1 Introduction
###### Abstract

A modified version of the conditional symmetry method, together with the classical method, is used to obtain new classes of elliptic solutions of the isentropic ideal compressible fluid flow in (3+1) dimensions. We focus on those types of solutions which are expressed in terms of the Weierstrass -functions of Riemann invariants. These solutions are of special interest since we show that they remain bounded even when these invariants admit the gradient catastrophe. We describe in detail a procedure for constructing such classes of solutions. Finally, we present several examples of an application of our approach which includes bumps, kinks and multi-wave solutions.

Elliptic solutions of isentropic ideal compressible fluid flow in (3+1) dimensions

R. Conte, A.M. Grundland, B. Huard

1. LRC MESO, École normale supérieure de Cachan (CMLA) et CEA-DAM,

61, avenue du Président Wilson, F-94235 Cachan Cedex, France.

2. Service de physique de l’état condensé (URA 2464), CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France

3. Centre de Recherches Mathématiques, Université de Montréal,

C.P. 6128, Succ. Centre-ville, Montréal, (QC) H3C 3J7, Canada,

4. Université du Québec, Trois-Rivières CP500 (QC) G9A 5H7, Canada

5. Département de mathématiques et de statistique, Université de Montréal C.P. 6128, Succ. Centre-ville, Montréal, (QC) H3C 3J7, Canada

Email addresses : Robert.Conte@cea.fr, grundlan@crm.umontreal.ca, huard@dms.umontreal.ca

## 1 Introduction

The purpose of this paper is to construct bounded elliptic solutions of a compressible isentropic ideal flow in () dimensions. Such solutions exist even in the case where the Riemann invariants admit the gradient catastrophe.

Let us first present a brief description of a procedure detailed in [10] for constructing rank- solutions in terms of Riemann invariants for the case of an isentropic compressible ideal fluid in dimensions. Such a model is governed by the equations

 uαt+4∑β=13∑j=1Ajαβ(u)uβj=0,α=1,2,3,4, (1.1)

where and are real-valued matrix functions of the form

 Aj=⎛⎜ ⎜ ⎜ ⎜⎝uiδi1κ−1aδi2κ−1aδi3κ−1aδi1κaui00δi2κa0ui0δi3κa00ui⎞⎟ ⎟ ⎟ ⎟⎠,j=1,2,3,

and is the adiabatic exponent of the medium under consideration. The independent and dependent variables are denoted by and , respectively, and stands for the first order partial derivatives of , i.e. , , . Here, the quantity stands for the velocity of sound in the medium and is the velocity vector field of the flow. Throughout this paper, we adopt the summation convention over repeated lower and upper indices.

The purpose of this article is to obtain rank- solutions of system (1.1) expressible in terms of Riemann invariants. To this end, we seek solutions of (1.1) defined implicitly by the following set of relations between the variables and ,

 u=f(r1(x,u),…,rk(x,u)),rA(x,u)=λAi(u)xi,ker(λA0I4+Ai(u)λAi)≠0, (1.2)

for some function and . Such a solution is called a rank- solution if . The functions are called the Riemann invariants associated with the wave vectors of the system (1.1). Here, denotes a direction of wave propagation and the eigenvalue is a phase velocity of the considered wave. Two types of admissible wave vectors for the isentropic equations (1.1) are obtained by solving the dispersion relation

 det(λ0(u)I4+λi(u)Ai(u))=[(λ0+→u⋅→λ)2−a2→λ2](λ0+→u⋅→λ)2=0. (1.3)

They are called the entropic and acoustic wave vectors and are defined by

 i)λE=(εa+→u⋅→e,−→e),ε=±1,ii)λS=(det(→u,→e,→m),−→e×→m),|→e|2=1, (1.4)

where and are unit and arbitrary vectors, respectively.

The construction of rank- solutions through the conditional symmetry method (CSM) is achieved by considering an overdetermined system, consisting of the original system (1.1) in independent variables together with a set of compatible first order differential constraints (DCs),

 ξia(u)uαi=0,λAiξia=0,a=1,…,4−k, (1.5)

for which a symmetry criterion is automatically satisfied. Such notions as conditional symmetry, conditional symmetry algebra and conditionally invariant solution for the original system (1.1) we use in accordance with definitions given in [10]. Under the above circumstances, the following result holds :

The isentropic compressible ideal fluid equations (1.1) admit a -dimensional conditional symmetry algebra if and only if there exists a set of linearly independent vector fields

