Shock Wave Solutions Research Articles

Shock waves: The Maxwell-Cattaneo case.

Several continuum theories for shock waves give rise to a set of differential equations in which the analysis of the underlying vector field can be done using the tools of the theory of dynamical systems. We illustrate the importance of the divergences associated with the vector field by considering the ideas by Maxwell and Cattaneo and apply them to study shock waves in dilute gases. By comparing the predictions of the Maxwell-Cattaneo equations with shock wave experiments we are lead to the following conclusions: (a) For low compressions (low Mach numbers: M) the results from the Maxwell-Cattaneo equations provide profiles that are in fair agreement with the experiments, (b) as the Mach number is increased we find a range of Mach numbers (1.27 ≈ M(1) < M < M(2) ≈ 1.90) such that numerical shock wave solutions to the Maxwell-Cattaneo equations cannot be found, and

Shock wave solution of the variable coefficient nonlinear partial differential equations

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)‐dimensional variable coefficient Kadomtsev–Petviashvili equation with time‐dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Similarity Solution of Spherical Shock Waves -Effect of Viscosity

In this paper we investigated self-similar Solutions for Magneto Hy-drodynamic shock waves for the equation of state of Mie-Gruneisen type. Solutions are obtained numerically and the effect of viscosity (K) and the non-idealness parameter (d) on the self-similar solutions are studied in detail. The findings confirmed that, the non-idealness parameter and the viscosity parameter have major effect on the shock strength and the flow variables. All discontinuities of the physical pa-rameters are removed by the viscosity and complete flow field depends upon the magnitude of the viscosity. The obtained results are in good agreement with the results obtained by some of the researchers. All the analysis is presented pictorially in this paper.

On critical behaviour in generalized Kadomtsev–Petviashvili equations

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves.

Properties of cylindrical and spherical heavy ion-acoustic solitary and shock structures in a multispecies plasma with superthermal electrons

A theoretical investigation on heavy ion-acoustic (HIA) solitary and shock structures has been accomplished in an unmagnetized multispecies plasma consisting of inertialess kappa-distributed superthermal electrons, Boltzmann light ions, and adiabatic positively charged inertial heavy ions. Using the reductive perturbation technique, the nonplanar (cylindrical and spherical) Kortewg–de Vries (KdV) and Burgers equations have been derived. The solitary and shock wave solutions of the KdV and Burgers equations, respectively, have been numerically analyzed. The effects of superthermality of electrons, adiabaticity of heavy ions, and nonplanar geometry, which noticeably modify the basic features (viz. polarity, amplitude, phase speed, etc.) of small but finite amplitude HIA solitary and shock structures, have been carefully investigated. The HIA solitary and shock structures in nonplanar geometry have been found to distinctly differ from those in planar geometry. Novel features of our present attempt may contribute to the physics of nonlinear electrostatic perturbation in astrophysical and laboratory plasmas.

Geometric Numerical Integration for Peakon b-Family Equations

AbstractIn this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however they also have the significant difference, for example there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

“Regularity singularities” and the scattering of gravity waves in approximate locally inertial frames

Abstract. It is an open question whether solutions of the EinsteinEuler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether “regularity singularities” can exist at such points. The term regularity singularity was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous (C), but no smoother, in any coordinate system of the C atlas. An existence theory for shock wave solutions in C admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but C is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves, within the larger C atlas. Our purpose here is to clarify and motivate the open problem of regularity singularities, and to prove that if locally inertial coordinates do not lie within the C atlas, then the scattering of gravitational radiation by a regularity singularity produces quantifiable physical e↵ects analogous to non-removable Coriolis forces.

Bäcklund transformation and N-shock-wave solutions for a (3+1)-dimensional nonlinear evolution equation

In this paper, a (3+1)-dimensional nonlinear evolution equation is investigated, which can be used to describe reacting mixtures and shallow water waves. Through the Hirota method and symbolic computation, bilinear forms and Backlund transformation are derived, which are different from those in the existing literature. Moreover, N-shock-wave solutions are obtained. Based on those shock-wave solutions, propagation and collision of the shock waves are discussed via the asymptotic and graphic analysis on different planes: (1) oblique elastic collisions between/among the two/three shock waves will arise on the x–y and y–z planes, while parallel elastic collisions exist on the x–z plane; (2) shock waves maintain their original directions, amplitudes and velocities except for some small phase shifts after each collision; (3) the shock wave with higher amplitude travels faster and moves across the slower.

