Exact Self-similar Solutions Research Articles

Enstrophy growth in the viscous Burgers equation

We study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy's rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the Poincare inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to prove that a larger growth of enstrophy can be achieved when the initial data to the viscous Burgers equation saturates the Poincare inequality up to a numerical constant. An exact self-similar solution of the Burgers equation is constructed to describe formation of a metastable viscous shock on the background of a linear rarefactive wave. When we consider the Burgers equation on an infinite line subject to the nonzero (shock-type) boundary conditions, we prove that the maximum enstrophy achieved in the time evolution is scaled as E 3/2 , where E is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as E −1/2 log(E). Similar but slower rates are proved on the unit circle.

Exact self-similar wave solutions for the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schröinger equation with distributed coefficients

In this paper, a similarity transformation is presented to reduce the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schrödinger equation with distributed coefficients to the related constant-coefficients one. Then a number of spatiotemporal self-similar wave solutions are constructed. Under the specific choice of the dispersion, cubic and quintic nonlinearities, phase modulation and the gain/loss, we investigate the dynamical behaviors of those spatiotemporal self-similar waves in an inhomogeneous optical fiber media.

Self-similar spatiotemporal solitary waves in Bessel—Hermite optical lattices

An exact self-similar solution to a (3+1)-dimensional nonlinear Schrödinger equation with gain in the Bessel-Hermite lattice is obtained analytically. The stability of the analytical solution is confirmed by using numerical simulation. It is shown that the light bullet has a stable ellipsoid or vortex profile and a linear spatiotemporal chirp.

Exact self-similar solution of countercurrent capillary imbibition

Consideration is given to the one-dimensional countercurrent capillary imbibition of a wetting phase (water) into a semiinfinite porous medium. A model based on consideration of the function of movement of self-similar saturation profiles is proposed. An equation is derived for it, and boundary conditions are determined. The solution of the equation gives the imbibition volume, the saturation and imbibition-flux profiles, and the parameters at entry into the medium and on the saturation front. Calculation of the imbibition indices for a sandstone sample is performed.

Ultrashort self-similar solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients in the inhomogeneous fiber

By means of the similarity transformation, we obtain exact self-similar solutions (similaritons) of the generalized cubic-quintic (CQ) nonlinear Schrödinger equation with spatially inhomogeneous group velocity dispersion, CQ nonlinearity and amplification or attenuation. Exact balance conditions between the dispersion, nonlinearity and the gain/loss have been obtained. As an example, we investigate their propagation dynamics in the dispersion decreasing fiber (DDF). Considering the fluctuation of the fiber parameter in real application, the exact balance conditions do not satisfy, and so we perform direct numerical analysis with initial 10% white noise for the bright similariton in both the DDF and the periodic distributed amplification system. Numerical calculations indicate stable propagation of the bright similariton over tens of dispersion lengths. These ultrashort self-similar optical waves are potentially useful for all-optical data-processing schemes and the design of beam compressors and amplifiers.

Exact self-similar solutions for axisymmetric wakes

This Note presents an analytical solution of two-dimensional axisymmetric wakes valid in the development region, before the ultimate equilibrium state. Based on the boundary layer equations in polar coordinates, assuming a small velocity defect, the problem reduces to a linear diffusive equation and can be expressed as an eigenvalue problem. Then a complete set of eigenfunctions is analytically obtained, which are damped and evolve self-similarly in space. The first mode corresponds to the Schlichtingʼs solution, in agreement with the downstream asymptotic behavior.

Chirped Self-Similar Solutions of a Generalized Nonlinear Schrödinger Equation

An improved homogeneous balance principle and an F-expansion technique are used to construct exact chirped self-similar solutions to the generalized nonlinear Schrödinger equation with distributed dispersion, nonlinearity, and gain coefficients. Such solutions exist under certain conditions and impose constraints on the functions describing dispersion, nonlinearity, and distributed gain function. The results show that the chirp function is related only to the dispersion coefficient, however, it affects all of the system parameters, which influence the form of the wave amplitude. As few characteristic examples and some simple chirped self-similar waves are presented.

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansion technique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation (1) ε u τ τ + u τ = ( a ( ξ ) u ξ ) ξ − | u | p − 1 u , ( ξ , τ ) ∈ R × R + , where ε > 0 , a ( ξ ) → 1 as | ξ | → + ∞ and 1 < p < 3 . We prove in this article that the exact self-similar solutions of the semi-linear parabolic equation obtained by setting ε = 0 and a ( ξ ) ≡ 1 in (1) are also asymptotically stable self-similar solutions of the Eq. (1). The proof of our result relies on various energy estimates rewritten in the variables ξ / τ , ln τ .

Theory of the energy separation of a compressible gas flow

Optimum gas parameters which ensure the maximum heat transfer across a flat plate separating two streams with different Mach numbers are found on the basis of an exact self-similar solution for a laminar boundary layer.