 Xa=ξia(u)∂∂xi,a=1,…,4−k,ker(Ai(u)λAi)≠0,λAiξia=0,A=1,…,k≤3, (1.6)

which satisfy on some neighborhood of the trace conditions

 i)tr(Aμ∂f∂rλ)=0,ii)tr(Aμ∂f∂rη(a1∂f∂r…ηas)∂f∂rλ)=0,μ=1,…,4, (1.7)

where

 λ=(λAi)∈Rk×4,r=(r1,…,rk)∈Rk,∂f∂r=(∂fα∂rA)∈R4×k, ηas=(∂λAas∂uα)∈Rk×4,s=1,…,k−1,

and denotes the symmetrization over all indices in the bracket. Solutions of the system which are invariant under the Lie algebra are precisely rank- solutions of the form (1.2).

This result is a special case of the proposition in [10]. Note that these symmetries are not symmetries of the original system, but they can be used to construct solutions of the overdetermined system composed of (1.1) and (1.5).

For the case of rank- entropic solution , the wave vector is a non-zero multiple of (1.4 i). Therefore, the corresponding vector fields and Riemann invariant become

 Xi=−(a+→e⋅→u)−1ei∂∂t+∂∂xi,i=1,2,3,r(x,u)=(a+→u⋅→e)t−→e⋅→x,|→e|2=1, (1.8)

where we chose in (1.4 i). Rank-1 solutions invariant under the vector fields are obtained through the change of coordinates

 ¯t=t,¯x1=r(x,u),¯x2=x2,¯x3=x3,¯a=a,¯u1=u1,¯u2=u2,¯u3=u3, (1.9)

on . Assuming that the direction of the wave vector is constant, the fluid dynamics equations (1.1) transform into the system

 ∂¯a∂¯x1=κ−1ei∂¯ui∂¯x1,∂¯ui∂¯x1=κei∂¯a∂¯x1,i=1,2,3, (1.10)

with the invariance conditions

 ¯a¯t=¯a¯xj=0,¯uα¯t=¯uα¯xj=0,j=2,3,α=1,2,3. (1.11)

The general rank- entropic solution takes the form

 ¯a(¯t,¯x)=¯a(¯x1),¯ui(¯t,¯x)=κei¯a(¯x1)+Ci,Ci∈R,i=1,2,3, (1.12)

where the Riemann invariant is given by

 r(x,u)=[(1+κ)a+→e⋅→C]t−→e⋅→x,→C=(C1,C2,C3)∈R3.

A similar procedure can be applied to the rank- acoustic solution . Here, the wave vector is a non-zero multiple of (1.4 ii) and the corresponding vector fields and Riemann invariant are

 r(x,u)=det(→u,→e,→m)t−(→e×→m)⋅→x,Xi=(→e×→m)idet(→u,→e,→m)∂∂t+∂∂xi,i=1,2,3. (1.13)

Again, the change of variables (1.9) leads to transformed dynamical equations, which we integrate in order to find rank- acoustic solution of the form

 ¯a(¯t,¯x)=a0,¯u1(¯t,¯x)=¯u1(¯x1),¯u2(¯t,¯x)=¯u2(¯x1),C∈R, (1.14) ¯u3(¯t,¯x)=(e1m2−e2m1)−1[C−(e2m3−e3m2)¯u1(¯x1)−(e3m1−e1m3)¯u2(¯x1)].

Here et are arbitrary functions of the Riemann invariant which has the explicit form

 r(x,u)=Ct−det(→x,→e,→m). (1.15)

In general, the overdetermined system composed of (1.7 i) and (1.7 ii) is nonlinear and cannot always be solved in a closed form. Nevertheless, particular rank- solutions for many physically interesting systems of PDEs are well worth pursuing. These particular solutions of (1.7 i) and (1.7 ii) can be obtained by assuming that the function is in the form of a rational function, which may also be interpreted as a truncated Laurent series in the variables . This method can work only for equations having the Painlevé property [4]. Consequently, these equations can be very often integrated in terms of known functions.

Applying a version of the conditional symmetry method to the isentropic model (1.1), several new classes of solutions have been constructed in a closed form [9, 10]. Comparing these results with the ones obtained via the generalized method of caracteristics (GMC) [20], it was shown that more diverse classes of solutions are involved in superpositions (i.e. rank- solutions) than in the case of the GMC [10].

