Riemann Problems and Exact Solutions for the p-System

Abstract

In this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended differential constraint, several model laws for the pressure $p\left(v\right)$ are characterised. Therefore, the analysis developed in the paper has been carried out in the case of physical interest where $p=p_0{v}^{-\gamma }$, and an exact solution that generalises simple waves is determined. This allows us to study and to solve a class of generalised Riemann problems (GRP). In particular, we proved that the solution of the GRP can be discussed in the $(p,v)$ plane through rarefaction-like curves and shock curves. Finally, we studied a Riemann problem with structure and we proved the existence of a critical time after which a GRP is solved in terms of non-constant states separated by a shock wave and a rarefaction-like wave.

1. Introduction

Within the theoretical framework of nonlinear wave propagation, the Riemann problem (RP), the generalised Riemann problem (GRP) and the Riemann problem with structure (RPS) are of great interest because of their wide range of applications. An RP is an initial value problem characterised by two constant states with a discontinuity in a point. It was considered for the first time by Riemann in [1] (see also [2]) for an ideal gas. He considered a tube filled by a gas separated by a thin wall. On the left and on the right of the wall, the gas is at rest with constant but different values of mass density and pressure. At $t=0$, the wall is broken, and the problem is to study the resulting wave propagation. Starting from the paper of Riemann, many contributions have been given on this subject (see, for instance, the books of Smoller [3] and Dafermos [4] and references therein quoted). A general theory of the RP for hyperbolic homogeneous systems was developed by Lax in the fundamental paper [5]. He proved that, under the hypothesis of “small” jumps for the initial states, a unique solution of the RP exists, and it is determined by constant states separated by shock waves, rarefaction waves and/or contact discontinuities. In particular, rarefaction waves are smooth solutions with discontinuities in the first derivatives characterised by the well-known exact solutions called simple waves. Unfortunately, the theory for the solution of RP fails for hyperbolic nonhomogeneous systems. In fact, such a class of models usually does not admit simple waves, meaning that rarefaction waves cannot be computed. Quite recently, within the framework of the theory of differential constraints, some attempts at generalising simple waves and, in turn, rarefaction waves for nonhomogeneous systems have been given [6,7,8].

A GRP is characterised by initial non-constant data with a discontinuity in a point. Very few results have been obtained on this subject, and they mainly concern existence and uniqueness theorems [9,10,11,12,13,14] as well as asymptotic solutions [15,16]. The main difficulty is to find exact solutions of the initial non-constant data in order to characterise generalised rarefaction (simple) waves and to fit them with shock waves. Quite recently, some particular exact solutions of GRP have been obtained for nonhomogeneous hyperbolic systems [17].

An RPS is an initial value problem characterised by smooth data such that its limiting values for $x\to \pm \infty$ exist. In some cases, it is a more realistic description of problems given by RP. In fact, an RPS can take into account the small thickness of initial shocks or oscillations in a narrow zone between two initial constant states. In [18,19], it has been proved that in the case of hyperbolic conservation laws, an RPS converges to the corresponding RP for a large time for non-degenerate waves, while, in the case of exceptional waves, it tends to travelling waves. In the nonhomogeneous case, there exists a conjecture for which an RPS tends to a combination of shock structures and rarefaction waves of a suitable equilibrium subsystem [20,21]. Such a conjecture until now was verified numerically, but it still needs an analytical proof.

Within such a framework, the main aim of this paper is to outline a possible strategy for solving GRP and RPS for the following $2\times 2$ homogeneous system:

$$\left\{\begin{array}{c}{v}_t-u_x=0\hfill \\ u_t+{\left(p\left(v\right)\right)}_x=0\hfill \end{array}\right.$$

(1)

along with the conditions

$$p^{\prime }\left(v\right)<0,p^{\prime \prime }\left(v\right)>0.$$

(2)

In (2) and in the following, the prime refers to ordinary differentiation. Equation (1) can describe an isentropic ideal gas in Lagrangian coordinates. In such a case, $v=\frac{1}{\rho }$ denotes the specific volume, $\rho$ the mass density and u the velocity, while t is the time and x the Lagrangian spatial coordinate. Furthermore, for a perfect gas,

$$p=p_0{v}^{-\gamma },$$

(3)

where $p\left(v\right)$ denotes the pressure, while $\gamma >1$ is the adiabatic gas constant (i.e., the ratio between the specific heats) and $p_0>0$ is a constant. Under the hypothesis (2), system (1) is strictly hyperbolic. The characteristic speeds are

$${\lambda }_1=-\sqrt{-p^{\prime }\left(v\right)},{\lambda }_2=\sqrt{-p^{\prime }\left(v\right)},$$

(4)

while the corresponding left eigenvectors are

$${\mathbf{l}}_1=\left({\lambda }_1\left(v\right),1\right),{\mathbf{l}}_2=\left({\lambda }_2\left(v\right),1\right).$$

(5)

The pair of Equation (1), along with (2), is the celebrated p-system for which a large amount of results are known in the literature, mainly for nonlinear wave propagation problems [22,23,24,25]. In particular, an exhaustive description of the solution of the RP for the homogeneous p-system is given in [3].

