Three-dimensional Nonlinear Equation Research Articles

On the Solution to the Inverse Problem for the Toda Chain

The "elliptic" version of a three-dimensional nonlinear integrable equation, the Toda chain equation, is studied by means of the inverse scattering method. Several classes of decaying solutions corresponding to both the continuous and discrete spectrum of the associated spectral problem are obtained. Conditions that guarantee reality and analyticity along with relevant estimates are also given. Finally a class of solutions with rotational symmetry is constructed.

An approximate two-flow solution to the Boltzmann equation

An explicit approximate solution to the three-dimensional nonlinear Boltzmann equation for rigid spheres is constructed. It has the form of a spatially inhomogeneous linear combination of two Maxwellians corresponding to different densities, temperatures, and mass velocities. It is shown that the integral norm of the discrepancy between the left- and right-hand sides of the equation can be made arbitrarily small by choosing appropriate values of the parameters entering the distribution.

On the fracture of the unidirectional composites in compression

The experimental studies of the fracture of unidirectional composite materials under loading along the reinforcing elements showed that one of the major mechanism of this fracture is the stability loss in material structure. Up to the present time, numerous theoretical and experimental investigations of the mentioned problems have been made. Note that in these investigations (carried out from a theoretical viewpoint), the problems considered have been studied in the framework of the Euler approach and those have been reduced to investigations of the corresponding eigenvalue problems. In the present paper, a method for investigating the problem of fracture (stability loss) of a unidirectional composite in compression has been given, which uses as a criterion the value of the compressive force for which the initial infinitesimal curvature given to the fibre or layer starts to increase and grows indefinitely. In these cases the investigations are carried out in the framework of the piecewise-homogeneous body model with the use of the exact three-dimensional non-linear equations of the deformable solid body mechanics.Note that the results obtained with the application of the proposed approach to the investigation of the mentioned fracture problems coincide with those which have been obtained in the framework of the Euler approach. Moreover, the approach is also employed successfully for the theoretical investigations of the problems of the fracture mechanics of composites in compression, where the study in the framework of the Euler approach are impossible. In the present paper the fracture (stability) problems of composites with viscoelastic components in compression are also studied by the use of the proposed approach.

A composite asymptotic model for the wave motion in a steady three-dimensional subsonic boundary layer

The problem for a thin near-wall region is reduced, within the triple-deck approach, to unsteady three-dimensional nonlinear boundary-layer equations subject to an interaction law. A linear version of the boundary-value problem describes eigenmodes of different nature (including crossflow vortices) coupled together. The frequency ω of the eigenmodes is connected with the components k and m of the wavenumber vector through a dispersion relation. This relation exhibits two singular properties. One of them is of basic importance since it makes the imaginary part Im(ω) of the frequency increase without bound as k and m tend to infnity along some curves in the real (k, m)-plane. The singularity turns out to be strong, rendering the Cauchy problem ill posed for linear equations.Accounting for the second-order approximation in asymptotic expansions for the upper and main decks brings about significant alterations in the interaction law. A new mathematical model leans upon a set of composite equations without rescaling the original independent variables and desired functions. As a result, the right-hand side of a modified dispersion relation involves an additional term multiplied by a small parameter ε=R−1/8, R being the reference Reynolds number. The aforementioned strong singularity is missing from solutions of the modified dispersion relation. Thus, the range of validity of a linear approximation becomes far more extended in ω, k and m, but the incorporation of the higher-order term into the interaction law means in essence that the Reynolds number is retained in the formulation of a key problem for the lower deck.

Three-dimensional global scale permanent-wave solutions of the nonlinear quasigeostrophic potential vorticity equation and energy dispersion

The three-dimensional nonlinear quasi-geostrophic potential vorticity equation is reduced to a linear form in the stream function in spherical coordinates for the permanent wave solutions consisting of zonal wavenumbers from 0 ton andr n vertical components with a given degreen. This equation is solved by treating the coefficient of the Coriolis parameter square in the equation as the eigenvalue both for sinusoidal and hyperbolic variations in vertical direction. It is found that these solutions can represent the observed long term flow patterns at the surface and aloft over the globe closely. In addition, the sinusoidal vertical solutions with large eigenvalueG are trapped in low latitude, and the scales of these trapped modes are longer than 10 deg. lat. even for the top layer of the ocean and hence they are much larger than that given by the equatorialΒ-plane solutions. Therefore such baroclinic disturbances in the ocean can easily interact with those in the atmosphere.