This paper is a continuation of the papers [9, 10]. The objective is to construct bounded elliptic solutions of the isentropic system (1.1) using the version of the CSM proposed in [10]. These types of solutions are obtained through a proper selection of differential constraints (DCs) compatible with the initial system of equations (1.1). That is, the solution should satisfy both the initial system (1.1) and the differential constraints (1.5). Among the new results obtained, we have rank-2 and rank-3 periodic bounded solutions expressed in terms of Weierstrass -functions. They represent bumps, kinks and multiple waves, all of which depend on Riemann invariants. These solutions remain bounded even when the invariants admit a gradient catastrophe.

The paper is organized as follows. In Section 2 we construct rank-2 and rank-3 elliptic solutions of the system, among which multiple waves and doubly periodic solutions are included, and we show that they remain bounded everywhere. Section 3 summarizes the results obtained and contains some suggestions for future developments.

## 2 Rank-2 and rank-3 solutions

The construction approach outlined in Section 1 has been applied to the isentropic flow equations (1.1) in order to obtain rank-2 and rank-3 solutions. The results of our analysis are summarized in Tables 1 and 2. Several of them possess a certain amount of freedom. They depend on one or two arbitrary functions of one or two Riemann invariants, depending on the case. The range of the types of solutions obtained depends on different combinations of the vector fields as given in (1.6). For convenience, we denote by ,  ,  ,  , etc, the solutions which are the result of nonlinear superpositions of rank- solutions associated with different types of wave vectors (1.4 i) and (1.4 ii). By and we denote the Riemann invariants which coincide with the group invariants of the differential operators of the solution under consideration.

The arbitrary functions appearing in the solutions listed in Tables 1 and 2 allow us to change the geometrical properties of the governed fluid flow in such a way as to exclude the presence of singularities. This fact is of special significance since, as is well known [16, 21], in most cases, even for arbitrary smooth and sufficiently small initial data at the magnitude of the first derivatives of Riemann invariants becomes unbounded in some finite time . Thus, solutions expressible in terms of Riemann invariants usually admit a gradient catastrophe. Nevertheless, we have been able to show that it is still possible in these cases to construct bounded solutions expressed in terms of elliptic functions, through the proper selection of the arbitrary functions appearing in the general solution. For this purpose it is useful to select DCs corresponding to a certain class of the nonlinear Klein-Gordon equation which is known to possess rich families of bounded solutions [1]. We choose elliptic solutions of the Klein-Gordon equation because a group theoretical analysis has already been performed [22]. The obtained results can be adapted to the isentropic ideal compressible fluid flow in () dimensions. Thus, we specify the arbitrary function(s) appearing in the general solutions listed in Tables 1 and 2, say , to the differential constraint in the form of the Klein-Gordon -field equation in three independent variables and which form the coordinates of the Minkowski space

 ϕr1r1−ϕr2r2−ϕr3r3=cϕ5,c∈R. (2.1)

Here, we choose to be timelike and to be spacelike coordinates. It is well known (see e.g. [22]) that equation (2.1) is invariant with respect to the similitude Lie algebra involving the following generators

 D=ri∂ri−12ϕ∂ϕ,Pi=∂ri,i=1,2,3,Lab=ra∂rb−rb∂ra,a≠b=2,3,K1a=−(r1∂ra−ra∂r1),a=2,3, (2.2)

where denotes a dilation, represents translations, stands for rotations and for Lorentz boosts. A systematic use of the subgroup structure [22] of the invariance group of (2.1) allows us to generate all symmetry variables in terms of the Riemann invariants . We concentrate here only on the case when symmetry variables are invariants of the assumed subgroups of having generic orbits of codimension one. For illustration purposes, we perform a symmetry reduction analysis on four selected members of the list of subalgebras given in ([22], Table IV) which involve dilations. For each selected subalgebra in Minkowski space , we compute the group invariants of the corresponding Lie subgroup and reduce the equation (2.1) to a second order ODE. The application of the symmetry reduction method to equation (2.1) leads to solutions of the form

 ϕ(r)=α(r)F(ξ(r)),r=(r1,r2,r3), (2.3)

where the multiplier and the symmetry variable are given explicitly by group theoretical considerations and satisfies an ODE obtained by substituting (2.3) into equation (2.1). The results of our computation are listed below.