The procedure we intend to develop in this paper is based on the use of the method of differential constraints. Such an approach was proposed and applied to gas dynamics by Yanenko [26] (see also [27,28,29]). It consists of appending to a given governing model a set of auxiliary equations that play the role of differential constraints because they select the class of exact solutions of the system under interest we are going to determine. In fact, after solving the compatibility conditions between the original field equations and the added differential constraints, solutions of such an overdetermined system will characterise particular exact solutions of the original equations. These solutions will be determined in terms of arbitrary functions whose number depends on the number of the differential constraints. Without making any further assumptions, the method has a great generality and, in fact, it includes many of the approaches proposed in the literature for determining exact solutions of PDEs. Unfortunately, because of such a generality, the method is not always useful for solving problems of interest in the applications, and further hypotheses are needed. For example, the requirement that the overdetermined system is in involution [30,31,32] is of great interest for solving nonlinear wave problems [33,34,35,36,37].

The paper is organised as follows. In Section 2, a brief sketch of the method of differential constraints is given. In Section 3, we determine a class of exact solutions of model (1). In Section 4 and Section 5, we solve different GRP and RPS for the p-system here considered. Some conclusions end the paper.

2. Method of Differential Constraints

In this section, for further convenience, we sketch the main steps of the method of differential constraints. Let us consider a strictly hyperbolic nonhomogeneous system

$${\mathbf{U}}_t+A\left(\mathbf{U}\right){\mathbf{U}}_x=\mathbf{B}\left(\mathbf{U}\right),$$

(6)

where $\mathbf{U}\in R^N$ denotes the vector field whose components are the unknown field variables, $A\left(\mathbf{U}\right)$ is the $N\times N$ matrix coefficients and $\mathbf{B}\left(\mathbf{U}\right)\in R^N$ is the vector source. Furthermore, in the following, we indicate with ${\lambda }^{\left(i\right)}$, ${\mathbf{l}}^{\left(i\right)}$ and ${\mathbf{d}}^{\left(i\right)}$, respectively, the eigenvalues (characterising the characteristic speeds of (6)) and the corresponding left and right eigenvectors of the matrix A. We add to the system (6) a set of further first order differential relations

$$F_k(x,t,\mathbf{U},{\mathbf{U}}_x)=0,k=1,...,M$$

(7)

and we look for solutions of (6) and (7). The Equation (7) are called differential constraints because they select the class of exact solutions of (6) which satisfy also (7). Without any further hypotheses, the form of relations (7) must be postulated. In order to overcome such a problem, it is useful to require the involutiveness of the system (6), (7) (i.e., to require that it is not possible to find any new independent differential equations by means of differentiation of (6) and (7)). It can be proved [38,39,40] (see also [29]) that if the overdetermined system (6), (7) is in involution, then the more general first-order differential constraints which can be appended to (6) assume the form

$${\mathbf{l}}^{\left(k\right)}·{\mathbf{U}}_x=p^{\left(k\right)}\left(x,t,\mathbf{U}\right),k=1,...,M$$

(8)

where the functions $p^{\left(k\right)}$ must be determined through the later consistency requirement with Equation (6).

In the following, we require $M=N-1$, which is of great interest for studying nonlinear wave problems. In fact, in such a case, after some algebra, the system (6) assumes the form

$${\mathbf{U}}_t+{\lambda }^{\left(\mathrm{N}\right)}{\mathbf{U}}_x=\mathbf{B}+\sum _{i=1}^{N-1}{p}^{\left(i\right)}\left({\lambda }^{\left(\mathrm{N}\right)}-{\lambda }^{\left(i\right)}\right){\mathbf{d}}^{\left(i\right)},$$

(9)

while constraints (8) specialise to

$${\mathbf{l}}^{\left(k\right)}\left({\mathbf{U}}_0\left(x\right)\right)·{\mathbf{U}}_0^{\prime }\left(x\right)=p^{\left(k\right)}\left(x,0,{\mathbf{U}}_0\right)k=1,..,N-1$$

(10)

with ${\mathbf{U}}_0\left(x\right)=\mathbf{U}(x,0)$ denoting the initial data. Finally, once the compatibility conditions between (6) and (8) are solved, exact particular solutions of (6) can be obtained by solving Equations (9) and (10).

In order to integrate (9), we notice that the left-hand side of such a system involves the derivatives of the vector field $\mathbf{U}$ alongside the characteristic curves associated to ${\lambda }^{\left(N\right)}$—namely, alongside the characteristics selected by the constraints (8). Therefore, exact solutions of (9) can be obtained by means of the method of characteristics and, because they must satisfy also (10), they are determined in terms of one arbitrary function. It is simple to verify that when $\mathbf{B}=0$ and $p^{\left(k\right)}=0$, such a class of solutions specialises to the classical simple wave solutions admitted by homogeneous hyperbolic systems. For this reason, the solutions of (9) and (10) are called generalised simple waves.

