Averaged Variational Principle Research Articles

Spacetime quasicrystals in Bose-Einstein condensates

An autoresonant approach for exciting spacetime quasicrystals in Bose-Einstein condensates is proposed by employing two-component chirped frequency parametric driving or modulation of the interaction strength within Gross-Pitaevskii equation. A weakly nonlinear theory of the process is developed using Whitham's averaged variational principle yielding reduction to a two-degrees-of-freedom dynamical system in action-angle variables. Additionally, the theory also delineates permissible driving parameters and establishes thresholds on the driving amplitudes required for autoresonant excitation. Published by the American Physical Society 2024

Autoresonant excitation of space-time quasicrystals in plasma

We demonstrate theoretically and numerically that a warm fluid model of a plasma supports space-time quasicrystalline structures. These structures are highly nonlinear, two-phase, ion acoustic waves that are excited autoresonantly when the plasma is driven by two small amplitude chirped-frequency ponderomotive drives. The waves exhibit density excursions that substantially exceed the equilibrium plasma density. Remarkably, these extremely nonlinear waves persist even when the small amplitude drives are turned off. We derive the weakly nonlinear analytical theory by applying Whitham's averaged variational principle to the Lagrangian formulation of the fluid equations. The resulting system of coupled weakly nonlinear equations is shown to be in good agreement with fully nonlinear simulations of the warm fluid model. The analytical conditions and thresholds required for autoresonant excitation to occur are derived and compared to simulations. The weakly nonlinear theory guides and informs numerical study of how the two-phase quasicrystalline structure "melts" into a single phase traveling wave when one drive is below a threshold. These nonlinear structures may have applications to plasma photonics for extremely intense laser pulses, which are limited by the smallness of density perturbations of linear waves.

On the spectral asymptotics for the buckling problem

We provide a direct proof of Weyl’s law for the buckling eigenvalues of the biharmonic operator on domains of Rd of finite measure. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean R2(z). Lower bounds are obtained by making use of the so-called “averaged variational principle.” Upper bounds are obtained in the spirit of Berezin–Li–Yau. Moreover, we state a conjecture for the second term in Weyl’s law and prove its correctness in two special cases: balls in Rd and bounded intervals in R.

Excitation and control of large amplitude standing ion acoustic waves

We study the formation of large-amplitude standing ion acoustic waves (SIAWs) by nonlinear phase-locking (autoresonance) with a weak, chirped frequency standing ponderomotive drive. These waves comprise a nonlinear two-phase solution, with each phase locked to one of the two traveling waves comprising the drive. The autoresonance in the system is guaranteed provided that the driving amplitude exceeds a threshold. The phenomenon is illustrated via water bag simulations within a nonlinear ion fluid model and analyzed using Whitham's averaged variational principle. The local ion and electron densities in the autoresonant SIAWs may significantly exceed the initial unperturbed plasma density and are only limited by kinetic wave-breaking.

Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians

Abstract We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin–Li–Yau and Kröger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.

Eigenvalue bounds of mixed Steklov problems

We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

Excitation and control of large-amplitude standing magnetization waves

A robust approach to excitation and control of large amplitude standing magnetization waves in an easy axis ferromagnetic by starting from a ground state and passage through resonances with chirped frequency microwave or spin torque drives is proposed. The formation of these waves involves two stages, where in the first stage, a spatially uniform, precessing magnetization is created via passage through a resonance followed by a self-phase-locking (autoresonance) with a constant amplitude drive. In the second stage, the passage trough an additional resonance with a spatial modulation of the driving amplitude yields transformation of the uniform solution into a doubly phase-locked standing wave, whose amplitude is controlled by the variation of the driving frequency. The stability of this excitation process is analyzed both numerically and via Whitham's averaged variational principle.

Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

We present asymptotically sharp inequalities for the eigenvalues $\mu\_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \[14]. For the Riesz mean $R\_1(z)$ of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$. In addition, we obtain two-sided bounds for individual $\mu\_k$, which are semiclassically sharp, and we obtain a Neumann version of Laptev’s result that the Pólya conjecture is valid for domains that are Cartesian products of a generic domain with one for which Pólya’s conjecture holds. In a final section, we remark upon the Dirichlet case with the same methods.