 1.{D,P1}:α={4c[(r2)2+(r3)2]}−1/4,ξ=12arctanr3r2,F′′+F+F5=0,2.{D,L31}:α={−c(r1)2/4}−1/4,ξ=(r2)2+(r3)2(r1)2,ξ(1+ξ)F′′+(2ξ+32)F′+316F+F5=0,3.{D+1+qqK12,L23}:α={−(2q+1)c}1/4(r1+r2)q/2,ξ=[(r1)2−(r2)2−(r3)2](r1+r2)q,F′′+3q+l2q+11ξF′+F5=0,q=−l/3,l−2,4−3l,l∈Z+,4.{D+12K12,L1−K13}:α=(9/4C)1/4{r2−(r1+r3)2/4}−1/2,ξ=6(r3−r1)+6r2(r1+r3)−(r1+r3)38(r2−(r1+r3)2/4)3/2,(1+ξ2)F′′+73F′+13F+F5=0. (2.4)

The parity invariance of (2.1) suggests the substitution

 F(ξ)=[H(ξ)]1/2

which transforms the equations listed in (2.4) to

 {D,P1} H′′=H′22H−2(H+H3), (2.5) {D,L31} H′′=H′22H−1ξ(1+ξ)[(2ξ+32)H′+38H+2H3], (2.6) {D+1+qqK12,L23} H′′=H′22H−[mξH′+2H3],m=3q+l2q+1=(0,4/3,2), (2.7) {D+12K12,L1−K13} H′′=H′22H−11+ξ2[73ξH′+23H+2H3], (2.8)

where the three admissible values for the scalar come from group theoretical considerations [22]. Each of these four equations possesses a first integral

 K′=14Gg2(gH)′2gH−c04(gH)3−3e0gH, (2.9)

in which the four sets of functions and constants obey the respective conditions

 G=−3c04, g2=4e0c0, G=−3c04ξ(ξ+1), g2=−64e0c0ξ, G=−3c04, (a,e0,g2)=(0,0,k1), (4/3,0,k1ξ4/3), (2,e0,−16e0c0ξ2), G=−3c04(ξ2+1), g=k1(1+ξ2)1/3, e0=0,

( denotes an arbitrary nonzero real constant). Under a transformation which preserves the Painlevé property,

 H(ξ)=U(ζ)/g(ξ), (dζdξ)2=1Gg2,

the equation (2.9) becomes autonomous

 U′2−c0U4−12e0U2−4K′U=0, c0≠0.

When , is either a sine, cosine, hyperbolic sine or a hyperbolic cosine function, depending on the signs of the constants, therefore bounded solutions are easily characterized.

When , it is convenient to first integrate this elliptic equation in terms of the Weierstrass function ,

 U(ζ)=K′℘(ζ)−e0, g2=12e20, g3=−8e30−c0K′2, ℘′2=4(℘−e1)(℘−e2)(℘−e3)=4℘3−g2℘−g3,

(where we abbreviate by ) then to use the classical formulae which connect and various bounded Jacobi functions.

This correspondence is quite easy to write down if one uses the symmetric notation of Halphen [14] to represent the Jacobi functions. Halphen introduces three basis functions

 hα(u)=√℘(u)−eα,α=1,2,3,

and the connection between the Weierstrass function and the Jacobi copolar trio is given by [14, p. 46]

 cs(z|k)h1(u)=ds(z|k)h2(u)=ns(z|k)h3(u)=uz=1√e1−e3,k2=e2−e3e1−e3,

where is the modulus of the Jacobi elliptic functions. For full details on Halphen’s symmetric notation, see [15]. We give here explicit solutions in terms of the Weierstrass function, leaving the conversion to Jacobi’s notation to the reader. These solutions are obtained by convenient choices of the normalization constants .

With the normalization , the general solution of (2.5) is

 F2(ξ)=C℘(ξ)+1/3, ζ=ξ, g2=43, g3=827+43C2, C∈R. (2.10)

With the normalization , the solution of (2.6) has the form

 F2(ξ)=Cξ−1/2℘(ζ)−148k20, ζ=−2k0argth√ξ+1, g2=1192k40, g3=−113824k60+4C23k20,

with .

The three cases for equation (2.7) associated with the subalgebra yield the respective solutions

 q=−k/3:F2(ξ)=C℘(ξ), ζ=ξ, g2=0, g3=4C23,q=4−3k:F2(ξ)=Cξ−2/3℘(ζ), ζ=3k0ξ1/3, g2=0, g3=4C23k20,q=k−2:F2(ξ)=Cξ−1℘(ζ)−112k20, ζ=k0logξ, g2=112k40, g3=−1216k60+4C23k20.

Finally, equation (2.8) integrates as (equation no in (2.4))

 F2(ξ)=C(ξ2+1)−1/3℘(ζ),ζ=ξ2F1(12,56;32;−ξ2), g2=0, g3=4C23k20,

where denotes the hypergeometric function.

