A new type of selfsimilarity is found in the problem of a planeparallel, ultrarelativistic blastwave, propagating in a powerlaw density profile of the form $ \rho \u221d z \u2212 k$. Selfsimilar solutions of the first kind can be found for k < 7∕4 using dimensional considerations. For steeper density gradients with k > 2, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate powerlaw indices $ 7 / 4 < k < 2$, the flow does not obey any of the known types of selfsimilarity. Instead, the solutions belong to a new class in which the selfsimilar dynamics are dictated by the nonselfsimilar part of the flow. We obtain an exact solution to the ultrarelativistic fluid equations and find that the nonselfsimilar flow is described by a relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of a fast material that coincides with the interiors of the blastwave and terminates at the shock. The dynamics of the piston itself are selfsimilar and universal and do not depend on the external medium. The exact solution of the nonselfsimilar flow is used to solve for the shock in the new class of solutions.
I. INTRODUCTION
Selfsimilar solutions offer great mathematical simplicity by reducing a set of partial differential equations (PDEs) to a set of ordinary differential equations (ODEs). They are used to characterize the asymptotic behavior of a physical system when its initial scales become unimportant and can sometimes be described by analytical solutions. Selfsimilar solutions are traditionally classified into one of two types: when the selfsimilar properties of the system can be found from dimensional considerations, the solutions are of first type. Second type solutions are obtained when one is required to solve an eigenvalue problem.^{1,2}
A canonical example for selfsimilarity in hydrodynamics is the strong explosion problem. Consider a powerlaw density profile of the form $ \rho \u221d r \u2212 k$, where r is the distance from the origin and k is a constant. The release of a large amount of energy at r = 0 produces a strong shock wave that scales with time like $ R \u221d t \alpha $, where α is a function of k. Dimensional analysis can be used when the energy is a relevant parameter of the selfsimilar flow, such that α is found by enforcing energy conservation. First type solutions for Newtonian spherically symmetric explosions were originally obtained by Taylor,^{3} von Neumann,^{4} and Sedov.^{5} Waxman and Shvarts^{6} showed that when the density falls sufficiently fast (k > 3) dimensional considerations give a wrong temporal scaling for the shock as it accelerates and loses causal contact with the bulk of the flow. Second type solutions are then obtained for k > 3.26 by eliminating a singularity from the equations, corresponding to a smooth crossing of the sonic line. Waxman and Shvarts^{6} concluded that when $ 3 < k < 3.26$ the solutions cannot be described by either of the known types of selfsimilarity. Gruzinov^{7} subsequently showed that the solutions in that parameter space belong to a new class (see also Kushnir and Waxman^{8}). The uniqueness of the new class will be explained when discussing its ultrarelativistic analogue.
The paper is organized as follows: after deriving the selfsimilar equations in Sec. II, we give an overview of first and second type solutions in the ultrarelativistic blastwave problem in Sec. III. We then study the asymptotic behavior of the flow at large distances behind the shock in Sec. IV and obtain exact solutions for expansion into vacuum in Sec. V. The selfsimilar scaling of the shock in new class of solutions is found in Sec. VI. We compare to numerical simulations in Sec. VII and summarize in Sec. VIII.
II. THE SELF SIMILAR EQUATIONS
The temporal scalings of $ \Gamma ( t )$ and P(t) are given by the ultrarelativistic jump conditions across the shock, such that taking $ g ( 1 ) = f ( 1 ) = h ( 1 ) = 1$ implies $ d \u2009 log \u2009 \Gamma 2 / d \u2009 log \u2009 t = \u2212 m$ and $ d \u2009 log \u2009 P / d \u2009 log \u2009 t = \u2212 m \u2212 k$. Substituting the definitions in Eqs. (7)–(8b) into equation set (6), we obtain the following selfsimilar equations:
III. OVERVIEW OF FIRST AND SECOND TYPE SELFSIMILAR SOLUTIONS
A. First type solutions
B. Second type solutions
The solutions in this case are implicit in $ g \chi $,
In order to study energy convergence, we obtain in Sec. IV the asymptotic properties of the flow in the far downstream of the shock.
IV. POWERLAW ASYMPTOTIC AT $ \chi \u226b 1$
Second type solutions exist only when the solution crosses the sonic line, i.e., $ g \chi as < g \chi +$. Equations (14) and (21) show that this happens only for k > 2, such that $ g \chi as = 4 + 2 3 ( 1 \u2212 k )$.
