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We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, Xs(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co-located with a moving boundary x=Xs(t), where Xs(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) Xs for non-uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.

We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.

This article presents errors, corrections, and additions to the research outlined in the following citation: Christov, I., Christov, C. I., & Jordan, P. M. (2007). Modeling weakly nonlinear acoustic wave propagation. The Quarterly Journal of Mechanics and Applied Mathematics, 60(4), 473-495.

We investigate numerically the collision dynamics of elliptically polarized solitons of the System of Coupled Nonlinear Schrodinger Equations (SCNLSE) for various different initial polarizations and phases. General initial elliptic polarizations (not sech-shape) include as particular cases the circular and linear polarizations. The elliptically polarized solitons are computed by a separate numerical algorithm. We find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision ('polarization shock'). We study in detail the effect of the initial phases of the solitons and uncover different scenarios of the quasi-particle behavior of the solution. In majority of cases the solitons survive the interaction preserving approximately their phase speeds and the main effect is the change of polarization. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction. This outlines the role of the phase, which has not been extensively investigated in the literature until now.

We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form' in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field' may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from' of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum., accepted

Boussinesq equation arises in shallow water flows and in elasticity of rods and shells. It contains non- linearity and fourth-order dispersion and has been one of the main soliton models in 1D. To find its 2D solutions, a perturbation series with respect to the small parameter e = c 2 is developed in the present work, where c is the phase speed of the localized wave. Within the order O(e 2 )= O(c 4 ), a hierarchy is derived consisting of one-dimensional fourth-order equations. The Bessel operators involved are refor- mulated to facilitate the creation of difference schemes for the ODEs from the hierarchy. The numerical scheme uses a special approximation for the behavioral condition in the singularity point (the origin). The results of this work show that at infinity the stationary 2D wave shape decays algebraically, rather than exponentially as in the 1D cases. The new result can be instrumental for understanding the interaction of 2D Boussinesq solitons, and for creating more efficient numerical algorithms explicitly acknowledging the asymptotic behavior of the solution.

We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.

The propagation of stationary solitary waves on an infinite elastic rod on elastic foundation equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solution of type of solitary wave, which is a bifurcation problem. The bifurcation is treated by prescribing the solution in the origin and introducing an unknown coefficient in the equation. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher-order boundary value problem. The latter is solved by means of an iterative difference scheme, which is thoroughly validated. Solitary waves with oscillatory tails are obtained for different values of tension and linear restoring force. Special attention is devoted to the case with negative tension, when the solutions have oscillatory tails.

In recent works of the author [Found. Phys. 36 (2006) 1701-1717; Math. Comput. Simul. 74 (2007) 93-103], the argument has been made that Hertz's equations of electrodynamics reflect the material invariance (indifference) of the latter. Then the principle of material invariance was postulated in lieu of Lorentz covariance, and the respective absolute medium was named the metacontinuum. Here, we go further to assume that the metacontinuum is a very thin but very stiff 3D hypershell in the 4D space. The equation for the deflection of the shell along the fourth dimension is the ''master'' nonlinear dispersive equation of wave mechanics whose linear part (Euler-Bernoulli equation) is nothing else but the Schrodinger wave equation written for the real or the imaginary part of the wave function. The wave function has a clear non-probabilistic interpretation as the actual amplitude of the flexural deformation. The ''master'' equation admits solitary-wave solutions/solitons that behave as quasi-particles (QPs). We stipulate that particles are our perception of the QPs (schaumkommen in Schrodinger's own words). We show the passage from the continuous Lagrangian of the field to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. We interpret the membrane tension as an attractive (gravitational?) force acting between the QPs. Thus, a self-consistent unification of electrodynamics, wave mechanics, gravitation, and the wave-particle duality is achieved.