3. Exact Solutions

Here, along the lines of the analysis sketched in the previous section, we find generalised simple waves for the homogeneous p-system. Taking (8) into account, the possible differential constraints that can be appended to (1) are

$$u_x-\lambda \left(v\right)v_x=q(x,t,v,u),$$

(11)

where we set

$$\lambda =\pm \sqrt{-p^{\prime }\left(v\right)}.$$

(12)

while the function $q(x,t,v,u)$ must be determined later alongside the reduction procedure here considered. By requiring the differential compatibility between (1) and (11), the following consistency conditions are obtained:

$$\left\{\begin{array}{c}2\lambda \left(\lambda q_u+q_v\right)+{\lambda }^{\prime }q=0\hfill \\ q_t+\lambda q_x+q\left(\lambda q_u+q_v\right)=0.\hfill \end{array}\right.$$

(13)

After some algebra, the general solution of (13) is given by

$$q={\left\{\left(c_{0}t+c_{1}x+k_0\right)\sqrt{\left|\lambda \right|}\right\}}^{-1}$$

(14)

provided that

$$\frac{d\lambda }{dv}+2\lambda \sqrt{\left|\lambda \right|}\left(c_0+c_1\lambda \right)=0,$$

(15)

where $c_0$, $c_1$ and $k_0$ are arbitrary constants. By integrating (15), the following cases are obtained.

(i) If $c_1=0$, we find

$$\sqrt{\left|\lambda \right|}=\frac{1}{c_{0}v},$$

(16)

so that, taking (12) into account, we get

$$p=\frac{1}{3c_0^4{v}^3}.$$

(17)

(ii) If $c_0=0$, we obtain

$$\left|\lambda \right|={\left(\pm 3c_{1}v\right)}^{-\frac{2}{3}},$$

(18)

which, in turn, gives

$$p=p_0{v}^{-\frac{1}{3}},p_0=\frac{1}{\sqrt[3]{3}{\left(\pm c_1\right)}^{\frac{4}{3}}}.$$

(19)

(iii) If $c_0^2+c_1^2\ne 0$ and $c_0{c}_1>0$, then, in the case $\lambda =\sqrt{-p^{\prime }}$ we have

$$\frac{1}{{ϵ}^3}\left(\frac{ϵ}{\mathsf{\Lambda }}+arctan\left(\frac{\mathsf{\Lambda }}{ϵ}\right)\right)=c_{1}v,$$

(20)

while, if $\lambda =-\sqrt{-p^{\prime }}$, we obtain

$$\frac{1}{2{ϵ}^3}\left(\frac{2ϵ}{\mathsf{\Lambda }}+ln\left(\frac{\mathsf{\Lambda }-ϵ}{\mathsf{\Lambda }+ϵ}\right)\right)=c_{1}v,$$

(21)

where we set $\frac{c_0}{c_1}={ϵ}^2$ and $\mathsf{\Lambda }=\sqrt{\left|\lambda \right|}$.

(iv) If $c_0^2+c_1^2\ne 0$ and $c_0{c}_1<0$, then in the case $\lambda =-\sqrt{-p^{\prime }}$ we have

$$\frac{1}{{ϵ}^3}\left(\frac{ϵ}{\mathsf{\Lambda }}+arctan\left(\frac{\mathsf{\Lambda }}{ϵ}\right)\right)=-c_{1}v,$$

(22)

while, if $\lambda =\sqrt{-p^{\prime }}$ we obtain

$$\frac{1}{2{ϵ}^3}\left(\frac{2ϵ}{\mathsf{\Lambda }}+ln\left(\frac{\mathsf{\Lambda }-ϵ}{L+ϵ}\right)\right)=-c_{1}v,$$

(23)

where we set $\frac{c_0}{c_1}=-{ϵ}^2$ and $\mathsf{\Lambda }=\sqrt{\left|\lambda \right|}$.

In order to consider a case of possible physical interest, in the following, we deal with the case (i). Therefore, owing to (14), the Equation (9) assume the form

$$\left\{\begin{array}{c}{v}_t-\lambda v_x=\frac{v}{t+k}\hfill \\ \\ u_t-\lambda u_x=\mp \frac{1}{c_0^2\left(t+k\right)v}\hfill \end{array}\right.$$

(24)

where, without loss of generality, we set $k=k_0{c}_0$ and we assume $k>0$. The integration of (24) by the method of characteristics leads to

$$\begin{array}{ccc}& & v=\frac{t+k}{k}{v}_0\left(\sigma \right)\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & u=u_0\left(\sigma \right)\mp \frac{t}{c_0^2\left(t+k\right)v_0\left(\sigma \right)}\hfill \end{array}$$

(26)

where the functions $v_0\left(x\right)=v(x,0)$ and $u_0\left(x\right)=u(x,0)$ denote the initial data, while the characteristic variable $\sigma$ is given implicitly by

$$x=\mp \frac{kt}{c_0^2(t+k)v_0^2\left(\sigma \right)}+\sigma$$

(27)

which defines the family of characteristics associated, respectively, to ${\lambda }_2$ or to ${\lambda }_1$. Finally, substituting (25) and (26) in the constraint (11), we get

$$u_0^{\prime }\left(x\right)=\pm \frac{v_0^{\prime }\left(x\right)}{c_0^2{v}_0^2\left(x\right)}+\frac{v_0\left(x\right)}{k}$$

(28)

which selects the class of initial value problems that can be solved by means of the present approach.