Although m and Q are unknown when $ 7 / 4 < k < 2$, upper and lower limits can be set on m. The asymptotic solution necessarily satisfies $ g \chi as = max { g \chi 1 , g \chi 2}$ and also $ g \chi as > g \chi +$. These two conditions give $ m \u2265 ( 3 \u2212 2 3 ) k$. Another restriction comes from the requirement that $ D \u2265 0$, which is satisfied for $ m \u2264 \u2212 4 3 ( k \u2212 1 ) + 3 k$ or $ m \u2265 4 3 ( k \u2212 1 ) + 3 k$. Only the first inequality is consistent with a continuous solution for m as a function of k. Therefore, m must lie in within $ ( 3 \u2212 2 3 ) k \u2264 m \u2264 \u2212 4 3 ( k \u2212 1 ) + 3 k$. These limits imply that inside the gap Q > 1 and the energy diverges asymptotically at $ x \u2192 \u221e$. One, therefore, faces a new challenge: the energy diverges, while a sonic point is not part of the solution. This necessarily means that the selfsimilar dynamics are dictated by the nonselfsimilar part of the flow, which contains most of the energy. In order to study the dynamics of the shock, one must first solve for the nonselfsimilar flow.
A clue to how one should address the problem comes from the fact that when the pressure increases behind the shock, the asymptotic solution satisfies $ p ( 0 ) = 0$ and $ q ( 0 ) \u2192 \u221e$. This kind of behavior is expected in the leading edge of a relativistically hot gas that expands into vacuum. We, therefore, expect the nonselfsimilar flow to follow expansion into vacuum and look for exact solutions that coincide with the power law asymptotic of Eq. (24) at $ x \u2192 0$. We impose initial conditions in which the energy in the flow converges, and then check what kind of shocks can be driven by these flows. The similarity index m will then be found from the condition for the coexistence of the shock with the fluid that expands into vacuum behind it. The associated selfsimilar solutions will naturally be consistent with energy conservation and convergence.
V. ULTRARELATIVISTIC EXPANSION INTO VACUUM
In this section, we obtain exact solutions describing the expansion of an ultrarelativistic gas into vacuum. Our goal is to find solutions in which the flow approaches the powerlaw asymptotic [(Eq. (24)] expected to appear at $ x \u2192 0$, while deviating from it at some large x in order to comply with energy convergence.
In the first stage, we reduce the two fluid equations (6) to a single PDE by performing a hodograph transformation. We proceed by finding solutions that approach the powerlaw asymptotic at $ x \u2192 0$, and in the final stage, construct a solution that satisfies specific boundary conditions imposed by initial profiles with finite energy.
A. Hodograph transformation
B. Solutions of the Klein–Gordon equation
The problem defined by Eq. (33) with boundary conditions on $ C \xb1$ characteristics (corresponding to fixed values of u and v) is a well posed problem, which can be translated to an initial condition problem in p and q. The solution to Eq. (33) is then a superposition of $ \psi pl$ and $ F ( u \lambda , v / \lambda )$ [and any other additional solution of Eq. (33)] that satisfies the boundary conditions. We obtain the solution that develops from specific initial profiles in Subsection V C.
C. Solution for specific initial conditions
The onset of perturbations to the initial profiles occurs upon the arrival of limiting $ C \xb1$ characteristics that emerged from x_{0} at the initial time t_{0}. The $ C +$ characteristic along which u = 0 propagates within the small x powerlaw asymptotic, while the $ C \u2212$ characteristic carrying the value v = 0 propagates within the largerx asymptotic. As illustrated in Fig. 1, the perturbed flow lies in the region between the two characteristics, while outside of it the fluid retains its original profiles. The boundary conditions of Eq. (33) are, therefore, defined on the limiting $ C \xb1$ characteristics that emerge from x_{0}. Equation (33) needs to be solved only in the first quadrant, where $ u , v > 0$; everywhere else the flow is unperturbed and the solution is given by Eq. (39).
We use Eqs. (31) and (41) with $ x 0 = 1$, λ = 5, and $ \mu = 2.8$ to solve for u(v) at time $ t = 10 80$. Figure 2 shows that in the limits $ u \u226a v / \lambda 2$ and $ u \u226b v / \mu 2$ the exact solution coincides with those of the small and large x asymptotics, respectively. Corresponding profiles of p(x) and q(x) are shown in Fig. 3 using Eqs. (28), (29), (31), and (32).
D. Ultrarelativistic piston
The width of the pressure maximum on the {u, v} plane can be estimated by $ \Delta v p = ( p d 2 p / d v 2 ) 1 / 2  v p$, yielding a narrowing relative width of $ \Delta v p / v p = 2 ( 2 \u2212 3 ) \u2009 ln \u2009 ( t ) \u2212 1 / 2$ and $ \Delta u p / u p = 2 ( 2 + 3 ) \u2009 ln \u2009 ( t ) \u2212 1 / 2$. The two characteristic lines $ u = v / \lambda 2$ and $ u p ( v )$ are shown in Fig. 4.
