arXiv subjects

Mark A. Hoefer

Publications and source records attributed to Mark A. Hoefer.

At least 19 recordsLinked to original sources

arXiv2026

Dry dam-break over parabolic bathymetry

Motivated by dry dambreak flows, a perturbative framework for solving the 1D shallow water equations (SWE) over parabolic bathymetry, together with the inviscid Burgers' equation, is developed. Expressed in Riemann variables, the solutions are expanded as analytic power series in the bathymetric curvature $\omega^2$ and summed in closed form in terms of trigonometric functions. The solutions, termed generalised rarefaction waves (GRWs), exhibit finite-time singularities and are shown to describe the asymptotic behaviour of the fluid near dry (vacuum) points. The GRWs bear direct relevance to wave run-up at a shore. A repulsive parabolic hill bathymetry is also considered, yielding solutions in terms of hyperbolic functions. The connection to the nonlinear Schr\"odinger/Gross-Pitaevskii equation that models Bose-Einstein condensates confined in a harmonic potential is discussed. The analytical results agree with direct numerical simulations of the Burgers' equation and of the SWE in different scenarios.

physics.gen-ph↗

arXiv2026

Modulation theory for lumps and interactions between lumps and a mean field in the Kadomtsev-Petviashvili equation

A (2+1)-dimensional hyperbolic system of four quasi-linear partial differential equations is derived that describes the modulations of lump solutions of the Kadomtsev-Petviashvili I (KPI) equation in the presence of a mean field. The system is then shown to satisfy the necessary conditions for integrability of hydrodynamic chains. Moreover, a suitable reduction of the resulting modulation system is applied to study the interactions between lumps and a rarefaction wave for the mean field. Precise conditions are derived that describe how the lump parameters change as a result of the interaction, and which in particular determine whether the lump is transmitted through or trapped inside the rarefaction wave. The theoretical predictions are compared to direct numerical simulations of the KPI equation, showing excellent agreement.

nlin.SI↗

arXiv2026

Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink, and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic traveling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic speeds is shown to lead to modulational instability of periodic traveling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic traveling waves and initial value problems that also reveal additional short-wave instabilities.

nlin.PS↗

arXiv2026

Homoclinic and heteroclinic solutions of the nonlinear Schr\"odinger equation with a complex Wadati potential

Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schr\"odinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a repulsive real part, generated by a Wadati potential function,that support the existence of homoclinic and heteroclinic solutions that asymptote to the same or different, respectively, nonlinear plane waves in the far field. Asymptotic analysis and numerical simulations are used to examine solution existence, bifurcations, and structure. Such solutions play an important role in resonant nonlinear wave generation of dispersive media with localized gain and loss.

nlin.PS↗

arXiv2025

Breaking a superfluid harmonic dam: Observation and theory of Riemann invariants and accelerating sonic horizons

An experimental and theoretical study of sonic horizons emerging from the dam-break problem in a Bose-Einstein condensate confined in an anisotropic harmonic trap is presented. Measurements, analysis, and numerics reveal the formation of a sonic horizon that undergoes acceleration due to harmonic confinement. The superfluid is characterized using a robust measurement technique to determine Riemann invariants. Experimental observations agree with an analytical solution of the Gross-Pitaevskii equation and computations. The collision and annihilation between two sonic horizons at long times is predicted.

cond-mat.quant-gas↗

arXiv2024

Mach reflection and expansion of two-dimensional dispersive shock waves

The oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions for the Kadomtsev-Petviashvili II equation. Various asymptotic wave patterns are identified, classified and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons. An eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle is demonstrated.

nlin.PS↗

arXiv2024

Hydrodynamics of a Discrete Conservation Law

The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

nlin.PS↗

arXiv2023

Whitham modulation theory and two-phase instabilities for generalized nonlinear Schr\"{o}dinger equations with full dispersion

The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch (New J. Phys. 21 (2019)). A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long-wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that, while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher order dispersion, including applications to finite depth water waves and the discrete NLS equation are presented and compared with direct numerical simulations.

nlin.PS↗

arXiv2023

Long-time asymptotics and the radiation condition for linear evolution equations on the half-line with time-periodic boundary conditions

The large time $t$ asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker acting upon an initially quiescent medium. The asymptotic Dirichlet-to-Neumann (D-N) map is constructed by expanding upon the recently developed $Q$-equation method. The D-N map is proven to be unique if and only if the radiation condition that selects the unique wavenumber branch of the dispersion relation for a sinusoidal, time-dependent boundary condition holds: (i) for frequencies in a finite interval, the wavenumber is real and corresponds to positive group velocity, (ii) for frequencies outside the interval, the wavenumber is complex with positive imaginary part. For fixed spatial location $x$, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time-periodic wave. Uniform-in-$x$ asymptotic solutions for the physical cases of the linearized Korteweg-de Vries and Benjamin-Bona-Mahony (BBM) equations are obtained via integral asymptotics. The linearized BBM asymptotics are found to quantitatively agree with viscous core-annular fluid experiments.

math.AP↗

arXiv2023

Two-dimensional reductions of the Whitham modulation system for the Kadomtsev-Petviashvili equation

