidad Catolica de Chile Chile Co-Author(s):    Rodrigo Cienfuegos, Damyan Santander

Abstract: We study the nonlinear Fourier spectrum of four shallow-water wave models --KdV, Serre-Green-Naghdi (SGN), Madsen-Sorensen, and Whitham-- using the periodic inverse scattering transform (pIST) of KdV as a spectral diagnostic, justified by a piecewise KdV-coupling approximation argument. Applying this framework to numerical simulations and laboratory data of multisoliton fission from Trillo et al. (Phys. Rev. Lett., 2016), we find that SGN best reproduces the KdV-theoretical soliton-spectrum amplitudes within 3-7% RMSE, yet all models exhibit substantially larger and increasingly variable errors against laboratory data, with up to 30-60% RMSE at the highest Ursell (Ur) numbers (84--26,000). This growing model-lab discrepancy is accompanied by a spatial drift of pIST soliton amplitudes along the channel, which would remain constant under perfect KdV integrability, providing direct spectral evidence of integrability breakdown driven by dissipation. We propose a physical conjecture for the dominant dissipation mechanism, supported by fitted power-law decay of pIST soliton amplitudes along the channel. These findings establish a spectral benchmark for shallow-water models and demonstrate that dissipation closures accounting for subtle energy losses are essential to capture multisoliton fission dynamics at high Ur.

Numerical Study of Boussinesq systems

Christian Klein University Burgundy Europe/IMB France Co-Author(s):    Vincent Duchenes, Theo Gaudry, Sergey Gavrilyuk, Jean-Claude Saut

Abstract: We present a numerical study of solutions to equations appearing in the theory of surface waves, concretely Boussinesq systems (integrable and non-integrable examples) and Serre-Green-Naghdi (SGN) equations. Solitary waves in 1D are constructed, and there stability is studied numerically. The time evolution of localised initial data is explored. Of special interest is the role of the non-cavitation condition. The appearence of dispersive shock waves, zones of rapidly modulated oscillations, in the vicinity of shocks of the corresponding dispersionless systems is studied. In the context of the SGN equations, these questions are also addressed in 2D.

Transverse instability of line periodic waves to the KP-I type equations

Wei Lian NTNU Norway Co-Author(s):    Erik Wahlen

Abstract: 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 very recently. In this talk, we present how such an adaptation works with the aid of exponential trichotomies and an approximate scheme. Joint work with Erik Wahlen.

A variational construction of solitary waves for the Babenko equation on finite depth

JOHANNA MARSTRANDER NTNU - Norwegian University of Science and Technology Norway Co-Author(s):    Mats Ehrnstrom, Tomas Dohnal

Abstract: We construct solitary waves for the two-dimensional steady water wave problem directly from the Babenko formulation in the case of pure gravity waves over finite depth. The proof is straight-forward in the sense that it is built on constrained minimization of a scalar nonlinear and nonlocal equation as in Weinstein (1987), and a limiting procedure from periodic waves as Ehrnstr\{o}m-Groves-Wahl\`{e}n (2012), using concentration--compactness. The novelty lies not the least in the nonlocal cubic terms appearing in the Babenko variational formulation, which carry negative order, and make nonlinear harmonic analysis an important part of the proof. The constructed smooth and localised solutions of the Babenko equation correspond to smooth and localised free-surface waves in the two-dimensional steady water wave problem over a flat bed with gravity. We show that the Babenko solitary waves are approximated by scalings of the classical Korteweg--De Vries solitary waves.

Solitary Waves in a Two-Layer Fluid with a Free Surface

Dag Nilsson Linnaeus University Sweden Co-Author(s):    Mark Groves

Abstract: We study two-dimensional internal travelling waves featuring both a free surface and an internal interface, propagating under the combined effects of gravity and surface and interfacial tension. By applying the method of spatial dynamics, the governing equations are reformulated as an infinite-dimensional dynamical system in which the unbounded spatial variable assumes the role of time: \[u_x=Lu+N(u)\] We focus on the special case in which the imaginary part of the spectrum of $L$ consists of the eigenvalues $0$, $\pm\mathrm{i}\kappa$, each with algebraic multiplicity two. A centre manifold reduction is then carried out, leading to a finite-dimensional system of reduced equations. At leading order, this reduced system takes the form of a coupled KdV-NLS system and we discuss the existence of homoclinic solutions of this system.

Mode interactions of surface gravity waves in triangular domains

Panayiotis Panayotaros IIMAS, Universidad Nacional Autonoma de Mexico Mexico Co-Author(s):    Eddaly Guerra Velasco, Boris A. Percino Figueroa, Rosa M. Vargas Maga\~na

Abstract: We present a model of nonlinear mode interactions of 2-D gravity surface water waves in a domain with inclined lateral boundaries, assuming free surface potential flow. We consider an isosceles triangular domain with known linear modes and frequencies, and describe a preliminary analysis of the lowest nonlinear mode using normal forms. We also report on recent progress in the computation of linear modes and frequencies in more general triangular domains.