Therefore, the relations (25)–(27), along with (28), characterise a generalised simple wave for the p-system (1) supplemented by (17). According to the results illustrated in the previous section, we remark that when the constraint (11) is homogeneous (i.e., $k\to +\infty$), the solution (25)–(27) specialises to the classical simple wave solution. Finally, it could be of a certain interest to look for a critical time $t_c$ in which the characteristic curves meet and the corresponding solution loses its regularity. In the present case, from (27), by requiring that $\frac{dx}{d\sigma }=0$, we obtain

$$t=-\frac{c_0^{2}k}{c_0^2\mp k\frac{d}{d\sigma }\left(\frac{1}{v_0^2\left(\sigma \right)}\right)},$$

(29)

so that, if the following condition holds,

$$-\frac{c_0^2}{k}\le \frac{d}{dx}\left(\frac{1}{v_0^2\left(x\right)}\right)\le \frac{c_0^2}{k},$$

(30)

a critical time $t_c$ does not exist, and the solution in points is smooth $\forall t\ge 0$.

Remark 1.

In [17], some exact solutions for the nonhomogeneous p-system have been obtained through the method of differential constraints. In particular, it was proved that when the relaxation term goes to zero as well as the source involved in the constraint, the exact solution there obtained tends to the corresponding solution of the homogeneous p-system. Such a solution is different from that given in (25)–(27). In fact, the differential constraint (11) here considered characterises a class of initial value problems that are different from those taken into account in [17]. Of course, when in (25)–(28) we take $k\to +\infty$, the solutions of the homogeneous system and of the nonhomogeneous one coincide because they specialise to simple waves.

4. Generalised Riemann Problem

Here, owing to the analysis developed in the (i) case of Section 3, our aim is to solve the following GRP:

$$v(x,0)=\left\{\begin{array}{c}{v}_l\left(x\right)\mathrm{for}x<0\hfill \\ v_r\left(x\right)\mathrm{for}x>0\hfill \end{array},\right.u(x,0)=\left\{\begin{array}{c}{u}_l\left(x\right)\mathrm{for}x<0\hfill \\ u_r\left(x\right)\mathrm{for}x>0,\hfill \end{array}\right.$$

(31)

where $v_l\left(x\right)$, $v_r\left(x\right)$, $u_l\left(x\right)$ and $u_r\left(x\right)$ are smooth functions such that

$$\begin{array}{ccc}& & v_L=\underset{x\to 0^{-}}{lim}{v}_l\left(x\right),v_R=\underset{x\to 0^{+}}{lim}{v}_r\left(x\right),\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & u_L=\underset{x\to 0^{-}}{lim}{u}_l\left(x\right),u_R=\underset{x\to 0^{+}}{lim}{u}_r\left(x\right)\hfill \end{array}$$

(33)

with $v_L\ne v_R$ and $u_L\ne u_R$. According to the approach outlined in Section 2, since the initial data must satisfy the constraint (11), by the substitution of (31) into (11) and by a further integration, we find

$$\begin{array}{ccc}& & u_l\left(x\right)=u_L\mp \frac{1}{c_0^2}\left(\frac{1}{v_l\left(x\right)}-\frac{1}{v_L}\right)+\frac{1}{k}{\int }_0^x{v}_l\left(z\right)dz,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & u_r\left(x\right)=u_R\mp \frac{1}{c_0^2}\left(\frac{1}{v_r\left(x\right)}-\frac{1}{v_R}\right)+\frac{1}{k}{\int }_0^x{v}_r\left(z\right)dz.\hfill \end{array}$$

(35)

Therefore, if we consider the initial data assigned for $x<0$, from (25) and (26), we obtain

$$\begin{array}{ccc}& & v=\frac{t+k}{k}{v}_l\left({\sigma }_l\right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & u=u_l\left({\sigma }_l\right)\mp \frac{t}{c_0^2\left(t+k\right)v_l\left({\sigma }_l\right)},\hfill \end{array}$$

(37)

where

$$x=\mp \frac{kt}{c_0^2(t+k)v_l^2\left({\sigma }_l\right)}+{\sigma }_l,\mathrm{with}{\sigma }_l<0,$$

(38)

while, taking the initial data for $x>0$ into account, we have

$$\begin{array}{ccc}& & v=\frac{t+k}{k}{v}_r\left({\sigma }_r\right),\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & u=u_r\left({\sigma }_r\right)\mp \frac{t}{c_0^2\left(t+k\right)v_r\left({\sigma }_r\right)},\hfill \end{array}$$

(40)

where

$$x=\mp \frac{kt}{c_0^2(t+k)v_r^2\left({\sigma }_r\right)}+{\sigma }_r,\mathrm{with}{\sigma }_r>0.$$

(41)

Because the characteristic parameters ${\sigma }_l<0$ and ${\sigma }_r>0$, the solution given by (36), (37) is defined in the region $x<x_L\left(t\right)$, while the solution (39), (40) exists in $x>x_R\left(t\right)$, where

$$\begin{array}{ccc}& & x_L\left(t\right)=\underset{{\sigma }_l\to 0^{-}}{lim}\left(\mp \frac{kt}{c_0^2(t+k)v_l^2\left({\sigma }_l\right)}\right)=\mp \frac{kt}{c_0^2(t+k)v_L^2},\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & x_R\left(t\right)=\underset{{\sigma }_r\to 0^{+}}{lim}\left(\mp \frac{kt}{c_0^2(t+k)v_r^2\left({\sigma }_r\right)}\right)=\mp \frac{kt}{c_0^2(t+k)v_R^2}.\hfill \end{array}$$

(43)

Then, in (42), (43) $x=x_L\left(t\right)$ and $x=x_R\left(t\right)$ denote, respectively, the left and the right limiting characteristics starting from the discontinuity point $(0,0)$ of the $(x,t)$ plane.

