Shock-like Solutions Research Articles

Macroscopic Equity Markets Model: Towards Predicting Financial Crashes

This research examines the structural properties of the macroscopic model introduced in [AlShelahi and Saigal, 2018]. We present a theoretical analysis of the behavior of the macroscopic variables. In particular, we show that the model exhibits shock-like solutions, providing a new narrative for financial shocks. To solve the system of stochastic nonlinear partial differential equations adaptively, an integrative algorithm is devised and tested on abnormal and normal trading days. The results suggest that abnormalities can be identified before crashes. Our findings warrant further investigation into the macroscopic structure of equity markets.

Nonlinear dispersive regularization of inviscid gas dynamics

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in one dimension, singularities in the Hopf equation can be non-dissipatively smoothed via Korteweg–de Vries (KdV) dispersion. In this paper, we develop a minimal conservative regularization of 3D ideal adiabatic flow of a gas with polytropic exponent γ. It is achieved by augmenting the Hamiltonian by a capillarity energy β(ρ)(∇ρ)2. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy, and entropy using the standard Poisson brackets is β(ρ) = β*/ρ for constant β*. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of ρ, which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary waves, and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine–Hugoniot conditions. Nevertheless, in one-dimension, for γ = 2, numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves, which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocusing nonlinear Schrödinger equation (cubically nonlinear for γ = 2), with β* playing the role of ℏ2. Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV and nonlinear Schrödinger equations to include the adiabatic dynamics of density, velocity, pressure, and entropy in any dimension.

Solitary and shocklike wave solutions for the Gardner equation in dusty plasmas

The propagation of dust ion acoustic waves (DIAWs) for two-temperature ions with isothermal electrons in dusty plasmas has been investigated. Using the reductive perturbation method, the Gardner equation is derived. Many analytical solutions have been presented for small but finite amplitude acoustic waves, e.g. solitary and shocklike waves. The effects of many parameters on the feature of the shocklike waves have been studied for the dusty plasma system. The results of this study could be helpful for understanding the existence of nonlinear potential structures those are observed in various cosmic dust-laden plasmas, such as Saturn's E-ring, noctilucent clouds, Halley's comet and interstellar molecular clouds.

Shocklike soliton because of an impinge of protons and electrons solar particles with Venus ionosphere

This paper introduces an investigation of shocklike soliton or small amplitude Double Layers (DLs) in a collisionless plasma, consisting of positive and negative ions, nonthermal electrons, as well as solar wind streaming protons and electrons. Gardner equation is derived and its shocklike soliton solution is obtained. The model is employed to recognize a possible nonlinear wave at Venus ionosphere. The results indicate that the number densities and velocities of the streaming particles play crucial role to determine the polarity and characteristic features (amplitude and width) of the shocklike soliton waves. An electron streaming speed modifies a negative shocklike wave profile, while an ion streaming speed modulates a positive shocklike wave characteristic.

CANARDS IN ADVECTION–REACTION–DIFFUSION SYSTEMS IN ONE SPATIAL DIMENSION

This thesis contains a mathematical investigation of the existence of travelling wave solutions to singularly perturbed advection-reaction-diffusion models of biological processes. An enhanced mathematical understanding of these solutions and models is gained via the identification of canards (special solutions of fast/slow dynamical systems) and their role in the existence of the most biologically relevant, shock-like solutions. The analysis focuses on two existing models. A new proof of existence of a whole family of travelling waves is provided for a model describing malignant tumour invasion, while new solutions are identified for a model describing wound healing angiogenesis.

Fixed points of a destabilized Kuramoto–Sivashinsky equation

We consider the family of destabilized Kuramoto–Sivashinsky equations in one spatial dimension ut+νuxxxx+βuxx+γuux=αu for α, ν ≥ 0 and β,γ∈R. For certain parameter values, shock-like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self-consistent bounds and validated numerics.

A finite volume approach for the simulation of nonlinear dissipative acoustic wave propagation

A form of the conservation equations for fluid dynamics is presented, deduced using slightly less restrictive hypothesis than those necessary to obtain the Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A two-dimensional finite volume method using the Roe linearization was implemented to obtain numerically the solution of the proposed equations. In order to validate the code, two different tests have been performed: one against a special Taylor shock-like analytic solution, the other against published results on a High Intensity Focused Ultrasound (HIFU) system, both with satisfactory results. The code, available under an open source license, is written for parallel execution on a Graphics Processing Unit (GPU), thus improving performance by a factor of over 60 when compared to the standard serial execution finite volume code CLAWPACK 4.6.1, which has been used as reference for the implementation logic as well.