A new high-order shallow-water modeling approach: variational derivation and numerical simulations

Christos Papoutsellis ECOLE NATIONALE DES PONTS ET CHAUSSEES France Co-Author(s):    Christos Papoutsellis, Michel Benoit

Abstract: We introduce a variational framework that generates a family of shallow-water wave models of increasing order of accuracy, unifying classical asymptotic long-wave theories with nonlocal Hamiltonian formulations. Starting from Luke`s variational principle, we approximate the velocity potential using vertical polynomial ans\atze motivated by the shallow-water structure of the underlying Dirichlet-to-Neumann operator. This construction leads to a hierarchy of models expressed in canonical variables, whose evolution equations preserve the canonical, nonlocal Hamiltonian structure of the full water-wave problem. In the case of uniform depth, we establish the equivalence with the Isobe--Kakinuma model and clarify connections with several classical and modern asymptotic models. We analyze the linear dispersion and stability properties of the resulting systems and validate the framework numerically through simulations of solitary-wave propagation and interactions. Finally, we extend the formulation to variable bathymetry and present results illustrating nonlinear wave--wave and wave--topography interactions.

Analysis of the adhesion model and reconstruction in cosmology

Robert Pego Carnegie Mellon University USA Co-Author(s):    Jian-Guo Liu

Abstract: The adhesion model is a multidimensional model of mass transport in cosmology in a perturbed Einstein-de Sitter universe, which reduces to sticky particle flow in 1D. Velocity is given by a Hopf-Lax formula and concentrates mass on sheets, filaments and points, but it is known that momentum is not conserved in general. Work in the astronomy literature by Brenier, Frisch and collaborators relates the problem of reconstructing the primeval velocity distribution to an optimal mass transport problem. We carry out a mathematical analysis of the initial value problem, establishing a number of basic facts, including the uniqueness of a `sticky` mass flow and a characterization of its absolutely continuous part in terms of a natural Monge-Amp\`ere equation. We find that the singular part is not necessarily determined correctly by optimal transport, and this can result in inexact reconstruction.

(Down)shifting Paradigms in Water Waves: What Processes Drive the Spectral Shifts?

DANIEL J RATLIFF Northumbria University England Co-Author(s):    T.J. Bridges, O. Tritchenko

Abstract: Despite centuries of study, there continue to be many open problems in dynamics of water waves. One such problem is that of frequency downshifting, a phenomenon in water waves where the spectral peak and/or mean of the wave spectra permanently moves downwards over the wave`s evolution. Typically this phenomenon is argued to be primarily driven by dissipative processes such as wavebreaking, with energy loss being the only pathway to permanent spectral movement. However, recent work by the authors have demonstrated that this phenomenon can happen conservatively (i.e. without energy loss) and can be mediated by wave-mean flow interactions. Since both processes are shown to contribute to downshifting, and both are present within water wave problems, this provokes the natural question - how do they interact and contribute to the overall process of downshifting in water waves? This talk will start with an overview of downshifting from a wave-mean flow perspective in order to demonstrate how such a process mediates downshifting, by using a Benney-Roskes system to illustrate the fundamentals of the process. We will then introduce (simple) dissipative effects and explore how this impacts movements in the spectral peak as a function of nondimensional depth. Ultimately we find that deep into the modulationally unstable regime dissipation leads to more monotonic and narrow-banded downshifting than that driven by purely mean flow effects, suggesting that dissipation plays a long-term role in arresting the downshifting phenomena.

Physical-Space Models for Unidirectional Deep-Water Waves

Paivo Simson Tallinn University of Technology Estonia Co-Author(s):

Abstract: We investigate unidirectional reductions of deep-water evolution equations without recourse to Fourier normal variables or complex canonical transformations. By closing the system entirely in terms of surface elevation, we obviate the need for velocity potential initial data, which is a significant advantage for simulating coherent structures such as breathers from experimental observations. The resulting non-local equations retain the exact linear dispersion relation and capture leading-order cubic nonlinearities. We show that both Hamiltonian and non-Hamiltonian models can be consistently formulated within this framework. Notably, lower-accuracy Hamiltonian variants reveal an unexpected integrable structure, while non-Hamiltonian versions achieve remarkable accuracy against fully nonlinear Euler computations up to moderate steepness. All models reduce to the nonlinear Schrodinger equation in the narrow-band limit with the correct deep-water coefficients.