In order to determine a central state solution that connects smoothly the left state (36), (37) with the right one (39), (40), we integrate the system (24), along with the constraint (11), with the initial data

$$v(0,0)=\overline{v}\left(a\right),u(0,0)=\overline{u}\left(a\right),\mathrm{with}a\in \left[0,1\right],$$

(44)

subjected to the conditions

$$\overline{v}\left(0\right)=v_L,\overline{v}\left(1\right)=v_R,\overline{u}\left(0\right)=u_L,\overline{u}\left(1\right)=u_R.$$

(45)

In (44,) we indicate with a the parameter characterising the fan of characteristics starting from the origin of the $(x,t)$ plane. Then, after some algebra, in the central region $x_L\left(t\right)\le x\le x_R\left(t\right)$, the following solution is obtained:

$$\begin{array}{ccc}& & v=\sqrt{\mp \frac{t(t+k)}{k{c}_0^{2}x}},\hfill \end{array}$$

(46)

$$\begin{array}{ccc}& & u=u_L\pm \frac{1}{c_0^2{v}_L}\mp \frac{2t+k}{c_0}\sqrt{\mp \frac{x}{kt(t+k)}},\hfill \end{array}$$

(47)

along with the condition

$$u_R\pm \frac{1}{c_0^2{v}_R}=u_L\pm \frac{1}{c_0^2{v}_L}.$$

(48)

Finally, by requiring $x_L\left(t\right)<x_R\left(t\right)$, we find the further conditions

$$\begin{array}{ccc}& & v_L<v_R\mathrm{in}\mathrm{the}\mathrm{case}\mathrm{where}\lambda =\sqrt{-p^{\prime }},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & v_L>v_R\mathrm{in}\mathrm{the}\mathrm{case}\mathrm{where}\lambda =-\sqrt{-p^{\prime }}.\hfill \end{array}$$

(50)

Therefore, provided that conditions (48) and (49) or (50) are satisfied, the central state (46), (47) connects smoothly the left state (36), (37) with the right one (39), (40), and it characterises a generalised rarefaction wave that solves the GRP (31). More in general, from (48), we find the generalised rarefaction curves

$$\begin{array}{ccc}& & u=R^{\left(1\right)}(v,v_L,u_L)=u_L-\frac{1}{c_0^2}\left(\frac{1}{v}-\frac{1}{v_L}\right),\mathrm{with}{v}_L<v,\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & u=R^{\left(2\right)}(v,v_L,u_L)=u_L+\frac{1}{c_0^2}\left(\frac{1}{v}-\frac{1}{v_L}\right),\mathrm{with}{v}_L>v,\hfill \end{array}$$

(52)

which in the $(v,u)$ plane characterise all the initial right states whose limiting values $(v_R,u_R)$ defined in (32) and (33), by belonging to the curve $u=R^{\left(1\right)}(v,v_L,u_L)$ or $u=R^{\left(2\right)}(v,v_L,u_L)$, permit us to solve the GRP (31) by means of the generalised rarefaction wave determined in (46) and (47).

The smooth exact solution of (1), supplemented by (31), obtained here is given in implicit form depending on the initial data $v(x,0)$ considered. In order to obtain an explicit solution that will be useful in the following, we chose the initial condition (31) as

$$v(x,0)=\left\{\begin{array}{c}{v}_L\mathrm{for}x<0\hfill \\ v_R\mathrm{for}x>0,\hfill \end{array}\right.$$

(53)

where the constants $v_L$ and $v_R$ are such that $v_L\ne v_R$, so that from (34) and (35) we obtain

$$u(x,0)=\left\{\begin{array}{c}{u}_l\left(x\right)=u_L+\frac{v_L}{k}x\mathrm{for}x<0\hfill \\ \\ u_r\left(x\right)=u_R+\frac{v_R}{k}x\mathrm{for}x>0.\hfill \end{array}\right.$$

(54)

In such a case, the corresponding left state solution (36)–(38) assumes the form

$$v={\tilde{v}}_l\left(t\right)=\frac{t+k}{k}{v}_L,u=u_l\left(x\right)\mathrm{in}x<x_L\left(t\right),$$

(55)

while the right state solution (39)–(41) specialises to

$$v={\tilde{v}}_r\left(t\right)=\frac{t+k}{k}{v}_R,u=u_r\left(x\right)\mathrm{in}x>x_R\left(t\right).$$

(56)

Furthermore, the central state connecting smoothly (55) with (56) is still characterised by (46), (47). Next, in order to discuss the general solution of the GRP here considered, we look for shock wave solutions for the p-system (1) with (53) and (54). Such an analysis is well known for the p-system, so we refer to [3] for more details. Therefore, by solving the Rankine–Hugoniot conditions for (1), we find

$$s=-\frac{u_r-u_l}{{\tilde{v}}_r-{\tilde{v}}_l},u_r=u_l\mp \sqrt{\left({\tilde{v}}_r-{\tilde{v}}_l\right)\left(p\left({\tilde{v}}_l)-p({\tilde{v}}_r\right)\right)},$$