Two-dimensional reductions of the KP-Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the Kadomtsev-Petviashvili equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent of x , independent of y , and of t (i.e., stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave generically has a nonzero speed and a nonzero slope in the x y plane. In all three of these reductions, the properties of the resulting systems of equations are studied. It is shown that the resulting reduced system is not integrable unless one enforces the compatibility of the system with all conservation of waves equations (or considers a reduction to the harmonic or soliton limit). In all cases, compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the Boussinesq equation is also explicitly obtained.

nlin.SI↗

arXiv2023

Observation of Traveling Breathers and Their Scattering in a Two-Fluid System

The observation of traveling breathers (TBs) with large-amplitude oscillatory tails realizes an almost 50-year-old theoretical prediction (Kuznetsov 1975) and generalizes the notion of a breather. Two strongly nonlinear TB families are created in a core-annular flow by interacting a soliton and a nonlinear periodic (cnoidal) carrier. Bright and dark TBs are observed to move faster or slower, respectively, than the carrier while imparting a phase shift. Agreement with model equations is achieved. Scattering of the TBs are observed to be physically elastic. The observed TBs generalize to many continuum and discrete systems.

nlin.PS↗

arXiv2023

Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions

The Whitham modulation equations for the defocusing nonlinear Schrodinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.

nlin.PS↗

arXiv2023

KdV breathers on a cnoidal wave background

Using the Darboux transformation for the Korteweg-de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schr\"odinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.

nlin.PS↗

arXiv2022

Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics

The propagation of localized solitons in the presence of large-scale waves is a fundamental problem, both physically and mathematically, with applications in fluid dynamics, nonlinear optics and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg-de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A comprehensive review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton-rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite-gap description are used to describe soliton-rarefaction wave and soliton-dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the Inverse Scattering Transform. For transmitted solitons, far-field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave-mean field interaction

nlin.PS↗

arXiv2022

Computation of High Frequency Magnetoelastic Waves in Layered Materials

The direct calculation of magnetoelastic wave dispersion in layered media is presented using an efficient, accurate computational technique. The governing, coupled equations for elasticity and magnetism, the Navier and Landau-Lifshitz equations, respectively, are linearized to form a quadratic eigenvalue problem that determines a complex web of wavenumber-frequency dispersion branches and their corresponding mode profiles. Numerical discretization of the eigenvalue problem via a spectral collocation method (SCM) is employed to determine the complete dispersion maps for both a single, finite-thickness magnetic layer and a finite magnetic-nonmagnetic double-layer. The SCM, previously used to study elastic waves in non-magnetic media, is fast, accurate, and adaptable to a variety of sample configurations and geometries. Emphasis is placed on the extremely high frequency regimes being accessed in ultrafast magnetism experiments. The dispersion maps and modes provide insight into how energy propagates through the coupled system, including how energy can be transferred between elastic- and magnetic-dominated waves as well as between different layers. The numerical computations for a single layer are further understood by a simplified analytical calculation in the high-frequency, exchange-dominated regime where the resonance condition required for energy exchange (an anticrossing) between quasi-elastic and quasi-magnetic dispersion branches is determined. Nonresonant interactions are shown to be well approximated by the dispersion of uncoupled elastic and magnetic waves. The methods and results provide fundamental theoretical tools to model and understand current and future magnetic devices powering spintronic innovation.

cond-mat.mes-hall↗

arXiv2022

Ultrafast perturbation of magnetic domains by optical pumping in a ferromagnetic multilayer

Ultrafast optical pumping of spatially nonuniform magnetic textures is known to induce far-from-equilibrium spin transport effects. Here, we use ultrafast x-ray diffraction with unprecedented dynamic range to study the laser-induced dynamics of labyrinth domain networks in ferromagnetic CoFe/Ni multilayers. We detected azimuthally isotropic, odd order, magnetic diffraction rings up to 5th order. The amplitudes of all three diffraction rings quench to different degrees within 1.6 ps. In addition, all three of the detected diffraction rings both broaden by 15% and radially contract by 6% during the quench process. We are able to rigorously quantify a 31% ultrafast broadening of the domain walls via Fourier analysis of the order-dependent quenching of the three detected diffraction rings. The broadening of the diffraction rings is interpreted as a reduction in the domain coherence length, but the shift in the ring radius, while unambiguous in its occurrence, remains unexplained. In particular, we demonstrate that a radial shift explained by domain wall broadening can be ruled out. With the unprecedented dynamic range of our data, our results provide convincing evidence that labyrinth domain structures are spatially perturbed at ultrafast speeds under far-from-equilibrium conditions, albeit the mechanism inducing the perturbations remains yet to be clarified.

cond-mat.mes-hall↗

arXiv2022

Experimental investigations of linear and nonlinear periodic traveling waves in a viscous fluid conduit

Conduits generated by the buoyant dynamics between two miscible, Stokes fluids with high viscosity contrast exhibit rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small and large amplitude periodic traveling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic traveling wave profiles. Measurements are favorably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber-frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is hypothesized to be explained by incorporating a weak recirculating flow into the full two-Stokes flow model which we observe to be operable in the experiment. When the boundary condition corresponds to the temporal profile of a nonlinear periodic traveling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.

physics.flu-dyn↗

arXiv2021

Modulation theory for soliton resonance and Mach reflection

Resonant Y-shaped soliton solutions to the Kadomtsev-Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine-Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.

nlin.PS↗