For the Coupled Nonlinear Schrodinger Equations (CNLSE) we construct a conservative fully implicit scheme (in the vein of the scheme with internal iterations proposed in [C.I. Christov, S. Dost, G.A. Maugin, Inelasticity of soliton collisions in system of coupled nls equations, Physica Scripta 50 (1994) 449-454.]). Our scheme makes use of complex arithmetic which allows us to reduce the computational time fourfold. The scheme conserves the ''mass'', momentum, and energy. We investigate collisions of solitary waves (quasi-particles or QPs) with linear polarization in the initial configuration. We elucidate numerically the role of nonlinear coupling on the quasi-particle dynamics. We find that the initially linear polarizations of the QPs change after the collision to elliptic polarizations. For large values of cross-modulation parameter, an additional QP is created during the collision. We find that although the total energy is positive and conserved, the energy only of the system of identifiable after the collision QPs is negative, i.e., the different smaller excitations and radiation carry away part of the energy. The effects found in the present work shed light on the intimate mechanisms of interaction of QPs.

A material-invariant (frame indifferent) version of the Maxwell–Cattaneo law is proposed in which the relaxation rate of the heat flux is given by Oldroyd’s upper-convected derivative. It is shown that the new formulation allows for the elimination of the heat flux, thus yielding a single equation for the temperature field. This feature is to be expected from a truly frame indifferent description.

A Fourier–Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.

The present paper deals with the effect of dissipation on the propagation of wave packets governed by a wave equation of Jeffrey type. We show that all packets undergo a shift of the central frequency (the mode with maximal amplitude) towards the lower frequencies (“redshift” in theory of light or “baseshift” in acoustics). Packets with Gaussian apodization function do not change their shape and remain Gaussian but undergo redshift and spread. The possible applications of the results are discussed.

Summary Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE’s performance being the poorest overall.

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).

We prove that, when linearized, the governing equations of an incompressible viscoelastic continuum can be rendered into a form identical to that of Maxwell's equations of electrodynamics. The divergence of deviator stress tensor is analogous to the electric field, while the vorticity (the curl of velocity field) is interpreted as the magnetic field. The elastic part of constitutive relation explains Maxwell's displacement current, and is responsible for the propagation of gradient (shear) waves. In turn, the viscous part is associated with the Ampere's and Ohm's laws for the current. This analogy is extended further and the nonlinearity of the material time derivative (the advective part of acceleration) is interpreted as the Lorentz force. The classical wave equations of electrodynamics are also derived as corollaries. Thus an interesting and far reaching analogy between the viscoelastic continuum and the electrodynamics is established.

We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special complete orthonormal system of functions in L^2(-~,~) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible.

We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for "relaxation" and "retardation." Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation., Comment: 5 pages, 1 figure, elsarticle format, v2 corrects some typograpgical errors, published in Mechanics Research Communications

Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we show that if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum.

A study of nonlinear acoustic waves in a homentropic perfect gas is presented. Conservation laws for the Euler and Lighthill–Westervelt equations are constructed and solved numerically using a Godunov-type finite-difference scheme. Simulations are carried out in the context of two initial-boundary-value problems (IBVP)s—one resulting in finite-time, and the other in infinite-time, blow-up at the wavefront. Additionally, analytical results are presented to support the numerical findings.

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher-order ultra-parabolic equation. The convergence with respect to the fictitious time (we call the latter 'internal iterations') allows one to obtain fully implicit nonlinear scheme in full time steps for the physical-time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second-order of approximation. For the solution of the internal iteration, a fractional-step scheme is proposed based on the splitting of the combination of the Laplace, bi-harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second-order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically.

The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is...