Eigenvalue sums of combinatorial magnetic Laplacians on finite graphs

We give a construction of a class of magnetic Laplacian operators on finite directed graphs. We study some general combinatorial and algebraic properties of operators in this class before applying the Harrell-Stubbe Averaged Variational Principle to derive several sharp bounds on sums of eigenvalues of such operators. In particular, among other inequalities, we show that if $G$ is a directed graph on $n$ vertices arising from orienting a connected subgraph of $d$-regular loopless graph on $n$ vertices, then if $\Delta_\theta$ is any magnetic Laplacian on $G$, of which the standard combinatorial Laplacian is a special case, and $\lambda_0\leq \lambda_1\leq ...\leq\lambda_{n-1}$ are the eigenvalues of $\Delta_{\theta},$ then for $k\leq \frac{n}{2},$ we have \[\frac{1}{k}\sum_{j=0}^{k-1}\lambda_j \leq d-1.\]

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [10] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kröger’s bound for Neumann spectra of Laplacians on Euclidean domains [15]. Among the operators we consider are the Laplace–Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [26] with a simplied proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrödinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

Extreme driven ion acoustic waves

The excitation of large amplitude, strongly nonlinear ion acoustic waves from trivial equilibrium by a chirped frequency drive is discussed. Under certain conditions, after passage through the linear resonance in this system, the nonlinearity and the variation of parameters work in tandem to preserve the phase-locking with the driving wave via excursion of the excited ion acoustic wave in its parameter space, yielding controlled growth of the wave amplitude. We study these autoresonant waves via a fully nonlinear warm fluid model and predict the formation of sharply peaked (extreme) ion acoustic excitations with local ion density significantly exceeding the unperturbed plasma density. The driven wave amplitude is bound by the kinetic wave-breaking, as the local maximum fluid velocity of the wave approaches the phase velocity of the drive. The Vlasov-Poisson simulations are used to confirm the results of the fluid model, and Whitham's averaged variational principle is applied for analyzing the evolution of autoresonant ion acoustic waves.

Parametric autoresonant excitation of the nonlinear Schrödinger equation.

Parametric excitation of autoresonant solutions of the nonlinear Schrodinger (NLS) equation by a chirped frequency traveling wave is discussed. Fully nonlinear theory of the process is developed based on Whitham's averaged variational principle and its predictions verified in numerical simulations. The weakly nonlinear limit of the theory is used to find the threshold on the amplitude of the driving wave for entering the autoresonant regime. It is shown that above the threshold, a flat (spatially independent) NLS solution can be fully converted into a traveling wave. A simplified, few spatial harmonics expansion approach is also developed for studying this nonlinear mode conversion process, allowing interpretation as autoresonant interaction within triads of spatial harmonics.

Autoresonant excitation of dark solitons.

Continuouslyphase-locked (autoresonant) dark solitons of the defocusing nonlinear Schrodinger equation are excited and controlled by driving the system by a slowly chirped wavelike perturbation. The theory of these excitations is developed using Whitham's averaged variational principle and compared with numerical simulations. The problem of the threshold for transition to autoresonance in the driven system is studied in detail, focusing on the regime when the weakly nonlinear frequency shift in the problem differs from the typical quadratic dependence on the wave amplitude. The numerical simulations in this regime show a deviation of the autoresonance threshold on the driving amplitude from the usual 3/4 power dependence on the driving frequency chirp rate. The theory of this effect is suggested.

Excitation and control of chirped nonlinear ion-acoustic waves.

Large-amplitude ion acoustic waves are excited and controlled by a chirped frequency driving perturbation. The process involves capturing into autoresonance (a continuous nonlinear synchronization) with the drive by passage through the linear resonance in the problem. The transition to autoresonance has a sharp threshold on the driving amplitude. The theory of this transition is developed beyond the Korteweg-de Vries limit by using the Whitham's averaged variational principle within the water bag model and compared with Vlasov-Poisson simulations.

Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons

An averaged variational principle is applied to analyze the nonlinear effect of transverse perturbations (including diffraction) on quasi-one-dimensional soliton propagation governed by various wave equations. It is shown that parameters of the spatiotemporal solitons described by the cubic Schrodinger equation and the Yajima-Oikawa model of interaction between long-and short-wavelength waves satisfy the spatial quintic nonlinear Schrodinger equation for a complex-valued function composed of the amplitude and eikonal of the soliton. Three-dimensional solutions are found for two-component “bullets” having long-and short-wavelength components. Vortex and hole-vortex structures are found for envelope solitons and for two-component solitons in the regime of resonant long/short-wave coupling. Weakly nonlinear behavior of transverse perturbations of one-dimensional soliton solutions in a self-defocusing medium is described by the Kadomtsev-Petviashvili equation. The corresponding rationally localized “lump” solutions can be considered as secondary solitons propagating along the phase fronts of the primary solitons. This conclusion holds for primary solitons described by a broad class of nonlinear wave equations.