Further intuition about the nature of the piston can be gained by a heuristic description of a hot, relativistic blob that undergoes free expansion, while exchanging internal energy with bulk kinetic energy. Let us associate $ \Gamma b$ with the bulk motion of the blob and $ \gamma th$ with the random motion of its particles (thermal energy). Since $ E \u223c \Gamma b ( \gamma th m c 2 ) = constant$ due to energy conservation, we have $ \gamma th \u221d \Gamma b \u2212 1$. Adiabatic expansion implies $ \gamma th m c 2 \u221d ( \Gamma b \Delta ) \u2212 1 / 3$, where $ \Delta \u221d t / \Gamma b 2$ is the volume of the expanding blob in the lab frame. This gives $ \Gamma b 2 \u221d t 1 / 2$. The scaling of the pressure can be found from energy conservation: $ E \u223c \Gamma b 2 p ( t / \Gamma b 2 ) = p \xb7 t = const \u2192 p \u221d t \u2212 1$. The evolution of the pressure is inferred solely from energy conservation and, therefore, agrees with our solution for the piston. However, this analysis returns a different scaling for q, which immediately implies that the expansion of the piston is not adiabatic. We conclude that the piston's entropy is not conserved, which suggests that it is not occupied by the same material throughout its evolution.
VI. CONSISTENCY WITH A SHOCK
In Sec. V, we obtained an exact solution for relativistic gas expanding into vacuum, whose leading front approaches the powerlaw asymptotic expected to form in the far downstream of a strong shock. Our objective is to smoothly connect the rear part of the selfsimilar blastwave solution with the leading front of expansion into vacuum. In order for both solutions to coexist with one another, one must find the appropriate values of the temporal index m for which the powerlaw asymptotic that forms by the piston at $ u , v > 0$ is not destroyed by the shock.
One might expect that the interpretation of a shock driven by expansion into vacuum continues to be correct also in the regime of second type solutions, as the external density gradient becomes even steeper. Indeed, in solving the inequality in Eq. (52), we have made the assumption that λ is finite. However, in second type solutions, all $ C +$ characteristics originate from the immediate vicinity of the sonic point, and therefore, u = constant in space. This corresponds to taking the limit $ \lambda \u2192 \u221e$ for which Eq. (53) returns $ m = m II = ( 3 \u2212 2 3 ) k$. The analysis in this paper, thus, applies to all k > 7∕4.
A characteristic plot summarizing the main features of the flow is shown in Fig. 5. A fluid element that crossed the shock at an arbitrary time t_{0} joins the asymptotic selfsimilar flow designated by the red region between the lines $ x sh$ and $ x pl$. As it is advected away from the shock, it leaves the selfsimilar flow after crossing $ x pl$ and eventually joins the bulk of the fluid at the piston when reaching $ x p$. A pair of $ C \xb1$ characteristics are also plotted to demonstrate that the flow is in causal contact. The position of the shock and the boundary of the powerlaw asymptotic have the same scaling and, therefore, the distance between them increases with time.
VII. DIRECT NUMERICAL SIMULATION
We run the simulation for k = 1.98 with initial conditions at t = 1 where a shock is placed at $ x s = 1$ and the pressure behind it decreases as $ p \u221d x \u2212 0.0231$. In Fig. 6, we show the profiles of p vs q at $ t = 10 15 , 10 25 , 10 35$, together with the theoretical curve (thick line). The numerical profiles approach the theoretical curve as time increases. At $ t = 10 35$, the numerical shock velocity satisfies $ m = d \u2009 log \u2009 x s \u0307 / d \u2009 log \u2009 t = \u2212 0.9138$, whereas the theoretical value given by Eq. (53) is $ m = \u2212 0.9186$. While our numerical results seem to agree with theory, the simulation converges very slowly and we are, therefore, not able to claim perfect agreement. We note that the choice of k = 1.98 for this example is arbitrary and that numerical results obtained for other values of k indicate a qualitatively similar behavior.
VIII. SUMMARY AND DISCUSSION
We study the propagation of a planeparallel, ultrarelativistic shock down a powerlaw density profile of the form $ \rho \u221d z \u2212 k$. In the parameter space $ 7 / 4 < k < 2$, the blastwave cannot be described by any of the known types of selfsimilar solutions; global conservation laws do not apply in the selfsimilar domain, and at the same time, no eigenvalue problem can be defined. Instead, the solution gap is populated by a new class of selfsimilar solutions. Following, we summarize our main results:

We find that within the solution gap the flow obeys selfsimilarity of the third type. The unique characteristic of this new class of solutions is that the similarity index m is determined by the part of the flow that does not participate in the selfsimilar solution. This is a consequence of the fact that most of the energy in the blastwave resides in the nonselfsimilar flow, which is in causal contact with the shock. Solving for the shock, therefore, requires a physical understanding of the solution in the entire space. In this work, we first obtained an exact solution for the nonselfsimilar flow, described by relativistic expansion into vacuum, and then solved for the similarity index m by requiring that the leading edge the expanding flow is not destroyed by the shock. We find that $ m = \u2212 4 3 ( k \u2212 1 ) + 3 k$ for $ 7 / 4 < k < 2$.

The flow that expands into vacuum has two general components: (1) an accelerating piston that contains most of the energy and (2) a leading edge described by a powerlaw asymptotic in t and x. The properties at the pressure maximum within the piston are characterized by universal exponents that are independent of k: $ p p \u221d t \u2212 1 , \u2009 q p \u221d t 3 / 4$, and $ x p \u221d t 1 / 4$. Despite containing most of the energy, the piston does not go through free expansion, so that its entropy is not conserved.

Expansion into vacuum drives shock waves that obey second type selfsimilarity when k > 2. Therefore, the analysis in this paper generally applies whenever k > 7∕4. Nonetheless, if k > 2 solving for the nonselfsimilar flow is not required to describe the shock and its near downstream; a loss of causal contact between the shock and the bulk of the blastwave means that the solution beyond the sonic point is unable to affect the shock and is, therefore, not important for that purpose.

Selfsimilar solutions are important because they describe the asymptotic behavior of a physical system for a large class of initial conditions. In addition, they also offer the technical benefit of having to solve a set of ODEs instead of a set of PDEs, which makes them a very useful tool. This simplifying feature was not used in this work, where an exact solution to the PDEs was obtained in order to derive the selfsimilar asymptotic thereafter. Nonetheless, the exact solution we find for expansion into vacuum does not probe the flow directly behind the shock, before it settles on the powerlaw asymptotic. In order to study the small scales behind the shock, one must indeed solve the selfsimilar equations (9) with the corresponding index m(k).

Numerical simulations seem to agree with our analytic results for the shock. However, due to the very slow numerical convergence we are not able to confirm perfect agreement with theory.

We note that the original classification to first and second type solutions,^{1,2} only makes the distinction between problems that can be solved by dimensional considerations versus the need to solve an eigenvalue problem. The definition of second type selfsimilarity should be refined to include the requirement that the eigenvalue problem is imposed exclusively by the selfsimilar flow.

Plane parallel, ultrarelativistic shock waves of the kind discussed in this work do not have obvious physical applications. The spherically symmetric analogue of this problem, on the other hand, is often discussed in the astrophysical context of gammaray burst afterglows. In addition, planeparallel implosions in which the shock wave is converging instead of diverging (explosions) describe the dynamics of ultrarelativistic shock waves that break out from a stellar edge.^{16,17} According to Sari,^{13} both first and second type solutions can be found for spherical explosions when $ 5 \u2212 3 / 4 < k < 17 / 4$ and for plane parallel implosions when $ 7 / 4 < k < 1 + 3 / 4$, such that there is an overlap rather than a gap. While numerical simulations can be used to break those degeneracies, exact solutions can be found by generalizing this work to spherical symmetry or to planeparallel implosions. We note that another solution gap exists in spherically symmetric implosions^{13} and is likely analogous to the plane parallel case.
ACKNOWLEDGMENTS
R.S. was partially supported by ISF, NSF/BSF, and MOS grants.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Tamar Faran: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Andrei Gruzinov: Formal analysis (equal); Writing – original draft (supporting). Re'em Sari: Conceptualization (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.