Asymmetric Travelling Capillary-Gravity Waves

Douglas Svensson Seth Norwegian University of Science and Technology Sweden Co-Author(s):

Abstract: Periodic travelling waves that solve the capillary-gravity Whitham equation have been fully characterised in the case of small and even waves. This characterisation 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. The method also generalises in a straightforward way to a class of similar equations for which we either can prove the existence of or non-existence of asymmetric solutions. However, the proof relies on some technical calculations that are different for each equation. We note how this could be done for the Babenko equation, which is equivalent to the full water wave problem, to determine the existence of small amplitude Capillary-Gravity Waves.

Improved existence time for dispersive water waves equations

Nadia Skoglund Taki University of Bergen Norway Co-Author(s):    Benjamin Melinand, Didier Pilod, Sigmund Selberg, and Achenef Tesfahun

Abstract: In this presentation, we will discuss techniques to improve the time of existence for dispersive water waves equations while keeping the small parameters $(\epsilon,\mu)$ decoupled. The proofs rely on the energy method and new Strichartz estimates. These techniques are robust and can be adapted to several dispersive equations and systems. In the two-dimensional case, we exceed the time of existence of order $\epsilon^{-1}$ in the long wave regime, which appears to be new. The talk is based on joint work with Benjamin Melinand, Didier Pilod, Sigmund Selberg, and Achenef Tesfahun.

Soliton Turbulence in Non-integrable Systems

Marcelo V. Flamarion Pontificia Universidad Catolica del Peru Peru Co-Author(s):    Ekaterina Didenkulova and Efim Pelinovsky

Abstract: We present a comparative study of the statistical properties of rarefied soliton gases within both integrable and non-integrable models from the Korteweg-de Vries (KdV) hierarchy. Focusing on multi-soliton solutions of the modified KdV equation and the modular Schamel equation, we show that bipolar soliton gases exhibit the formation of rogue waves, a feature absent in unipolar gases. This distinction is reflected in the evolution of higher-order statistics, with kurtosis increasing in the bipolar case and decreasing in the unipolar one. In integrable settings, the statistical characteristics relax to stationary states, whereas in non-integrable systems they remain time-dependent. We also observe inelastic energy transfer from smaller to larger solitons, leading to increasingly extreme wave events. The emergence of anomalously large structures, referred to as champion solitons, is discussed in the context of non-integrable dynamics.

Inertial particle transport under weakly modulated nearshore waves

Rosa Maria RM Vargas-Magana University of Bergen Norway Co-Author(s):

Abstract: The ocean surface is intrinsically nonlinear, with resonant wave-wave interactions continuously shaping the wave field. As waves shoal toward the coast, dispersive effects weaken while nonlinearities become more strongly expressed. In this regime, weak-to-moderate amplitude modulation, spanning timescales from seconds to minutes, is readily observed in both Lagrangian and Eulerian measurements of the surface elevation. The transport of inertial particles within such wave fields remains a significant open problem in many disciplines. In this talk, we examine inertial particle trajectories under weakly modulated wave conditions representative of the nearshore zone. The dynamics reveal a clear two-scale structure: particles undergo oscillatory spiral motion at the carrier-wave frequency while simultaneously experiencing a slow back-and-forth drift governed by the group modulation. The amplitude, period, and net displacement of this drift depend sensitively on the modulation parameters and initial conditions. These findings reveal how the nonlinear group structure of nearshore waves regulates particle transport in ways not captured by current models, with direct implications for sediment transport, pollutant dispersion, and a more reliable assessment of the fate of buoyant material along coastlines.

Dynamics of inviscid ferrofluid jets: the Hamiltonian framework

Zhan Wang Institute of Mechanics, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: This talk investigates the surface dynamics and stability of solitary waves on an inviscid ferrofluid jet. We begin by establishing a proof of the Hamilton principle for the axisymmetric system and deriving a unique homogeneous expansion of the Dirichlet-Neumann operator (DNO), offering a formulation distinct from that of Guyenne & Parau (2016). Within a unified Hamiltonian/Lagrangian framework, we introduce a systematic methodology for deriving reduced model equations across multiple scales. In particular, we propose a nonlinear model with full dispersion by truncating the DNO expansion at the cubic order. We demonstrate that this model accurately captures the speed-amplitude and speed-energy bifurcations inherent in the full Euler equations. Utilizing this reduced model, we examine the stability of solitary waves under longitudinal disturbances. Our findings indicate that stability exchange occurs at the stationary points of the speed-energy bifurcation curve, providing an axisymmetric parallel to the stability criteria established by Saffman (1985). Extensions of these results to the full Euler equations will be discussed as time allows.

Stability of damped wave equation

Runzhang Xu Harbin Engineering University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with the description of global quantitative stability of wave equations with linear strong damping and linear or nonlinear weak damping. By giving some energy decay estimates, we obtain several conclusions about the continuous dependence of the global solution on the initial data and the coefficients of the strong damping term and linear or nonlinear weak damping term. This work also establishes a new idea to use the dissipative effect to obtain the better continuous dependence conclusions, which also reflect the dissipative properties of the solution.