Self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients

We present a systematic analysis of the self-similar propagation of optical pulses within the framework of the generalized cubic-quintic nonlinear Schr\"odinger equation with distributed coefficients. By appropriately choosing the relations between the distributed coefficients, we not only retrieve the exact self-similar solitonic solutions, but also find both the approximate self-similar Gaussian-Hermite solutions and compact solutions. Our analytical and numerical considerations reveal that proper choices of the distributed coefficients could make the unstable solitons stable and could restrict the nonlinear interaction between the neighboring solitons.

Self-similar rupture of viscous thin films in the strong-slip regime

We consider rupture of thin viscous films in the strong-slip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and van der Waals forces are dominant and that there are self-similar solutions of the second kind. For a corresponding simplified model the self-similar behaviour is analysed rigorously. There exists a continuous one-parameter family of exact self-similar solutions. In a certain parameter regime necessary and sufficient conditions for convergence to a self-similar solution are established. A conjecture on the domains of attraction of all self-similar solutions is presented and supported by numerical simulations.

On Flows in Porous Media with Moving Boundaries Arising as Scaling Limits of Discrete Aggregation Models

We study porous medium (or, equivalently, Hele-Shaw) flows with a moving boundary which arise as scaling (i.e., continuous) limits of certain discrete aggregation models. Specifically, we study the scaling limit of the internal DLA model with a killing and a one-way passing condition for particles on the negative semi-axis. These models were recently studied by L. Levine and Y. Peres. We give an exact self-similar solution for the corresponding porous medium flow in the first case (which matches perfectly the numerical data obtained by L. Levine and Y. Peres), and derive moment properties of such solution in the second case.

Theory of self-similar propagation of two coupled optical pulses in nonlinear optical fiber amplifiers: Coexistence and separation

We present a systematic method to investigate the self-similar propagation of two coupled optical pulses in nonlinear optical fiber amplifiers. By taking into account the intrinsic self-similarity of the two coupled nonlinear Schr\"odinger equations, we cast them into two simple algebraic equations. We then show that the determining of the most stable power profiles for the self-similar evolution of each pulse is equivalent to the finding of the ground-state solution of binary Bose-Einstein condensate mixtures. Based on this, we find a family of asymptotically exact self-similar solutions, including the parabolic similaritons and the piecewise parabolic similaritons. Further, we propose an approximate method to determine the intermediate evolution of initial pulses toward these similaritons. Both the analytical and numerical results show that when the parameter of cross-phase modulation nonlinearity $b$ is smaller (larger) than 1, the two coupled pulses favor the coexistence (separation) state, while when $b=1$, the collisionless shock waves could emerge.