A nonlinear model for magnetoacoustic waves in dense dissipative plasmas with degenerate electrons

The properties of nonlinear fast magnetoacoustic waves in dense dissipative plasmas with degenerate electrons are studied theoretically in the framework of the Zabolotskaya-Khokhlov (ZK) equation for small but finite amplitude excitations. Shock-like solutions of the ZK equation are obtained and are applied to parameters relevant to white dwarf stars.

A numerical scheme based on compact integrated-RBFs and Adams–Bashforth/Crank–Nicolson algorithms for diffusion and unsteady fluid flow problems

Abstract This paper presents a high-order approximation scheme based on compact integrated radial basis function (CIRBF) stencils and second-order Adams–Bashforth/Crank–Nicolson algorithms for solving time-dependent problems in one and two space dimensions. We employ CIRBF stencils, where the RBF approximations are locally constructed through integration and expressed in terms of nodal values of the function and its derivatives, to discretise the spatial derivatives in the governing equation. We adopt the Adams–Bashforth and Crank–Nicolson algorithms, which are second-order accurate, to discretise the temporal derivatives. The performance of the proposed scheme is investigated numerically through the solution of several test problems, including heat transfer governed by the diffusion equation, shock wave propagation and shock-like solution governed by the Burgers' equation, and torsionally oscillating lid-driven cavity flow governed by the Navier–Stokes equation in the primitive variables. Numerical experiments show that the proposed scheme is stable and high-order accurate in reference to the exact solution of analytic test problems and achieves a good agreement with published results for other test problems.

Dynamics of the Zakharov-Kuznetsov-Burgers equations in dusty plasmas

In this paper, we investigate the Zakharov-Kuznetsov-Burgers (ZKB) equations for the dust-ion-acoustic waves in dusty plasmas. Shock-like and soliton solutions are both constructed through the introduction of an auxiliary function and variable. ZKB-soliton propagation is plotted, and from those figures, we find that energy of the solitons increases when the number of electrons in a dust particle decreases or the mass of such dust particle becomes larger. Considering the external perturbations in the dusty plasmas, we study the perturbed ZKB equation via some qualitative and quantitative methods. To corroborate that the perturbed ZKB equation can indeed give rise to the chaos, we make use of the power spectrum and Lyapunov exponents. Then, we investigate the phase projections, and find that both the weak and developed chaos can be observed. Weak chaos occur when the absolute value of damped coefficient (l1) is stronger than the strength of perturbed term (g1), or else, the developed one occurs. Ranges of l1 and g1 are given via the largest Lyapunov exponents when the perturbed ZKB equation is in different chaotic states. Therefore, we can find that the chaotic motion of the perturbed ZKB equation will be enhanced with the number of electrons in a dust particle or the mass of such a dust particle decreasing.

Dust-acoustic shock formation in dusty plasmas with non-thermal ions

In this study, the nonlinear Burgers equation in the presence of the dust charge fluctuation is derived and the shock-like solution is determined. It is well known that in order to have a monotonic or oscillatory shock wave, a source of dissipation is needed. By using the experimental data reported in the laboratory observation of self-excited dust-acoustic shock waves [Heinrich et al., Phys. Rev. Lett. 103, 115002 (2009)], it is shown that dust charge fluctuation can be considered as a candidate for the source of dissipation needed for the dust-acoustic shock formation. By examining the effects of non-thermal ions on dust-acoustic shock's characteristics, a possible theoretical explanation for the discrepancies observed between theory and experiment is proposed.

Creating sharp features by colliding shocks on uniformly irradiated surfaces

Using a theoretical analysis of the ion beam sputtering dynamics, we demonstrate how ion bombardment on an initially sloped surface can create knife-edge-like ridges on the surface. These ridges arise as nonclassical shocklike solutions that are undercompressive on both sides and appear to control the dynamics over a large range of initial conditions. The slope of the ridges is selected uniquely by the dynamics and can be up to 30 or more depending on the orientation dependence of the sputtering yield. For 1 keV Ar${}^{+}$ on Si(001), the scale of the ridge is $\ensuremath{\approx}\phantom{\rule{-0.16em}{0ex}}2$ nm. This is much smaller than the most unstable length scale and suggests a method for creating very steep, very sharp features on a surface spontaneously, by prepatterning the surface to contain relatively modest slopes on the macroscale.

Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation

Nonlinear Burgers' equation is solved using polynomial based differential quadrature method (PDQ). Numerical simulations are studied for three well known test problems, namely shock-like solution, travelling wave and sinusoidal disturbance solutions of Burgers' equation. Obtained numerical results of the first and the third test problems are compared with some earlier numerical results. Discrete root mean square error norm and maximum error norm are computed for the first two test problems and a comparison with some earlier works is given.