An efficient model for coupling structural vibrations with acoustic radiation

In this paper, the problem of coupling between panel vibration and near and far field acoustic radiation is studied. The panel vibration is governed by the non-linear plate equations while the loading on the panel, which is the pressure difference across the panel, depends on the reflected and transmitted waves. Two models are used to solve this structural-acoustic interaction problem. One solves the three-dimensional non-linear Euler equations for the acoustic field coupled with the non-linear plate equations (the fully coupled model). The second uses the linear wave equation for the acoustic field and expresses the load as a double integral involving the panel oscillation (the decoupled model). The panel oscillation governed by a system of integro-differential equations is solved numerically and the acoustic field is then defined by an explicit formula. Numerical results are obtained using the two models for linear and non-linear panel vibration regimes excited by incident waves having different sound pressure levels. The predictions given by these two models are in good agreement, but the computational time needed for the “fully coupled model” is 60 times longer than that for “the decoupled model”.

Multidimensional behavior of an electrostatic ion wave in a magnetized plasma

Three-dimensional nonlinear evolution equations of an electrostatic ion wave in the short wave region in a magnetized plasma are derived by means of the reductive perturbation method. It is shown that, in some cases, the evolution equations reduce to the Davey–Stewartson 1 equations, which are known to admit solutions with localized structure in higher dimension. It is also shown that there is a possibility of collapse of localized structure in the case of wave propagation parallel to a magnetic field.

Effects of a poloidal divertor separatrix on resistive magnetohydrodynamic instabilities in a tokamak

A computer code package has been developed to simulate the linear and nonlinear evolution of long-wavelength resistive magnetohydrodynamic (MHD) instabilities in a four-node poloidal divertor tokamak (e.g., Wisconsin Tokapole II [Nucl. Fusion 19, 1509 (1979)]). Distinguishing features of this package include the use of a full set of three-dimensional (3-D) nonlinear resistive MHD equations and the inclusion of the divertor separatrix and the plasma outside the divertor separatrix in the computational domain. The present numerical results suggest that the plasma current outside the divertor separatrix tends to linearly stabilize the resistive MHD instability dominated by the m=2, n=1 mode, and, to a lesser extent, that dominated by the m=1, n=1 mode. (Here, m and n are poloidal and toroidal mode numbers, respectively.) However, the nonlinear evolution of the m=1, n=1 dominant instability is not significantly affected by the divertor configuration; the m=1, n=1 island is shown to reconnect totally by developing a large region of magnetic stochasticity. Hence, the cause of the partial reconnection observed in Tokapole II seems to lie beyond the scope of the classical resistive MHD model.

Three-dimensional neoclassical nonlinear kinetic equation for low collisionality axisymmetric tokamak plasmas

The three-dimensional nonlinear kinetic equation for low collisionality tokamak plasmas with consistent consideration of neoclassical effects is obtained using an approach differing from the standard neoclassical theory technique. This allows treatment of large banana widths and large inverse aspect ratios. The equation is suitable for computer modeling of bootstrap currents and other phenomena arising from non-Maxwellian distributions. The formalism described in this paper, which is for noncanonical variables, might also be of use for the consistent derivation of three-dimensional kinetic equations that treat other effects, for example, additional heating.

Local regularity of nonlinear wave equations in three space dimensions

This note examines the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for three-dimensional nonlinear wave equations of the form [partial derivative][sub t][sup 2]u [minus] [Delta]u = G(u, Du), (1), u(0) = f, [partial derivative][sub t]u(0) = g, (2), where Du = ([partial derivative][sub t]u, [del][sub x]u). For nonintegral s, this requires the use of an inequality. Thus it is crucial to gain control of the L[sup [ell][minus]1] norm in time of the maximum norm of the gradient of the solution in space. This is usually done by using the Sobolev imbedding theorem which leads to the restriction s > n/2+1 for the Sobolev exponent. The authors show that in three space dimensions (the case to which they restrict themselves throughout the paper), the lower bound for the Sobolev exponent can be reduced from 5/2 to s([ell]) [identical to] max[l brace]2, (5[ell] [minus] 7)/(2[ell] [minus] 2)[r brace] when the nonlinearity G in (1) grows no faster than order [ell] in Du. Thus, as [ell] [yields] [infinity] the classical result s > 5/2 is approached. They also show that this is sharp in the sense that the quantity [parallel]f[parallel][sub H[sup s([ell])]] + [parallel]g[parallel][sub H[supmore » s([ell])[minus]1]] is not sufficient, in general, to control the local existence time of solutions (for [ell] [ge] 3). The authors return to the spherically symmetric case at the end of the paper. For general nonlinearities, such a result just fails. It is shown how to apply instead a space-time estimate for the free solution due to Marshall and Pecher, as an extension of the work of Strichartz. 9 refs.« less

Analysis of the solution of nonlinear three-dimensional equations of magnetohydrodynamics by projection on an unstable manifold

Analysis of the solution of nonlinear three-dimensional equations of magnetohydrodynamics by projection on an unstable manifold

A hybrid spectral method for the three-dimensional numerical modelling of nonlinear flows in shallow seas

A spectral method employing eddy-viscosity eigenfunctions is used to solve the full three-dimensional nonlinear hydrodynamic equations for the numerical computation of flows caused by tides or meteorological forcing. An explicit finite element method is developed to compute the nonlinear advective terms and an explicit treatment of bottom friction is used. This leads to a rapidly convergent expansion and relatively few eigenfunctions are required to obtain accurate solutions. An Arakawa B-grid is used in the horizontal directions and leapfrog time-stepping. The eigenfunctions are computed at the beginning of the program, for an arbitrary spatial dependence of eddy viscosity, using the SLEIGN subroutine. Several model problems have been used to test the accuracy, stability, and computational efficiency of the method.