(57)

where $\left({\tilde{v}}_l,u_l\right)$ is the state on the left of the shock determined by (55), $\left({\tilde{v}}_r,u_r\right)$ is the state on the right of the shock characterised by (56) and s is the shock velocity. Now, by requiring that the Lax conditions are satisfied, after some algebra, we find two shock families: the $1-$shocks in which $v_R<v_L$ and $s<0$ and the $2-$shock family where $v_R>v_L$ and $s>0$. In both cases, owing to (55) and (56), we easily find the shock curve

$$x=x_s\left(t\right)=-\frac{u_R-u_L}{v_R-v_L}\frac{kt}{t+k}$$

(58)

so that, from (57)${}_1$ with (55) and (56), the shock speed specialises to

$$s=s\left(t\right)=-\frac{u_R-u_L}{v_R-v_L}{\left(\frac{k}{t+k}\right)}^2.$$

(59)

Finally, owing to (57)${}_2$ and taking (17), (55), (56) and (58) into account, for $1-$shocks, the following shock curve is obtained:

$$u=S^{\left(1\right)}(v,v_L,u_L)=u_L-\frac{1}{\sqrt{3}{c}_0^2}\sqrt{\left(v-v_L\right)\left(\frac{1}{v_L^3}-\frac{1}{v^3}\right)}\mathrm{with}v<v_L,$$

(60)

while for $2-$shocks, we find

$$u=S^{\left(2\right)}(v,v_L,u_L)=u_L-\frac{1}{\sqrt{3}{c}_0^2}\sqrt{\left(v-v_L\right)\left(\frac{1}{v_L^3}-\frac{1}{v^3}\right)}\mathrm{with}v>v_L.$$

(61)

The curves (60), (61) characterise in the $(v,u)$ plane all the right initial states which allow us to solve the GRP (53), (54) by a $1-$shock or by a $2-$shock. It is relevant to notice that both the generalised rarefaction curves (51) and (52) as well the shock curves (60) and (61) involve the limiting values of the initial data for $x\to 0$ (i.e., $v_L$, $v_R$, $u_L$, $u_R$). Therefore, such a curves coincide with those of the RP for the p-system (see [3]). It follows that the general discussion of the solution of the RP for the p-system is useful here for characterising the general solution of the GRP given in (53), (54). In fact, by referring to Figure 1, if $(v_R,u_R)$ belongs to one of the curves $R^{(1,2)}$, $S^{(1,2)}$, then the solution is given in terms of the non-constant states (55) and (56) separated, respectively, by the generalised rarefaction wave (46), (47) or by a shock wave ($1-$shock or $2-$shock). If, on the other hand, $(v_R,u_R)$ belongs to one of the regions I, $II$, $III$ or $IV$, then, as in the case of the RP, the solution of (53) and (54) is determined in terms of three non-constant states separated by generalised rarefaction waves and/or shock waves (the interested reader can find a detailed discussion corresponding to the RP in [3]). Finally, taking (59) into account, we notice that, for large t, both the $1-$shocks or the $2-$shocks tend to a stationary shock.

5. Riemann Problem with Structure

In this section, our aim is to study the following RPS:

$$v(x,0)=\left\{\begin{array}{c}{v}_L\mathrm{for}x<0\hfill \\ v_0\left(x\right)\mathrm{for}0\le x\le L\hfill \\ v_R\mathrm{for}x>L\hfill \end{array}\right.$$

(62)

where

$$v_0\left(x\right)=\frac{v_L}{\sqrt{1+\alpha x}},\mathrm{with}\alpha =\frac{v_L^2-v_R^2}{L{v}_R^2}.$$

(63)

Taking (11) into account, from (62) we find

$$u(x,0)=\left\{\begin{array}{c}{u}_L+\frac{v_L}{k}x\mathrm{for}x<0\hfill \\ \\ u_0\left(x\right)\mathrm{for}0\le x\le L\hfill \\ \\ u_R+\frac{v_R}{k}\left(x-L\right)\mathrm{for}x>L\hfill \end{array}\right.$$

(64)

with

$$u_0\left(x\right)=u_L+\frac{1}{c_0^2{v}_L}\left(\frac{2c_0^2{v}_L^2}{k\alpha }\mp 1\right)\left(\sqrt{1+\alpha x}-1\right).$$

(65)

Moreover, in order that the initial condition (64) is smooth, we require

$$u_R=u_L+\frac{v_L-v_R}{c_0^2{v}_L{v}_R}\left(\frac{2c_0^2{v}_L^2}{k\alpha }\mp 1\right).$$

(66)

In passing, we notice that in both cases where $\alpha >0$ or $\alpha <0$, $1+\alpha x>0$ results. Taking (25)–(27) into account, after some algebra, the solution of the initial value problem (62), (64) is given by the left state

$$v=\frac{t+k}{k}{v}_L,u=u_L+\frac{v_L}{k}x,\mathrm{for}x<x_l\left(t\right)=\mp \frac{kt}{c_0^2(t+k)v_L^2},$$

(67)

by the central state

$$\left\{\begin{array}{c}v=\frac{t+k}{k}\frac{v_L}{\sqrt{1+\alpha \eta }}\hfill \\ \mathrm{for}{x}_l\left(t\right)\le x\le x_r\left(t\right)\hfill \\ \\ u=u_L-\frac{1}{c_0^2{v}_L}\left(\frac{2c_0^2{v}_L^2}{k\alpha }\mp 1\right)+\frac{1}{c_0^2{v}_L}\left(\frac{2c_0^2{v}_L^2}{k\alpha }\mp \frac{2t+k}{t+k}\right)\sqrt{1+\alpha \eta }\hfill \end{array}\right.$$