Similarity Solutions of Cylindrical Shock Waves in Magnetogasdynamics With Thermal Radiation

Using Lie group of transformations, here we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of an ideal gas in the presence of radiative cooling and idealized azimuthal magnetic field. The similarity solutions are investigated behind a cylindrical shock wave which is produced as a result of a sudden explosion or driven out by an expanding piston. The shock is assumed to be strong and propagates into a medium which is at rest, with nonuniform density. The total energy of the wave is assumed to be time dependent obeying a power law. Indeed, with the use of the similarity solution, the problem is transformed into a system of ordinary differential equations (ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

Uniqueness of rarefaction waves in multidimensional compressible Euler system

We show that 1D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

Constraining the pickup ion abundance and temperature through the multifluid reconstruction of the Voyager 2 termination shock crossing

AbstractVoyager 2 observations revealed that the hot solar wind ions (the so‐called pickup ions) play a dominant role in the thermodynamics of the termination shock and the heliosheath. The number density and temperature of this hot population, however, have remained unknown, since the plasma instrument on board Voyager 2 can only detect the colder thermal ion component. Here we show that due to the multifluid nature of the plasma, the fast magnetosonic mode splits into a low‐frequency fast mode and a high‐frequency fast mode. The coupling between the two fast modes results in a quasi‐stationary nonlinear wave mode, the “oscilliton,” which creates a large‐amplitude trailing wave train downstream of the thermal ion shock. By fitting multifluid shock wave solutions to the shock structure observed by Voyager 2, we are able to constrain both the abundance and the temperature of the undetected pickup ions. In our three‐fluid model, we take into account the nonnegligible partial pressure of suprathermal energetic electrons (0.022–1.5 MeV) observed by the Low‐Energy Charged Particle Experiment instrument on board Voyager 2. The best fitting simulation suggests a pickup ion abundance of 20 ± 3%, an upstream pickup ion temperature of 13.4 ± 2 MK, and a hot electron population with an apparent temperature of ~0.83 MK. We conclude that the actual shock transition is a subcritical dispersive shock wave with low Mach number and high plasma β.

Positron-Acoustic Shock Waves in a Degenerate Multi-Component Plasma

A theoretical investigation on the propagation of positron-acoustic shock waves (PASWs) in an unmagnetized, collisionless, dense plasma (containing non-relativistic inertial cold positrons, non-relativistic or ultra-relativistic degenerate electron and hot positron fluids and nondegenerate positively charged immobile ions) is carried out by employing the reductive perturbation method. The Burgers equation and its stationary shock wave solution are derived and numerically analyzed. It is observed that the relativistic effect (i.e., the presence of non/ultra-relativistic electrons and positrons) and the plasma particle number densities play vital roles in the propagation of PASWs. The implications of our results in space and interstellar compact objects including non-rotating white dwarfs, neutron stars, etc. are briefly discussed.

Shock wave solution in a hot adiabatic dusty plasma having negative and positive non-thermal ions with trapped electrons

In this report an investigation of the properties of a dust acoustic (DA) shock wave propagating in an adiabatic dusty plasma, including the effect of the negative-ion-rich non-thermal ions and trapped electrons, is presented. The reductive perturbation method is employed to derive the modified Burgers equation and a new form of the lower-order nonlinear modified Burgers equation for DA shock waves in a homogeneous, unmagnetized and collisionless plasma whose constituents are electrons, singly-charged positive ions, singly-charged negative ions and massive, charged dust particles. The stationary analytical solution of the Burgers equation and the new analytical solution of the lower-order nonlinear modified Burgers equation are numerically analyzed, and the effects of various dusty plasma constituents on the DA shock wave’s propagation are taken into account. Both positive and negative ions in the dusty plasma are observed to play a key role in the formation of both positive, as well as negative, DA shock waves, and the ion concentration can be used to control the transformation from negative to positive potentials of the waves.

Shock wave solution for a class of nonlocal nonlinear singularly perturbed boundary value problems with turning point

The nonlocal nonlinear boundary value problem with turning point is considered. Leading the stretched variables the formal asymptotic solution of the original problem is constructed. And by using the theory of differential inequalities the existence and uniformly validity of solution for the original boundary value problems are proved. The contribution of this paper is that, the asymptotic behavior of solution is studied by using a simple and special method.