The motion of particles in a viscous fluid close to a porous membrane is modelled for the case when particles are large compared with the size of pores of the membrane. The hydrodynamic interactions of one particle with the membrane are detailed here. The model involves Stokes equations for the fluid motion around the particle together with Darcy equations for the flow in the porous membrane and Stokes equations for the flow on the other side of the membrane. Boundary conditions at the fluid‐membrane interface are the continuity of pressure and velocity in the normal direction and the Beavers and Joseph slip condition on the fluid side in the tangential directions. The no‐slip condition applies on the particle. This problem is solved here by two different methods.The first one is an extended boundary integral method (EBIM). A Green function is derived for the flow close to a porous membrane. This function is non‐symmetric, leading to difficulties hindering the application of the classical boundary integral method (BIM). Thus, an extended method is proposed, in which the unknown distribution of singularities on the particle surface is not the stress, like in the classical boundary integral method. Yet, the hydrodynamic force and torque on the particle are obtained by integrals of this distribution on the particle surface.The second method consists in searching the solution as an asymptotic expansion in term of a small parameter that is the ratio of the typical pore size to the particle size. The various boundary conditions are taken into account at successive orders: order (0) simply represents an impermeable wall without slip and order (1) an impermeable wall with a peculiar slip prescribed by order (0); at least the 3rd order is necessary to enforce all boundary conditions.The methods are applied numerically to a spherical particle and comparisons are made with earlier works in particular cases.The motion of particles in a viscous fluid close to a porous membrane is modelled for the case when particles are large compared with the size of pores of the membrane. The hydrodynamic interactions of one particle with the membrane are detailed here. The model involves Stokes equations for the fluid motion around the particle together with Darcy equations for the flow in the porous membrane and Stokes equations for the flow on the other side of the membrane. Boundary conditions at the fluid‐membrane interface are the continuity of pressure and velocity in the normal direction and the Beavers and Joseph slip condition on the fluid side in the tangential directions. The no‐slip condition applies on the particle. This problem is solved here by two different methods.The first one is an extended boundary integral method (EBIM). A Green function is derived for the flow close to a porous membrane. This function is non‐symmetric, leading to difficulties hindering the application of the classical boundary integra...

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

Using a complete orthonormal system of functions in L 2(−∞ ,∞) a Fourier-Galerkin spectral technique is developed for computing of the localized solutions of equations with cubic nonlinearity. A formula expressing the triple product into series in the system is derived. Iterative algorithm implementing the spectral method is developed and tested on the soliton problem for the cubic Boussinesq equation. Solution is obtained and shown to compare quantitatively very well to the known analytical one. The issues of convergence rate and truncation error are discussed.

In this paper, we consider a nonlinear generalized diffusion equation called the Swift-Hohenberg equation (SH) for which a Lyapunov functional is known. We develop a computationally efficient second-order in time implicit difference scheme based on the operator-splitting method. Internal iterations are used to make the scheme both nonlinear and implicit. We prove that the scheme allows strict (independent of the truncation error) implementation of a discrete approximation of the Lyapunov functional. The new scheme is used to investigate the pattern formation from random initial conditions, and spatially chaotic states are found.

The flow of inviscid liquid in a shallow layer with free surface is revisited in the framework of the Boussinesq approximation. The unnecessary approximations connected with the moving frame are removed and a Boussinesq model is derived which is Galilean invariant to the leading asymptotic order. The Hamiltonian structure of the new model is demonstrated. The conservation and/or balance laws for wave mass, energy and wave momentum (pseudo-momentum) are derived. A new localized solution is obtained analytically and compared to the classical Boussinesq sech. Numerical simulation of the collision of two solitary waves is conducted and the impact of Galilean invariance on phase shift is discussed.

The convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is considered for very large Rayleigh numbers. Gravity is taken to be vertical and to consist of both a mean and a harmonic modulation (‘jitter’) at a given frequency and amplitude. The time-dependent Boussinesq equations governing the two-dimensional convection are solved numerically. To this end an economic operator-splitting scheme is devised combined with internal iterations within a given time step. The approximation of the nonlinear terms is conservative and no scheme viscosity is present in the approximation. The flow is investigated for a range of Prandtl numbers from Pr = 1000 when fluid inertia is insignificant and only thermal inertia plays a role to Pr = 0.73 when both are significant and of the same order. The flow is governed by several parameters. In the absence of jitter, these are the Prandtl number, Pr, the Rayleigh number, Ra, and the dimensionless critical stratification, τB. Simulations are reported for Pr = 103 and a range of τB and Ra, with emphasis on mode selection and finite-amplitude states. The presence of jitter adds two more parameters, i.e. the dimensionless jitter amplitude e and frequency ω, rendering the flow susceptible to new modes of parametric instability at a critical amplitude ec. Stability maps of ec vs. ω are given for a range of ω. Finally we investigate the response of the system to jitter near the neutral curves of the various instability modes.

In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.