(68)

where

$$\eta =\frac{v_L^2{c}_0^{2}x(t+k)\pm kt}{v_L^2{c}_0^2(t+k)\mp \alpha kt},$$

(69)

and by the right state

$$v=\frac{t+k}{k}{v}_R,u=u_R+\frac{v_R}{k}(x-L),\mathrm{for}x>x_r\left(t\right)=\mp \frac{kt}{c_0^2(t+k)v_R^2}+L.$$

(70)

It is simple to verify that in the case $\lambda =\sqrt{-p^{\prime }}$, if

$$v_L>v_R\mathrm{and}\frac{c_0^2}{k}<\frac{\alpha }{v_L^2},$$

(71)

the limiting characteristics $x_l\left(t\right)$ and $x_r\left(t\right)$ meet at the critical time $t_c$ given by

$$t_c=-\frac{c_0^{2}k{v}_L^2}{c_0^2{v}_L^2-\alpha k},$$

(72)

while, in the case $\lambda =-\sqrt{-p^{\prime }}$, under the conditions

$$v_L<v_R\mathrm{and}\frac{c_0^2}{k}<-\frac{\alpha }{v_L^2}$$

(73)

the curves $x_l\left(t\right)$ and $x_r\left(t\right)$ meet at the critical time

$$t_c=-\frac{c_0^{2}k{v}_L^2}{c_0^2{v}_L^2+\alpha k}.$$

(74)

In both cases the the left and right characteristics $x_l\left(t\right)$, $x_r\left(t\right)$ meet in the point

$$x_c=-\frac{1}{\alpha }.$$

(75)

Of course, if the condition (71) or (73) is not satisfied, then the solution (67)–(70) is smooth $\forall t\ge 0$. In passing, we notice that the initial data (62)${}_1$ and (62)${}_3$ satisfy the relation (30) while, if condition (71) or (73) is satisfied, then the initial datum (62)${}_2$ fulfills (30) until $t_c$, where all the characteristics of the central region (68) meet in $x_c$.

In the following, we consider the case characterised by (71). Of course, similar results can be obtained in the remaining case. Since the solution (67)–(70) is regular until $t_c$, we have now to study the following GRP:

$$v(x,t_c)=\left\{\begin{array}{c}\frac{t_c+k}{k}{v}_L\mathrm{for}x<x_c\hfill \\ \\ \frac{t_c+k}{k}{v}_R\mathrm{for}x>x_c\hfill \end{array}\right.,u(x,t_c)=\left\{\begin{array}{c}{u}_L+\frac{v_L}{k}x\mathrm{for}x<x_c\hfill \\ \\ u_R+\frac{v_R}{k}(x-L)\mathrm{for}x>x_c.\hfill \end{array}\right.$$

(76)

In order to solve (76), it is convenient to set

$$\tau =t-t_c,\xi =x-x_c,$$

(77)

along with

$$\begin{array}{ccc}& & \widehat{k}=t_c+k,{\widehat{v}}_L=\frac{t_c+k}{k}{v}_L,{\widehat{v}}_R=\frac{t_c+k}{k}{v}_R,\hfill \end{array}$$

(78)

$$\begin{array}{ccc}& & {\widehat{u}}_L=u_L+\frac{v_L}{k}{x}_c,{\widehat{u}}_R=u_R+\frac{v_R}{k}(x_c-L),\hfill \end{array}$$

(79)

so that, in the variables $(\xi ,\tau )$, the initial data (76) assume the form (53) and (54). As consequence, the results obtained in Section 4 can be useful for solving the present GRP. In particular, by referring to Figure 2, it is possible to prove that the point $({\widehat{v}}_R,{\widehat{u}}_R)$, which is obtained from the initial right state (76) when $x\to x_c$ (or $\xi \to 0$), belongs to region $III$, so that the solution of (76) is given by means of a back-shock and a forward generalised rarefaction wave. In fact, because of (71), we have ${\widehat{v}}_R<{\widehat{v}}_L$. Therefore, let $({\widehat{v}}_R,{\widehat{u}}_1)$ and $({\widehat{v}}_R,{\widehat{u}}_2)$ the points of abscissa ${\widehat{v}}_R$ which belong, respectively to $S^{\left(1\right)}$ and $R^{\left(2\right)}$, since ${\widehat{u}}_1=S^{\left(1\right)}({\widehat{v}}_R,{\widehat{v}}_L,\widehat{u_L})$ and ${\widehat{u}}_2=R^{\left(2\right)}({\widehat{v}}_R,{\widehat{v}}_L,\widehat{u_L})$, taking (66), (78) and (79) into account; after some algebra, ${\widehat{u}}_1<{\widehat{u}}_R<{\widehat{u}}_2$ results, meaning that $({\widehat{v}}_R,{\widehat{u}}_R)$ belongs to region $III$.