Dust-acoustic shock formation in adiabatic hot dusty plasmas with variable charge

Shocklike structure can be formed after a certain transient time due to the self-steepening of the negative potential. In order to have a monotonic or oscillatory shock wave, it is known that a source of dissipation is needed. In this study, we considered the variation of dust charge as a source of dissipation. By using the reductive perturbation technique, the nonlinear Burgers equation is derived and the shocklike solution is determined. The effects of dust temperature on different characteristics of dust-acoustic shock structure are discussed. It is found out that the dust thickness is not affected by dust temperature. By considering a dusty plasma system with a set of parameters, it is shown that there exists a specific dust critical temperature Tdc which gives maximum height for the dust-acoustic shock structure. The effects of the plasma species temperature on shock formation are also investigated.

Dust ion-acoustic shock waves in charge varying dusty plasmas with electrons having vortexlike velocity distribution

A weakly nonlinear analysis is carried out to investigate the properties of dust ion-acoustic shock waves in a charge varying dusty plasma with vortexlike electron distribution. We use the ionization model, hot ions with equilibrium streaming speed and a trapped electron charging current derived from the well-known orbit limited motion theory. A new modified Burger equation is derived. Besides nonlinear trapping, this equation involves two kinds of dissipation (the anomalous one inherent to nonadiabatic dust charge fluctuation and the one due to the particle loss and ionization). These two kinds of dissipation can act concurrently. The traveling wave solution has been acquired by employing the modified extended tanh-function method. The shocklike solution is numerically analyzed based on the typical numerical data from laboratory dusty plasma devices. It is found that ion temperature, trapped particles, and weak dissipations significantly modify the shock structures.

Shock-wave solutions in two-layer channel flow. I. One-dimensional flows

We study the dynamics of an interface separating two immiscible layers in an inclined channel. Lubrication theory is used to derive an evolution equation for the interface position that models the two-dimensional flow in both co- and countercurrent configurations. This equation is parameterized by viscosity and density ratios, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are treated as additional parameters. In the present work, which corresponds to part I of a two-part paper, we focus on one-dimensional flows. We use an entropy-flux analysis to delineate the existence of various types of shocklike solutions, which include compressive Lax shocks, pairs of Lax and under-compressive shocks, and rarefaction waves. Flows characterized by unstably stratified layers are accompanied by the formation of propagating, large-amplitude interfacial waves, which are not shocklike in nature. The results of our transient numerical simulations accord with our analytical predictions and elucidate the mechanisms underlying spatio-temporal development of the various types of waves; the stability of these waves to spanwise perturbations is investigated in part II of this work.

Obliquely propagating shock-like structures and nonlinear solitary holes in weakly relativistic magneto-plasma

A theoretical investigation is carried out for understanding the properties of obliquely propagating shock-like structures in weakly relativistic magneto-plasma. By using the Sagdeev's pseudo-potential method, we have found the obliquely propagating shock-like solution and the relation between the amplitude, the inverse scale length, the relativistic effects and the effects of obliqueness. It is shown that the shock-like structure is nonlinear extension of the solitary hole having negative trapping parameter in weakly relativistic magneto-plasma.

Ion-acoustic shock waves undergoing Landau damping

The Korteweg–de Vries equation with an additional term of Landau damping is numerically and analytically investigated. It is shown that the equation has a shock-like solution for an initial ramp signal. The temporal evolution of waveforms with various magnitudes of the Landau damping is studied for several values of the initial amplitude. Dependences of widths and velocities of the leading part on initial conditions are shown. It is found that a steepening is suppressed due to the Landau damping even when its coefficient is two orders less than those of nonlinear and dispersive terms. There is a critical relation for such a steepening to take place for a fixed height of the initial ramp. An analytical estimate of the magnitude of temporal Landau damping is given for a linear sinusoidal wave.

Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods

Strong-stability-preserving (SSP) time discretization methods (also known as total-variation-diminishing or TVD methods) are popular and effective algorithms for the simulation of partial differential equations having discontinuous or shock-like solutions. Optimal SSP Runge–Kutta (SSPRK) schemes have been previously found for methods with up to five stages and up to fourth order. In this paper, we present new optimal fourth-order SSPRK schemes with mild storage requirements and up to eight stages. We find that these schemes are ultimately more efficient than the known fourth-order SSPRK schemes because the increase in the allowable time-step more than offsets the added computational expense per step. We demonstrate these efficiencies on a pair of scalar conservation laws.

Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto–Sivashinsky equation

We investigate an extension of the Kuramoto–Sivashinsky (KS) model for complex spatiotemporal dynamics, containing an additional linear stabilizing or destabilizing term. Generalizing previous results, we prove dissipativity and analyticity. In the destabilized case, a stable, attracting shock-like transition layer solution is observed numerically, and we obtain its asymptotic scaling. This “viscous shock” solution does not exhibit extensive scaling, which sheds light on the difficulties in obtaining optimal bounds and proving extensivity for the KS equation.