Modelling flow in a pressure-sensitive, heterogeneous medium

Modeling ow in a pressure-sensitive, heterogeneous medium D. W. Vasco 1 and Susan E. Minkoff 2 Berkeley Laboratory, University of California, Berkeley, California 94720 University of Maryland, Department of Mathematics and Statistics, 1000 Hilltop Circle, Baltimore, Maryland, 21250 SUMMARY Using an asymptotic methodology, including an expansion in inverse powers of ω, where ω is the frequency, we derive a solution for flow in a medium with pressure dependent properties. The solution is valid for a heterogeneous medium with smoothly varying properties. That is, the scale length of the heterogeneity must be significantly larger then the scale length over which the pressure increases from it initial value to its peak value. The resulting asymptotic expres- sion is similar in form to the solution for pressure in a medium in which the flow properties are not functions of pressure. Both the expression for pseudo-phase, which is related to the ’travel time’ of the transient pressure disturbance, and the expression for pressure amplitude contain modifications due to the pressure dependence of the medium. We apply the method to synthetic and observed pressure variations in a deforming medium. In the synthetic test we model one- dimensional propagation in a pressure-dependent medium. Comparisons with both an analytic self-similar solution and the results of a numerical simulation indicate general agreement. Fur- thermore, we are able to match pressure variations observed during a pulse test at the Coaraze Laboratory site in France. Key words: fluid flow, permeability, asymptotic methods, pressure 1 INTRODUCTION The introduction of a volume of fluid into the Earth will induce some degree of deformation within the geologic material compris- ing the subsurface. In most cases the resulting deformation is not significant, or even observable, and may be safely ignored. How- ever, in some situations, such as poorly consolidated sediments, fractured media (Gale 1975, Jones 1975, Noorishad et al. 1992, Cappa et al. 2008), and large pressure changes and flow rates (Fatt 1958, Raghavan et al. 1972, Rutqvist et al. 1998), the changes within the matrix may impact the flow in important ways. For ex- ample, large pressure changes can modify the flow properties such as porosity and permeability. Also, due to the transmission of stress within the medium, pressure changes can lead to non-local effects. A comprehensive approach to this problem involves coupled modeling of the fluid flow and the deformation. Such coupled mod- eling can be complicated, making numerical methods attractive. While the utility of numerical modeling is well established (Noor- ishad et al. 1992, Rutqvist et al. 2002, Minkoff et al. 2003, Minkoff et al. 2004, Dean et al.2006), analytic solutions can aid in our un- derstanding of the factors, such as medium parameters, contributing to the calculated pressure and deformation. Thus, analytic solutions may provide valuable insight, complementing existing purely nu- merical approaches. To date, analytic studies of the coupled prob- lem have been restricted to relatively simple cases, such as for a ho- mogeneous medium in which the flow properties do not change as a function of pressure. A classic example is a homogeneous poroe- lastic medium (Booker and Carter 1986, Rudnicki 1986) which can exhibit the non-local effects noted above (Segall 1985). Recently, a semi-analytic solution was developed for a poroelastic medium with heterogeneous flow properties which also contained non-local effects due to the coupling of the diffusive Biot wave and the ’fast’ elastic wave (Vasco 2008, 2009). While coupled modeling provides a more complete and satis- factory approach for understanding flow in a deformable medium, it is also useful to examine particular aspects of this problem. In fact, the studies just mentioned focused on the coupling between elastic deformation in the medium and the fluid flow, neglecting the change in flow properties due to changes in the effective stress. In this paper we examine the alternative situation, in which the flow properties are functions of the fluid pressure and we neglect the transmission of elastic deformation throughout the medium. As such, this paper is a complement to the studies cited above. Allowing for pressure dependent flow properties leads to a non-linear diffusion equation (Wu and Pruess 2000). The non- linear diffusion equation has been used to model the flow of flu- ids in a deformable medium (Barenblatt 1952), flow in rock joints with pressure-dependent openings (Murphy et al. 2004), tempera- ture and pressure waves in fluid saturated rock (Natale and Salusti 1996), and compaction in sedimentary basins (Audet and Fowler 1992) among other things (Newman 1983). The non-linear diffu- sion equation has been studied rather extensively from a mathe- matical perspective (Crank 1975, Hayashi et al. 2006), particularly for a homogeneous medium and the case that the permeability is proportional to the pressure raised to a power. In this instance one may derive an exact self-similar solution to the non-linear diffu-

The interior structure of slowly rotating black holes

The internal structure of a slowly rotating, steadily accreting, charged black hole that is undergoing mass inflation at its inner horizon is derived. The approach is to introduce a small rotational (axial dipole) perturbation to a spherically symmetric black hole. The equations governing the angular (dipole) behaviour decouple from the radial (monopole) behaviour, so all conclusions regarding inflation in the spherical charged black hole carry through unchanged for the slowly rotating black hole. Quantities inflate only in the radial direction, not in the angular direction. Exact self-similar solutions are obtained. For sufficiently small accretion rates, the instantaneous angular motion of the accretion flow has negligible effect on the angular spacetime structure of the black hole, even if the instantaneous angular momentum of the accretion flow is large and arbitrarily oriented.

Size distribution of convective thermals in an unstable stratified turbulent surface layer

An ensemble of convective thermals is considered in the surface layer of penetrative turbulent convection over a homogeneous heated horizontal surface. An integral model of an unsteady spontaneous jet having an exact self-similar solution is proposed to describe the dynamics of an isolated convective element. A statistical model for an ensemble of convective elements using a hydrodynamic analogy of the isolated spontaneous jet equations is suggested. It is supposed that motion of the elements of an ensemble corresponds to a statistic invariant that combines the squared velocity and the diameter of the jet. Using the combination of the statistic invariant of an ensemble and the Boltzmann distributions on squares of velocities, the size distribution of spontaneous jets in a convective surface layer of the atmosphere is constructed, which agrees with available experimental data.

Grey Self-similar Solitary Waves in Inhomogeneous Nonlinear Media

This paper analyzes spatial grey self-similar solitary waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities. New exact self-similar solutions are found using a novel transformation and their main features are investigated by using direct computer simulations.

Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S d - 1 . The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403–417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: <http://arxiv.org/abs/math-ph/0608035>]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, Journal of Statistical Physics 109 (2002) 407–432; S.J. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683–700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503–541(39)] and [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651–1682].

Exact analytic self-similar solution of a wave attractor field

Stratified and rotating fluids support obliquely propagating internal waves. A symmetry-breaking shape of the fluid domain focuses them on a wave attractor. For a trapezoidal basin, it is here shown how to determine the internal wave field analytically. This requires solving the wave equation on a closed domain–an ill-posed Cauchy problem–whose solution exhibits a remarkable self-similar spatial structure. These results are relevant for mixing and mean flow generation in oceans, atmospheres and stars whose symmetry is generally broken and where internal waves are tidally forced.