The resulting solution after $t_c$ is given by three non-constant states separated by a shock and by a generalised rarefaction wave. In particular, by referring to Figure 3 where, in the $(x,t)$ plane, the solution of (62), (64) is given $\forall t\ge 0$, after the critical time $t_c$, we find the left state (67), which is separated by the central state

$$v={\tilde{v}}_c\left(t\right)=\frac{t+k}{t_c+k}{v}_0,u={\tilde{u}}_c\left(x\right)=u_0+\frac{v_0}{t_c+k}(x-x_c)$$

(80)

by a back shock whose line, taking (77) into account, is given by

$$x={\widehat{x}}_s\left(t\right)=-\frac{u_0-{\widehat{u}}_L}{v_0-{\widehat{v}}_L}\frac{\widehat{k}\left(t-t_c\right)}{k+t}+x_c.$$

(81)

The state (80) is connected smoothly to the right state (70) by a forward generalised rarefaction wave which, owing to (46) and (47), in terms of the $(\xi ,\tau )$ variables, assumes the form

$$\begin{array}{ccc}& & v=\sqrt{\frac{\tau (\tau +\widehat{k})}{\widehat{k}{c}_0^2\xi }},\hfill \end{array}$$

(82)

$$\begin{array}{ccc}& & u=u_0-\frac{1}{c_0^2{v}_0}+\frac{2\tau +\widehat{k}}{c_0}\sqrt{\frac{\xi }{\widehat{k}\tau (\tau +\overline{k})}}.\hfill \end{array}$$

(83)

Owing to (77), the left and the right characteristics limiting the generalised rarefaction wave (82), (83) are

$$x={\widehat{x}}_l\left(t\right)=\frac{\widehat{k}\left(t-t_c\right)}{c_0^2{v}_0^2\left(t+k\right)}+x_c,x={\widehat{x}}_r\left(t\right)=\frac{\widehat{k}\left(t-t_c\right)}{c_0^2{\widehat{v}}_R^2\left(t+k\right)}+x_c.$$

(84)

Furthermore, since the point $(v_0,u_0)\in S^{\left(1\right)}(v,{\widehat{v}}_L,{\widehat{u}}_L)$ and $({\widehat{v}}_R,{\widehat{u}}_R)\in R^{\left(2\right)}(v,v_0,u_0)$ (see Figure 3), we have

$$\begin{array}{ccc}& & u_0={\widehat{u}}_L-\frac{1}{\sqrt{3}{c}_0^2}\sqrt{\left(v_0-{\widehat{v}}_L\right)\left(\frac{1}{{\widehat{v}}_L^3}-\frac{1}{v_0^3}\right)},\hfill \end{array}$$

(85)

$$\begin{array}{ccc}& & {\widehat{u}}_R=u_0-\frac{1}{c_0^2}\left(\frac{1}{v_0}-\frac{1}{{\widehat{v}}_R}\right).\hfill \end{array}$$

(86)

Therefore, the values $(v_0,u_0)$ that are involved in the central state (80) are defined by (85) and (86).

6. Conclusions

The main aim of this paper is to show how the method of differential constraints has been useful for solving nonlinear wave problems for the celebrated p-system. Such a model, apart from its physical meaning, has been considered as a prototype of more general hyperbolic systems for describing nonlinear wave propagation.

After classifying all the possible differential constraints that can be appended to the governing model under interest, we solved the consistency conditions (13) arising from the differential compatibility between (1) and (11). As a consequence, the material law for the pressure $p\left(v\right)$ must obey one of the relations for $\lambda \left(v\right)$ characterised in cases (i)–(iv). Since for a perfect gas, we have $p=p_0{v}^{-\gamma }$ with $\gamma >1$, the unique case that has a physical meaning is determined by (16). Therefore, in the paper, we develop our analysis in the case (i), and an exact solution that generalises the classical simple wave admitted by homogeneous systems is obtained. In fact, when the source term q involved in the constraint (11) is zero, then the solution (25)–(27) specialises to a simple wave. Such a solution was useful for solving a class of generalised Riemann problems as well as of Riemann problems with structure.

In fact, we have been able to obtain the general solution of the GRP (53), (54) in terms of non-constant states separated by generalised rarefaction waves and/or by shock waves. In characterising such a solution, the generalised rarefaction curves $u=R^{\left(1\right)}(v,v_L,u_L)$, $u=R^{\left(2\right)}(v,v_L,u_L)$ play a prominent role as well as the shock curves $u=S^{\left(1\right)}(v,v_L,u_L)$, $u=S^{\left(2\right)}(v,v_L,u_L)$. Indeed, the analysis in the $(v,u)$ plane that can be carried on for the Riemann problem can be developed also for the GRP under interest, and it involves the point $(v_R,u_R)$ determined by the limiting values for $x\to 0$ of the initial data. Furthermore, owing to (17), we have

$${\int }_{v_L}^{\infty }\sqrt{-p^{\prime }\left(v\right)}dv=\frac{1}{c_0^2{v}_L},$$

(87)

so that if the point $(v_R,u_R)$ belongs to region $IV$ of the $(v,u)$ plane (see Figure 1), then a vacuum zone can be formed, as happens for the RP.

A Riemann problem with structure was also solved by means of the generalised simple wave determined in Section 3. In such a case, it was interesting to notice that, if the condition (71) or (73) is satisfied, then at a critical time $t_c$, a shock is formed and, in turn, a new GRP must be solved. By means of the analysis of the $(v,u)$ plane (see Figure 2), we have proved that such a GRP is solved by three non-constant states separated by a back shock and a forward generalised rarefaction wave. The corresponding full solution is given in Figure 3.