arXiv subjects

Pavel M. Lushnikov

Publications and source records attributed to Pavel M. Lushnikov.

At least 19 recordsLinked to original sources

arXiv2025

Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation

We present exact pole dynamics solutions to the generalized Constantin-Lax-Majda (gCLM) equation in a periodic geometry with dissipation $-Λ^σ$, where its spatial Fourier transform is $\widehat{Λ^σ}=|k|^σ$. The gCLM equation is a simplified model for singularity formation in the 3D incompressible Euler equations. It includes an advection term with parameter $a$, which allows different relative weights for advection and vortex stretching. There has been intense interest in the gCLM equation, and it has served as a proving ground for the development of methods to study singularity formation in the 3D Euler equations. Several exact solutions for the problem on the real line have been previously found by the method of pole dynamics, but only one such solution has been reported for the periodic geometry. We derive new periodic solutions for $a=0$ and $1/2$ and $σ=0$ and $1$, for which a closed collection of (periodically repeated) poles evolve in the complex plane. Self-similar finite-time blow-up of the solutions is analyzed and compared for the different values of $σ$, and to a global-in-time well-posedness theory for solutions with small data presented in a previous paper of the authors. Motivated by the exact solutions, the well-posedness theory is extended to include the case $a=0$, $σ\geq 0$. Several interesting features of the solutions are discussed.

math.AP↗

arXiv2022

The instability of near-extreme Stokes waves

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.

physics.flu-dyn↗

arXiv2022

Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda equation with dissipation $-Λ^σ$, where $\widehat {Λ^σ}=|k|^σ$, both for the problem on the circle $x \in [-π,π]$ and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $σ\geq 1$ and all real values of an advection parameter $a$ when the data is small. We also derive new analytical solutions in both geometries when $a=0$, and on the real line when $a=1/2$, for various values of $ σ$. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for $a=0$ and $σ=2$, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of $σ$ that are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion singularities in the complex plane. The computations validate and extend the analytical theory.

math.AP↗

arXiv2022

Superharmonic Instability of Stokes Waves

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.

physics.flu-dyn↗

arXiv2021

Comparison of Split-Step and Hamiltonian Integration Methods for Simulation of the Nonlinear Schrödinger Equation

We provide a systematic comparison of two numerical methods to solve the widely used nonlinear Schrödinger 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.

physics.comp-ph↗

arXiv2021

Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457\ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at $a=1/2$ as well as prove the existence of a leading order complex singularity for general values of $a$ in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For $a_c<a\leq1$, we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For $a \gtrsim 1.3$, we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for $a<a_c$ which are similar to the real line case. For $a_c<a\le0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $ a\ge 1,$ we identify a global existence of solutions.

nlin.PS↗

arXiv2019

Statistical properties of the laser beam propagating in a turbulent medium

We examine statistical properties of a laser beam propagating in a turbulent medium. We prove that the intensity fluctuations at large propagation distances possess Gaussian probability density function and establish quantitative criteria for realizing the Gaussian statistics depending on the laser propagation distance, the laser beam waist, the laser frequency and the turbulence strength. We calculate explicitly the laser envelope pair correlation function and corrections to its higher order correlation functions breaking Gaussianity. We discuss also statistical properties of the brightest spots in the speckle pattern.

cond-mat.stat-mech↗

arXiv2017

Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial and Outlook

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrodinger equation. This numerical solver is freely available, and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded- and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact.

physics.optics↗

arXiv2017

Toward defeating diffraction and randomness for laser beam propagation in turbulent atmosphere

A large distance propagation in turbulent atmosphere results in disintegration of laser beam into speckles. We find that the most intense speckle approximately preserves both the Gaussian shape and the diameter of the initial collimated beam while loosing energy during propagation. One per 1000 of atmospheric realizations produces at 7km distance an intense speckle above 20\% of the initial power. Such optimal realizations create effective extended lenses focusing the intense speckle beyond the diffraction limit of vacuum propagation. Atmospheric realizations change every several milliseconds. We propose to use intense speckles to greatly increase the time-averaged power delivery to the target plane by triggering the pulsed laser operations only at times of optimal realizations. Resulting power delivery and laser irradiance at the intense speckles well exceeds both intensity of diffraction-limited beam and intensity averaged over typical realizations.

physics.optics↗

arXiv2017

Exact solutions for nonlinear development of Kelvin-Helmholtz instability for counterflow of superfluid and normal components of Helium II

A relative motion of the normal and superfluid components of Helium II results in Kelvin-Helmholtz instability (KHI) at their common free surface. We found the exact solutions for the nonlinear stage of the development of that instability. Contrary to the usual KHI of the interface between two fluids, the dynamics of Helium II free surface allows decoupling of the governing equations with their reduction to the Laplace growth equation which has the infinite number of exact solutions including the formation of sharp cusps at free surface in a finite time.

physics.flu-dyn↗

arXiv2017

New conformal mapping for adaptive resolving of the complex singularities of Stokes wave

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.

physics.flu-dyn↗

arXiv2017

Langmuir wave filamentation in the kinetic regime. I. Filamentation instability of Bernstein-Greene-Kruskal modes in multidimensional Vlasov simulations

