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We consider steady surface waves in an infinitely deep two--dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 degree corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

We develop a numerical method based on canonical conformal variables to study two eigenvalue problems for operators fundamental to finding a Stokes wave and its stability in a 2D ideal fluid with a free surface in infinite depth. We determine the spectrum of the linearization operator of the quasiperiodic Babenko equation, and provide new results for eigenvalues and eigenvectors near the limiting Stokes wave identifying new bifurcation points via the Fourier-Floquet-Hill (FFH) method. We conjecture that infinitely many secondary bifurcation points exist as the limiting Stokes wave is approached. The eigenvalue problem for stability of Stokes waves is also considered. The new technique is extended to allow finding of quasiperiodic eigenfunctions by introduction of FFH approach to the canonical conformal variables based method. Our findings agree and extend existing results for the Benjamin-Feir, high-frequency and localized instabilities. For both problems the numerical methods are based on Krylov subspaces and do not require forming of operator matrices. Application of each operator is pseudospectral employing the fast Fourier transform (FFT), thus enjoying the benefits of spectral accuracy and $O(N \log N)$ numerical complexity. Extension to nonuniform grid spacing is possible via introducing auxiliary conformal maps.

We study the stability of Stokes waves on a free surface of an ideal fluid of infinite depth. For small steepness the modulational instability dominates the dynamics, but its growth rate is vastly surpassed for steeper waves by an instability due to disturbances localized at the wave crest, explaining why long propagating ocean swell consists of small-amplitude waves. The dominant localized disturbances are either co-periodic with the Stokes wave, or have twice its period. The nonlinear stage of instability for steep wave evolution reveals the formation of a plunging breaker.

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.

A stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behaviour of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website., Comment: 20 pages, 9 figures

In the present work we compute numerical solutions of an integro-differential equation for traveling waves on the boundary of a $2$D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudo--differential equation by means of the Newton-Conjugate Residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to $N = 65536$., Comment: 11 pages, 4 figures

We provide a systematic comparison of two numerical methods to solve the widely used nonlinear Schr\"odinger equation. The first one is the standard second order split-step (SS2) method based on operator splitting approach. The second one is the Hamiltonian integration method (HIM). It allows the exact conservation of the Hamiltonian at the cost of requiring the implicit time stepping. We found that numerical error for HIM method is systematically smaller than the SS2 solution for the same time step. At the same time, one can take orders of magnitude larger time steps in HIM compared with SS2 still ensuring numerical stability. In contrast, SS2 time step is limited by the numerical stability threshold.

We find a new class of solutions that are traveling waves on the boundary of two--dimensional droplet of ideal fluid. We assume that the free surface is subject only to the force of surface tension, and the fluid flow is potential. We use the canonical Hamiltonian variables discovered in the work of V.E. Zakharov in 1968, and conformally map lower complex plane to the interior of a fluid droplet. We write the equations in the form originally discovered in A.I. Dyachenko in (2001) for infinitely deep water, and adapted to bounded fluid in the work of S.A. Dyachenko (2019). The new class of solutions satisfies a pseudodifferential equation which is similar to the Babenko equation for the Stokes wave.

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., Comment: In an upcoming title in Tutorials, Schools, and Workshops in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1903.00097

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.

We derive a set of equations in conformal variables that describe a potential flow of an ideal inviscid fluid with free surface in a bounded domain. This formulation is free of numerical instabilities present in the equations for the surface elevation and potential derived by A. I. Dyachenko et al in 1996 with some of the restrictions on analyticity relieved. We illustrate with the results of a comparison of the numerical simulations with the exact solution, the Dirichlet ellipse. In presence of surface tension, we demonstrate the oscillations of the free surface of a unit disc droplet about its equilibrium, the disc shape.

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudo-differential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton-GMRES method. It works well with a fast Fourier transform and is more effective than a boundary integral method. The result is in excellent agreement, qualitatively and quantitatively, with earlier ones. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the steepness increases and tend to a touching wave, whose profile self-intersects somewhere along the trough line, trapping an air bubble; the numerical solutions become unphysical as the steepness increases further and make a gap in the wave speed versus steepness plane; a touching wave then takes over and the physical solutions follow in the wave speed versus steepness plane until they ultimately tend to an extreme wave, which exhibits a sharp corner at the crest. Overhanging waves of nearly maximum heights are found to approach rigid body rotation of a fluid disk as the strength of positive vorticity increases.

A new highly efficient method is developed for computation of traveling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularites above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.

We develop an operator splitting method to simulate flows of isothermal compressible natural gas over transmission pipelines. The method solves a system of nonlinear hyperbolic partial differential equations (PDEs) of hydrodynamic type for mass flow and pressure on a metric graph, where turbulent losses of momentum are modeled by phenomenological Darcy-Weisbach friction. Mass flow balance is maintained through the boundary conditions at the network nodes, where natural gas is injected or withdrawn from the system. Gas flow through the network is controlled by compressors boosting pressure at the inlet of the adjoint pipe. Our operator splitting numerical scheme is unconditionally stable and it is second order accurate in space and time. The scheme is explicit, and it is formulated to work with general networks with loops. We test the scheme over range of regimes and network configurations, also comparing its performance with performance of two other state of the art implicit schemes., Comment: 21 pages, 11 figures, submitted to SIAM Journal of Scientific Computing

We describe a broad class of bounded non-periodic potentials in one-dimensional stationary quantum mechanics having the same spectral properties as periodic potentials. The spectrum of the corresponding Schroedinger operator consists of a finite or infinite number of allowed bands separated by gaps. In this letter we consider the simplest class of potentials, whose spectra consist of an interval on the negative semiaxis and the entire positive axis. The potentials are reflectionless, and a particle with positive energy moves freely in both directions. The potential is constructed as a limit of Bargmann potentials and is determined by a Riemann-Hilbert problem, which is equivalent to a pair of singular integral equations that can be efficiently solved using numerical techniques.

A reduced Keller-Segel equation (RKSE) is a parabolic-elliptic system of partial differential equations which describes bacterial aggregation and the collapse of a self-gravitating gas of brownian particles. We consider RKSE in two dimensions, where solution has a critical collapse (blow-up) if the total number of bacteria exceeds a critical value. We study the self-similar solutions of RKSE near the blow-up point. Near the collapse time, $t=t_c$, the critical collapse is characterized by the $L\propto (t_c-t)^{1/2}$ scaling law with logarithmic modification, where $L$ is the spatial width of collapsing solution. We develop an asymptotic perturbation theory for these modifications and show that the resulting scaling agrees well with numerical simulations. The quantitative comparison of the theory and simulations requires to take into account several terms of the perturbation series.

We study the catastrophic stationary self-focusing (collapse) of laser beam in nonlinear Kerr media. The width of a self-similar solutions near collapse distance $z=z_c$ obeys $(z_c-z)^{1/2}$ scaling law with the well-known leading order modification of loglog type $\propto (\ln|\ln(z_c-z)|)^{-1/2}$. We show that the validity of the loglog modification requires double-exponentially large amplitudes of the solution $\sim {10^{10}}^{100}$, which is unrealistic to achieve in either physical experiments or numerical simulations. We derive a new equation for the adiabatically slow parameter which determines the system self-focusing across a large range of solution amplitudes. Based on this equation we develop a perturbation theory for scaling modifications beyond the leading loglog. We show that for the initial pulse with the optical power moderately above ($\lesssim 1.2$) the critical power of self-focusing, the new scaling agrees with numerical simulations beginning with amplitudes around only three times above of the initial pulse., Comment: 9 pages, 5 figures; extended text compare with previous version