Solitary and shock structures in a strongly coupled cryogenic quantum plasma

The quantum ion-acoustic (QIA) solitary and shock structures formed in a strongly coupled cryogenic quantum plasma (containing strongly coupled positively charged inertial cold ions and Fermi electrons as well as positrons) have been theoretically investigated. The generalized quantum hydrodynamic model and the reductive perturbation method have been employed to derive the Korteweg-de Vries (K-dV) and Burgers equations. The basic features of the QIA solitary and shock structures are identified by analyzing the stationary solitary and shock wave solutions of the K-dV and Burgers equations. It is found that the basic characteristics (e.g., phase speed, amplitude, and width) of the QIA solitary and shock structures are significantly modified by the effects of the Fermi pressures of electrons and positrons, the ratio of Fermi temperature of positrons to that of electrons, the ratio of effective ion temperature to electron Fermi temperature, etc. It is also observed that the effect of strong correlation among extremely cold ions acts as a source of dissipation, and is responsible for the formation of the QIA shock structures. The results of this theoretical investigation should be useful for understanding the nonlinear features of the localized electrostatic disturbances in laboratory electron-positron-ion plasmas (viz., super-intense laser-dense matter experiments).

Revisiting the method of characteristics via a convex hull algorithm

We revisit the method of characteristics for shock wave solutions to nonlinear hyperbolic problems and we propose a novel numerical algorithm—the convex hull algorithm (CHA)—which allows us to compute both entropy dissipative solutions (satisfying all entropy inequalities) and entropy conservative (or multi-valued) solutions. From the multi-valued solutions determined by the method of characteristics, our algorithm “extracts” the entropy dissipative solutions, even after the formation of shocks. It applies to both convex and non-convex flux/Hamiltonians. We demonstrate the relevance of the proposed method with a variety of numerical tests, including conservation laws in one or two spatial dimensions and problem arising in fluid dynamics.

Linear and nonlinear analysis of dust acoustic waves in dissipative space dusty plasmas with trapped ions

The propagation of linear and nonlinear dust acoustic waves in a homogeneous unmagnetized, collisionless and dissipative dusty plasma consisted of extremely massive, micron-sized, negative dust grains has been investigated. The Boltzmann distribution is suggested for electrons whereas vortex-like distribution for ions. In the linear analysis, the dispersion relation is obtained, and the dependence of damping rate of the waves on the carrier wave number \(k\), the dust kinematic viscosity coefficient \(\eta _{d}\) and the ratio of the ions to the electrons temperatures \(\sigma _{i}\) is discussed. In the nonlinear analysis, the modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation is derived via the reductive perturbation method. Bifurcation analysis is discussed for non-dissipative system in the absence of Burgers term. In the case of dissipative system, the tangent hyperbolic method is used to solve mKdV–Burgers equation, and yield the shock wave solution. The obtained results may be helpful in better understanding of waves propagation in the astrophysical plasmas as well as in inertial confinement fusion laboratory plasmas.

Planar and Nonplanar Shock Waves in a Degenerate Quantum Plasma

AbstractThe nonlinear propagation of modified electron‐acoustic (mEA) shock waves in an unmagnetized, collisionless, relativistic, degenerate quantum plasma (containing non‐relativistic degenerate inertial cold electrons, both nonrelativistic and ultra‐relativistic degenerate hot electron and inertial positron fluids, and positively charged static ions) has been investigated theoretically. The well‐known Burgers type equation has been derived for both planar and nonplanar geometry by employing the reductive perturbation method. The shock wave solution has also been obtained and numerically analyzed. It has been observed that the mEA shock waves are significantly modified due to the effects of degenerate pressure and other plasma parameters arised in this investigation. The properties of planar Burgers shocks are quite different from those of nonplanar Burgers shocks. The basic features and the underlying physics of shock waves, which are relevant to some astrophysical compact objects (viz. non‐rotating white dwarfs, neutron stars, etc.), are briefly discussed. (© 2015 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

No regularity singularities exist at points of general relativistic shock wave interaction between shocks from different characteristic families.

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C0,1 to C1,1 in a neighbourhood of points of shock wave collision in general relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier Proc. R. Soc. A publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Thus, our result implies that regularity singularities (a type of mild singularity introduced in our Proc. R. Soc. A paper) do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original Proc. R. Soc. A publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein-Euler equations remains open.

Traveling waves for conservation laws with cubic nonlinearity and BBM type dispersion

Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax–Oleinik construction. We consider dispersive terms of a BBM type and show that the calculation of traveling waves is somewhat more intricate than for a KdV-type dispersion. The explicit calculation is based upon the calculation of parabolic invariant manifolds for the associated ODE describing traveling waves. The results extend to the p-system of one-dimensional elasticity with a cubic stress–strain law.