arXiv subjects

Vera Mikyoung Hur

Publications and source records attributed to Vera Mikyoung Hur.

At least 19 recordsLinked to original sources

arXiv2026

Stable and unstable capillary-gravity waves

We develop a rigorous theory of spectral stability and instability for traveling periodic capillary-gravity waves of sufficiently small amplitude in a fluid of finite depth, encompassing unimodal waves and bimodal Wilton ripples. Our analysis builds upon a periodic Evans function framework developed by the authors for Stokes' gravity waves, but requires substantial extensions and new analytical tools to accommodate the considerably richer dynamics of the problem. Particularly, we show that the spectrum away from resonances remains confined to the imaginary axis for sufficiently small amplitude. A key new ingredient in the analysis for unimodal waves near nonzero resonances is an application of the Weierstrass preparation theorem to the discriminant of the Weierstrass polynomial associated with the periodic Evans function. This yields an explicitly computable real-analytic function of the amplitude, whose sign determines spectral stability or instability. Particularly, we obtain the first rigorous spectral stability and instability criteria for traveling periodic capillary-gravity waves away from the origin of the complex plane. We further show that, to leading order in the amplitude, the unstable spectrum near a nonzero resonance traces either an ellipse or a circle. We then investigate modulational instability near the origin of the complex plane for unimodal waves. Our stability index agrees with that obtained from formal asymptotic analysis and recovers the Benjamin--Feir instability for Stokes waves in the zero surface-tension limit. We extend the analysis to bimodal Wilton ripples and obtain the first rigorous spectral stability and instability criteria away from the origin. We numerically evaluate the resulting stability indices and identify parameter regimes for spectral instability either near the origin or near a nonzero resonance of order one or two.

math.AP↗

arXiv2026

Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model

We study front-type solutions of nonlinear dispersive-dissipative PDEs modeling the propagation of undular bores in a channel in two dimensions. The system extends the Korteweg-de Vries-Burgers (KdVB) equation by incorporating weak transverse motion, and admits the KdVB fronts as one-dimensional solutions. We investigate their stability under general two-dimensional perturbations. We prove that a one-dimensional front is a global asymptotic attractor in the weakly two-dimensional setting when the channel is sufficiently narrow in the transverse direction and the relative dispersion parameter lies in a range for stability to one-dimensional perturbations. Particularly, the front is unique up to spatial translations. The proof extends the energy method for temporally-modulated perturbed solutions, developed previously in the one-dimensional setting, to accommodate the transverse dynamics.

math.AP↗

arXiv2025

Asymptotic stability of sharp fronts: Analysis and rigorous computation

We investigate the stability of traveling front solutions to nonlinear diffusive-dispersive equations of Burgers type, with a primary focus on the Korteweg-de Vries-Burgers (KdVB) equation, although our analytical findings extend more broadly. Manipulating the temporal modulation of the translation parameter of the front and employing the energy method, we establish asymptotic, nonlinear, and orbital stability, provided that an auxiliary Schrödinger equation possesses precisely one bound state. Notably, our result is independent of the monotonicity of the profile and does not necessitate the initial condition to be close to the front. We identify a sufficient condition for stability based on a functional that characterizes the 'width' of the traveling wave profile. Analytical verification for the KdVB equation confirms that this sufficient condition holds for the relative dispersion parameter within an open interval $ν\in [-0.25,0.25]$, encompassing all monotone profiles. Utilizing validated numerics or rigorous computation, we present acomputer-assisted proof demonstrating that the stability condition itself holds for parameter values within the interval [0.2533, 3.9].

math.AP↗

arXiv2025

Stokes waves in rotational flows: internal stagnation and overhanging profiles

Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from a fixed rectangular one, where the governing equations along with the conformal mapping are solved using a finite difference scheme. This approach accommodates internal stagnation points, critical layers, and overhanging profiles, thereby overcoming limitations of previous studies. The numerical method is validated through comparisons with known solutions for zero and constant vorticity. Novel solutions are presented for affine vorticity functions and a two-layer constant vorticity scenario.

physics.flu-dyn↗

arXiv2024

Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models

Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics. We develop a general approach to constructing gauge invariant or anyonic symmetric autoregressive neural network quantum states, including a wide range of architectures such as Transformer and recurrent neural network (RNN), for quantum lattice models. These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint. We prove that our methods can provide exact representation for the ground and excited states of the 2D and 3D toric codes, and the X-cube fracton model. We variationally optimize our symmetry incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models. We simulate the dynamics and the ground states of the quantum link model of $\text{U(1)}$ lattice gauge theory, obtain the phase diagram for the 2D $\mathbb{Z}_2$ gauge theory, determine the phase transition and the central charge of the $\text{SU(2)}_3$ anyonic chain, and also compute the ground state energy of the SU(2) invariant Heisenberg spin chain. Our approach provides powerful tools for exploring condensed matter physics, high energy physics and quantum information science.

cond-mat.str-el↗

arXiv2023

Unstable Stokes waves

We investigate the spectral instability of a $2π/κ$ periodic Stokes wave of sufficiently small amplitude, traveling in water of unit depth, under gravity. Numerical evidence suggests instability whenever the unperturbed wave is resonant with its infinitesimal perturbations. This has not been analytically studied except for the Benjamin--Feir instability in the vicinity of the origin of the complex plane. Here we develop a periodic Evans function approach to give an alternative proof of the Benjamin--Feir instability and, also, a first proof of spectral instability away from the origin. Specifically, we prove instability near the origin for $κ>κ_1:=1.3627827\dots$ and instability due to resonance of order two so long as an index function is positive. Validated numerics establishes that the index function is indeed positive for some $κ<κ_1$, whereby there exists a Stokes wave that is spectrally unstable even though it is insusceptible to the Benjamin--Feir instability. The proofs involve center manifold reduction, Floquet theory, and methods of ordinary and partial differential equations. Numerical evaluation reveals that the index function remains positive unless $κ=1.8494040\dots$. Therefore, we conjecture that all Stokes waves of sufficiently small amplitude are spectrally unstable. For the proof of the conjecture, one has to verify that the index function is positive for $κ$ sufficiently small.