Three-dimensional optical structures described by the nonlinear Helmholtz equation

New distributions of the optical field described by the three-dimensional nonlinear Helmholtz equation correspond both to periodic (in the space of standing waves realized in a nonlinear resonator) and spherical self-localized optical bunches.

Similarity solutions to three-dimensional nonlinear diffusion equations

The similarity transformation method has been used to solve three-dimensional nonlinear diffusion equations. The general Lie group is calculated. Exact solutions to the cylindrical and spherical symmetry cases are found, and their relations to some real physical processes are discussed.

Dynamics of localized ion-acoustic waves in a magnetized plasma

The evolution of negative potential pulses in a magnetized plasma is studied. A three-dimensional (3-D) nonlinear ion-acoustic wave equation, including nonstationary effects of reflected electrons, has been derived from the Poisson–Vlasov equations with uniform magnetic field. In the low temperature limit, the equation is the Zakharov–Kuznetsov equation. Computer simulations of 1-D and 2-D versions of the equation have been performed. The studies show that a negative potential pulse can be enhanced by drifting electrons. The growing pulse develops asymmetrically with an oscillatory precursor and a local potential jump resembling the early phase of weak double layer formation. Two-dimensional pulses also show different scale lengths along the magnetic field and in the transverse dimension. Comparisons are made with results from particle simulations and spacecraft observations.

Nonlinear regular structures of drift magneto-acoustic waves

Nonlinear regular structures in a magnetized plasma connected with drift magneto-acoustic waves (DMA) are investigated theoretically. Three-dimensional nonlinear equations of weakly dispersive DMA waves are obtained. These equations contain both the scalar nonlinearity and the vector one, and generalize the two-dimensional Kadomtsev–Petviashvili (KP) equation. The existence is shown of regular stationary structures due to the scalar nonlinearity: one-dimensional solitons, two-dimensional rational solitons, chains of solitons and so-called ‘crosses’. The stability of one-dimensional DMA solitons is investigated. It is shown that soliton stability depends on the sign of the wave dispersion as in the case of systems described by the KP-type equation.

Nonlinear theory of a quadrupole free-electron laser

The nonlinear motion of electrons in a quadrupole free-electron laser is analyzed. If the betatron frequency is close to the mismatch frequency, the three-dimensional nonlinear equations can be reduced to an integrable one-dimensional equation. To enhance the efficiency, a tapered wiggler is introduced and the electron orbits in a tapered quadrupole free-electron laser are analyzed. The conditions for the particles to remain trapped as the heat wave decelerates are found.

Quasi classical regime of collapse in the three-dimensional nonlinear Schroedinger equation

Quasi classical regime of collapse in the three-dimensional nonlinear Schroedinger equation

On a multiaxial non-linear hereditary constitutive law for non-ageing materials with fading memory

A hereditary type complementary power potential is postulated which permits definition of the strain rate tensor in terms of the history and current magnitude of a certain combination of stress and stress deviator components. The strain rate tensor is decomposed into elastic and viscous portions, the latter being further separated into reversible and irreversible parts with the reversible portion being represented by Volterra type integrals. For a material obeying Norton's power law, the three-dimensional non-linear constitutive equations are derived in terms of two elastic constants and five viscous material parameters which may be determined from creep tests. These equations are applied to the derivation of general non-linear viscoelastic constitutive relations for a plate element which, in turn, are used to solve the cylindrical time-deflection problem of a long simply supported ice plate subjected to in-plane compressive forces applied along the longitudinal edges. The governing equations are solved numerically, using an incremental approach. An approximate method for the effective numerical treatment of problems involving hereditary type constitutive relations is discussed in detail.

Three-dimensional nonlinear Schrödinger equation for finite-amplitude gravity waves in a fluid

The averaged Lagrangian method is used to derive a new nonlinear Schrodinger equation in order to describe the evolution of three-dimensional modulations superposed on finite-amplitude gravity waves in a fluid. An interesting feature of this equation is that the wave energy initially confined to a narrow band of wave numbers continues to be confined to a limited region in the wave number space. Unlike the three-dimensional nonlinear evolution equation available in the literature, the equation obtained in this paper may, therefore, be adequate for the description of the evolution of the three-dimensional modulations.