Excitation of multiphase waves of the nonlinear Schrödinger equation by capture into resonances

A method for adiabatic excitation and control of multiphase ( N -band) waves of the periodic nonlinear Schrödinger (NLS) equation is developed. The approach is based on capturing the system into successive resonances with external, small amplitude plane waves having slowly varying frequencies. The excitation proceeds from zero and develops in stages, as an (N+1) -band (N=0,1,2,...) , growing amplitude wave is formed in the (N+1) th stage from an N -band solution excited in the preceding stage. The method is illustrated in simulations, where the excited multiphase waves are analyzed via the spectral approach of the inverse scattering transform method. The theory of excitation of 0- and 1-band NLS solutions by capture into resonances is developed on the basis of a weakly nonlinear version of Whitham's averaged variational principle. The phenomenon of thresholds on the driving amplitudes for capture into successive resonances and the stability of driven, phase-locked solutions in these cases are discussed.

On the Effect of Transverse Perturbations on the Propagation of Soliton-Like Two-Component Pulses in a Uniaxial Crystal

The combined approach including the method of analytic continuation of dispersion parameters to the complex plane and an averaged variational principle of the Ritz-Whitham type is used for studying the propagation of a soliton-like two-component pulse in a uniaxial crystal in the direction perpendicular to its optic axis. The ordinary component of the soliton is a pulse with an arbitrary number of light oscillations (down to one period), and the extraordinary component is a video pulse. Such a pulse is shown to be stable against small transverse perturbations, including small-scale ones.

Dispersion-managed solitons via an averaged variational principle

Constrained minimization is used as a computational strategy to approximate and study dispersion-managed solitons through their characterization as minima of an averaged variational principle. A basis of Hermite–Gaussian functions is used and the constrained minimization procedure is carried out to find such pulses. This method produces more accurate pulse shapes than the usual Gaussian approximations, propagating with less noise and providing a far better representation of the tail of the pulse. The success of this procedure provides confirmation that the dispersion-managed soliton is truly a minimum of the constrained variational principle. When the residual dispersion is negative, the dispersion-managed soliton can no longer be a minimum—a fact that is confirmed by our numerical simulations. Even in this case, however, approximate pulse shapes can be obtained using this finite dimensional approximation. Additionally, we find critical points other than the ground state. The most interesting is an excited bound state corresponding to a symmetric bisoliton.

On the quasi-soliton propagation of few-cycle optical pulses in isotropic dielectrics

A new combined approach for constructing approximate soliton-like solutions of nonlinear wave equations is proposed. The approach includes the method of analytic continuation of dispersion parameters to the complex plane and the averaged variational principle of the Ritz-Whitham type. Based on this approach, the solution of the nonlinear equation describing the propagation of an optical pulse in a transparent isotropic dielectric is found. The obtained solution involves both envelope solitons and breather-like pulses with duration down to one period of electromagnetic oscillations.

Optical solitons as ground states of NLS in the regime of strong dispersion management

The dispersion-managed nonlinear Schrödinger equation (DM-NLS) is considered. After applying an exact transformation, the so-called lens transformation, we derive a new Schrödinger-type equation with additional quadratic potential. In the case of strong dispersion management it is shown how the scales transform into the resulting equation. By constructing ground states of the averaged variational principle we can prove the existence of a standing wave solution of the averaged equation in the case of positive residual dispersion. In contrast to some former discussions of the problem we show the existence of a region where the potential is of trapping type. Due to the potential the solution has a faster decay than the traditional soliton. Moreover we explain why the shape of the DM-soliton changes with increasing pulse energy from a sech-profile to a behavior with Gaussian core and oscillating tails and further to a flatter profile. Furthermore, we illustrate why the DM-soliton can propagate only for small values of the initial pulse width in the case of vanishing or even negative residual dispersion. This results seem to be a new analytical description of what is well known from numerical simulations.