Interaction between a base thermocapillary flow and a time-dependent buoyant force is studied for a slot geometry. A temperature gradient applied along a fluid-filled slot with thermocapillarity at a free surface produces a base parallel flow. The system is subjected to streamwise gravitational acceleration that varies harmonically in time. Grassia and Homsy [Phys. Fluids. 10, 1273 (1998)] have shown that in the limit of zero frequency modulation, coupling of the thermocapillary flow with long wave convective modes leads to singularities at critical points corresponding to the Rayleigh–Benard eigenvalues. In the case of small but finite frequency modulation studied here, inertial effects moderate the singularities which are replaced by a response that scales exponentially with the inverse of the dimensionless modulation frequency. An O(1) delay is observed in the onset of the resonant response even for small modulation frequencies. The response is also found to scale exponentially with the inverse Prandtl number for large Prandtl numbers and to be independent of Prandtl number for small Prandtl numbers. Relaxation oscillations are observed in certain parameter ranges as a result of the coupling between the fluid and thermal fields. A Galerkin approximation is used to reduce the problem to an equivalent dynamical system, the analysis of which gives analytical support to and insight into the numerical results.

The inverse parabolic problem of coefficient identification from over-posed data is embedded into a fourth order in space and second order in time elliptic boundary value problem. The latter is well posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. A difference scheme of splitting-type is employed, and featuring examples are elaborated numerically.

The inverse problem of identification of boundary-layer thickness is replaced by the higher-order boundary value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original system (Method of Variational Imbedding – MVI). The imbedding problem is correct in the sense of Hadamard and consists of an explicit differential equation for the boundary-layer thickness. The existence and uniqueness of solution of the linearized imbedding problem is demonstrated and a difference scheme of splitting type is proposed for its numerical solution. The performance of the technique is demonstrated for three different boundary-layer problems: the Blasius problem, flow in the vicinity of plane stagnation point and the flow in the leading stagnation point on a circular cylinder. Comparisons with the self-similar solutions where available are quantitatively very good.

A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Benard convection and results are obtained for very large system sizes and for very long times on small computational platform.

We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.

Pattern formation in a finite layer of fluid induced either by buoyancy or by a surface-tension gradient is considered. The fluid is confined between poor conducting horizontal boundaries, leading to patterns with a characteristic horizontal scale much larger than the fluid depth. The evolution of the system is studied by numerical integration of the (1+2)D equation introduced by Knobloch [1990]: [Formula: see text] Here µ is the scaled bifurcation parameter, κ=1, and a represents the effects of a heat transfer finite Biot number. The coefficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0, δ≠0) or when non-Boussinesq effects are taken into account (γ≠0). When the conductive state becomes unstable due to surface-tension inhomogeneities, it is shown that the system evolves towards a stationary pattern of hexagons with up or down flow depending on the relative value of the coefficients β and δ. In the case of buoyancy-driven convection (β=δ≠0), the system displays a tesselation of steady squares. Knobloch’s equation also describes time-dependent patterns at high thermal gradients, including spatio-temporal chaos, due to the non-variational character of the equation.

The Klein—Gordon equation with cubic nonlinearity (the φ 4 equation) is considered and an energy-conserving difference scheme is proposed for its solution. The scheme, extended to finite time increments and spacing, is then used to define a coupled map lattice for which an energy-like functional is conserved. The case of linear instability of the vacuum state is considered when this energy is not positive definite and found to lead, under certain additional conditions, to spatio-temporal chaos. The statistical properties of this type of solution such as probability densities and correlation functions are calculated. Strong intermittency, whereby the process wanders between two sub-manifolds, is found and studied in detail.

A new class of solutions of three-dimensional equations from the Boussinesq paradigm are considered. The corresponding profiles are not localized functions in the sense of the integrability of the square over an infinite domain. For the new type of solutions, the gradient and the Hessian/Laplacian are square integrable. In the linear limiting case, analytical expressions for the profiles of the pseudolocalized solutions are found. The nonlinear case is treated numerically with a special approximation of the differential operators with spherical symmetry that allows for automatic acknowledgement of the behavioral conditions at the origin of the coordinate system. The asymptotic boundary conditions stem from the $1/r$ behavior at infinity of the pseudolocalized profile. A special approximation is devised that allows us to obtain the proper behavior for much smaller computational box. The pseudolocalized solutions are obtained for both quadratic and cubic nonlinearity., Comment: 8 pages, 3 figures