math.AP↗

arXiv2023

Floquet theory and stability for Hamiltonian partial differential equations

We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

math.CA↗

arXiv2023

Convolutional GRU Network for Seasonal Prediction of the El Niño-Southern Oscillation

Predicting sea surface temperature (SST) within the El Niño-Southern Oscillation (ENSO) region has been extensively studied due to its significant influence on global temperature and precipitation patterns. Statistical models such as linear inverse model (LIM), analog forecasting (AF), and recurrent neural network (RNN) have been widely used for ENSO prediction, offering flexibility and relatively low computational expense compared to large dynamic models. However, these models have limitations in capturing spatial patterns in SST variability or relying on linear dynamics. Here we present a modified Convolutional Gated Recurrent Unit (ConvGRU) network for the ENSO region spatio-temporal sequence prediction problem, along with the Niño 3.4 index prediction as a down stream task. The proposed ConvGRU network, with an encoder-decoder sequence-to-sequence structure, takes historical SST maps of the Pacific region as input and generates future SST maps for subsequent months within the ENSO region. To evaluate the performance of the ConvGRU network, we trained and tested it using data from multiple large climate models. The results demonstrate that the ConvGRU network significantly improves the predictability of the Niño 3.4 index compared to LIM, AF, and RNN. This improvement is evidenced by extended useful prediction range, higher Pearson correlation, and lower root-mean-square error. The proposed model holds promise for improving our understanding and predicting capabilities of the ENSO phenomenon and can be broadly applicable to other weather and climate prediction scenarios with spatial patterns and teleconnections.

physics.ao-ph↗

arXiv2022

Almost extreme waves

Numerically computed with high accuracy are periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are: (i) a boundary layer where the angle rises sharply from $0^\circ$ at the crest to a local maximum, which converges to $30.3787\dots^\circ$ as the amplitude increases toward that of the extreme wave, independently of the vorticity, (ii) an outer region where the angle descends to $0^\circ$ at the trough for negative vorticity, while it rises to a maximum, greater than $30^\circ$, and then falls sharply to $0^\circ$ at the trough for large positive vorticity, and (iii) a transition region where the angle oscillates about $30^\circ$, resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.

physics.flu-dyn↗

arXiv2022

Overhanging and touching waves in constant vorticity flows

We show the existence of periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper's celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity.

math.AP↗

arXiv2021

Traveling water waves -- the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.

math.AP↗

arXiv2021

Spacetime Neural Network for High Dimensional Quantum Dynamics

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schrödinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schrödinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

cond-mat.dis-nn↗

arXiv2021

Superharmonic instability for regularized long-wave models

We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney--Luke, and Benjamin--Bona--Mahony equations. Of particular interest is a striking new instability phenomenon -- spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg--de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We carry out asymptotic spectral analysis to short wavelength perturbations, distinguishing whether the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments to corroborate our analytical findings.

math.AP↗

arXiv2021

The Benjamin-Feir instability in the infinite depth

We prove that a Stokes' periodic wave of sufficiently small amplitude, traveling under gravity at the free surface of a two dimensional, infinitely deep, and irrotational flow, is spectrally unstable to slow modulation, rigorously justifying Benjamin and Feir's formal argument.

math.AP↗

arXiv2021

Numerical bifurcation and stability for the capillary-gravity Whitham equation

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic traveling waves of the capillary-gravity Whitham equation, which combines the dispersion in the linear theory of capillary-gravity waves and a shallow water nonlinearity. We employ a highly accurate numerical method for space discretization and time stepping, to address orbital stability and instability for a rich variety of the solutions. Our findings can help classify capillary-gravity waves and understand their long-term dynamics.

physics.flu-dyn↗

arXiv2020

A new application of Crapper's exact solution to waves in constant vorticity flows

In 1957 Crapper found an exact solution for capillary waves propagating at the surface of an irrotational flow of infinite depth. Here we provide conclusive analytical and numerical evidence that a Crapper wave makes the profile of a periodic traveling wave propagating, in the absence of the effects of gravity and surface tension, in a constant vorticity flow. This is achieved by constructing a Stokes expansion up to the third order in a small amplitude parameter and by numerically computing large amplitude waves.

physics.flu-dyn↗

arXiv2019

Stokes waves in a constant vorticity flow

The Stokes wave problem in a constant vorticity flow is formulated via a conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.

physics.flu-dyn↗

arXiv2019

Stokes waves with constant vorticity: II. folds, gaps and fluid bubbles

The Stokes wave problem in a constant vorticity flow is formulated, by virtue of conformal mapping techniques, as a nonlinear pseudodifferential equation, involving the periodic Hilbert transform, which becomes the Babenko equation in the irrotational flow setting. The associated linearized operator is self-adjoint, whereby the modified Babenko equation is efficiently solved by means of the Newton-Conjugate Gradient method. For strong positive vorticity, a `fold' appears in the wave speed versus amplitude plane, and a `gap' as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself at the trough line, enclosing a bubble of air. More folds and gaps follow for stronger vorticity. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while the profile encloses a circular bubble of fluid in rigid body rotation at the ends of the gaps. Touching waves at the beginnings of the second gaps tend to the circular vortex wave on top of the limiting Crapper wave in the infinite vorticity limit, and the circular vortex wave on top of itself at the ends of the gaps. Touching waves for higher gaps accommodate more circular bubbles of fluid.

physics.flu-dyn↗