A nonlinear Langmuir wave in the kinetic regime $kλ_D\gtrsim0.2$ may have a filamentation instability, where $k$ is the wavenumber and $λ_D$ is the Debye length. The nonlinear stage of that instability develops into the filamentation of Langmuir waves which in turn leads to the saturation of the stimulated Raman scattering in laser-plasma interaction experiments. Here we study the linear stage of the filamentation instability of the particular family \cite{RoseRussellPOP2001} of Bernstein-Greene-Kruskal (BGK) modes \cite{BernsteinGreeneKruskal1957} that is a bifurcation of the linear Langmuir wave. Performing direct $2+2D$ Vlasov-Poisson simulations of collisionless plasma we find the growth rates of oblique modes of the electric field as a function of BGK's amplitude, wavenumber and the angle of the oblique mode's wavevector relative to the BGK's wavevector. Simulation results are compared to theoretical predictions.

physics.plasm-ph↗

arXiv2017

Langmuir wave filamentation in the kinetic regime. II. Weak and Strong Pumping of Nonlinear Electron Plasma Waves as the Route to Filamentation

We consider two kinds of pumped Langmuir waves (LWs) in the kinetic regime, $kλ_D\gtrsim0.2,$ where $k$ is the LW wavenumber and $λ_D$ is the Debye length. They are driven to finite amplitude by a coherent external potential whose amplitude is either weak or strong. These dynamically prepared nonlinear LWs develop a transverse (filamentation) instability whose nonlinear evolution destroys the LW's transverse coherence. Instability growth rates in the weakly pumped regime are the same as those of BGK modes considered in Part I, while strongly pumped LWs have higher filamentation grow rates.

physics.plasm-ph↗

arXiv2016

Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface

Stokes wave is a finite amplitude periodic gravity wave propagating with constant velocity in inviscid fluid. Complex analytical structure of Stokes wave is analyzed using a conformal mapping of a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. There is one square root branch point per spatial period of Stokes located in the upper complex half-plane at the distance $v_c$ from the real axis. The increase of Stokes wave height results in approaching $v_c$ to zero with the limiting Stokes wave formation at $v_c=0.$ The limiting Stokes wave has $2/3$ power law singularity forming $2π/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but nonzero $v_c$ making the limit of zero $v_c$ highly nontrivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of Riemann surface to find coupled square root singularities at the distances $\pm v_c$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of Stokes wave into all these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that non-limiting Stokes wave at the leading order consists of the infinite number of nested square root singularities which also implies the existence in the third and higher sheets of the additional square root singularities away from the real and imaginary axes. These nested square roots form $2/3$ power law singularity of the limiting Stokes wave as $v_c$ vanishes.

nlin.PS↗

arXiv2015

Nonlinear combining of multiple laser beams in Kerr medium

We consider combining of multiple laser beams into a single near diffraction-limited beam by beam self-focusing (collapse) in a Kerr medium. The beams with the total power above critical are combined in the near field and then propagated in the Kerr medium. Nonlinearity results in self-focusing event, combining multiple beams into nearly a diffraction-limited beam that carries the critical power. Beam quality of the combined beam is analyzed as a function of the number of combining beams and the level of random fluctuations of the combining beams phases.

nlin.PS↗

arXiv2014

Nonlinear combining of laser beams

We propose to combine multiple laser beams into a single diffraction-limited beam by the beam self-focusing (collapse) in the Kerr medium. The beams with the total power above critical are first combined in the near field and then propagated in the optical fiber/waveguide with the Kerr nonlinearity. Random fluctuations during propagation eventually trigger strong self-focusing event and produce diffraction-limited beam carrying the critical power.

nlin.PS↗

arXiv2013

Vlasov multi-dimensional model dispersion relation

A hybrid model of the Vlasov equation in multiple spatial dimension $D>1$ [H. A. Rose and W. Daughton, Physics of Plasmas v. 18, 122109 (2011)], the Vlasov multi dimensional model (VMD), consists of standard Vlasov dynamics along a preferred direction, the $z$ direction, and $N$ flows. At each $z$ these flows are in the plane perpendicular to the $z$ axis. They satisfy Eulerian-type hydrodynamics with coupling by self-consistent electric and magnetic fields. Every solution of the VMD is an exact solution of the original Vlasov equation. We show convergence of the VMD Langmuir wave dispersion relation in thermal plasma to that of Vlasov-Landau as $N$ increases. Rotational symmetry about the $z$ axis in $3D$ of small perpendicular wavenumber Langmuir fluctuations is demonstrated for $N\geq 6$, with flows arranged uniformly over the azimuthal angle.

physics.plasm-ph↗

arXiv2013

Beyond the random phase approximation: Stimulated Brillouin backscatter for finite laser coherence times

We develop a statistical theory of stimulated Brillouin backscatter (BSBS) of a spatially and temporally partially incoherent laser beam for laser fusion relevant plasma. We find a new collective regime of BSBS (CBSBS) with intensity threshold controlled by diffraction, an insensitive function of the laser coherence time, $T_c$, once light travel time during $T_c$ exceeds a laser speckle length. The BSBS spatial gain rate is approximately the sum of that due to CBSBS, and a part which is independent of diffraction and varies linearly with $T_c$. We find that the bandwidth of KrF-laser-based fusion systems would be large enough to allow additional suppression of BSBS.

physics.plasm-ph↗