Using these results, we construct bounded rank-3 solutions of the equations (1.1). For this purpose, for each general solution appearing in Tables 2 and 3, we introduce the arbitrary functions into the Klein-Gordon equation (2.1) and select only the solutions expressed in terms of the Weierstrass -function.

For illustration, let us now discuss the case of the rank- entropic solution which represents a superposition of three rank- entropic solutions given by (1.12). We assume that the entropic wave vectors , and are linearly independent and take the form

 λEi=(a+→ei⋅→u,−→ei),|→ei|2=1,i=1,2,3.

Hence the corresponding vector fields and Riemann invariants are given by

 X=∂∂x3−det(→e1,→e2,→e3)β3∂∂t+β1β3∂∂x1+β2β3∂∂x2,ri(x,u)=(a+→ei⋅→u)t−→ei⋅→x, (2.11)

where . The rank- entropic solutions invariant under the vector field are obtained through the change of coordinates

 ¯t=t,¯x1=r1(x,u),¯x2=r2(x,u),¯x3=r3(x,u),¯a=a,¯u1=u1,¯u2=u2,¯u3=u3, (2.12)

on . Specifying the form of the solution as a linear superposition of rank- solutions (1.12),

 ¯a=¯a1(r1)+¯a2(r2)+¯a3(r3),→u=κ(→e1¯a1(r1)+→e2¯a2(r2)+→e3¯a3(r3)) (2.13)

the fluid dynamics equations (1.1) transform to

 2∑i=13∑j=i+1[κ(→ei⋅→ej)2+(1−κ)(→ei⋅→ej)−1]¯a′i(ri)¯a′j(rj)=0, (2.14)

while the invariance conditions have the form

 ¯a¯t=¯u1¯t=¯u2¯t=¯u3¯t=0. (2.15)

This solution exists if and only if the three entropic wave vectors intersect at a certain specific angle given by

 cosϕij=−1κ,i≠j=1,2,3,

where denotes the angle between the wave vectors and [10, 20]. Imposing the condition that each of the functions , obeys the ODE

 F′′+F+F5=0,

then according to (2.10), the rank- entropic solution takes the form

 a=3∑i=1Ci(℘(ri,43,827+43C4i)+13)1/2,→u=κ3∑i=1Ci→λi(℘(ri,43,827+43C4i)+13)1/2,ri=−(1+κ)Ci(℘(ri,43,827+43C4i)+13)1/2t+→λi⋅→x,i=1,2,3. (2.16)

Making use of an explicit expression for the zeros of the -function, we show that the values for which the denominator in solution (2.16) vanish are not located on the real axis for a specific choice of the constants of integration . Then we have the following result.

If the constants of integration are equal to , then the elliptic rank- entropic solution (2.16) of the isentropic ideal compressible fluid flow equations (1.1) is bounded.

Indeed, according to recent results obtained by Duke and Imamoglu in [6], the location of the zeros of the -function can be given explicitly in terms of generalized hypergeometric functions.

Considering a lattice , , the doubly periodic Weierstrass -function is defined by

 ℘(z;τ)=1z2+∑ω≠0(1(z−ω)2−1ω2),

where the sum ranges over all . Note that the -function assumes every value of the extended complex plane exactly twice in and since it is even, its zeros are of the form . The value of can be determined from the following theorem [6].

The zeros of the -function are given by

 z0=1+τ2+c2s1/43F2(13,23,1;34,54;s)2F1(112,512;1;1−s),c2=−i√63π,|s|<1,|1−s|<1, (2.17)

where denotes the generalized hypergeometric functions and is a function of the modular discriminant and the Eisenstein series ,

 s=1−1728ΔE34,Δ(τ)=q∏n≥1(1−qn)24,E4(τ)=1+240∞∑n=1n3qn1−qn,q=e2πiτ. (2.18)

It is understood that the principal branch is to be taken in the radical expression .

We now illustrate the application of this theorem by showing that the function is always strictly positive for real . This case is the specific case of solution (2.10) for which . From given invariants and , by finding the roots of the cubic polynomial , one can determine the values of the periods and . In the case when and , we obtain the periods and , hence .

For any pair of periods , the lattice can be rescaled in such a way that is normalized to by using the well-known formula for the -function

 ℘(z,ω1,ω2)=℘(z/ω1,ω2/ω1)/ω21. (2.19)

Introducing the numeric value of into (2.18), we can evaluate from (2.17) and (2.19) the zeros of the Weierstrass function , which are indeed complex. Note that the -function always possesses a double pole at and that it tends to at this point. Since it is continuous for all real , this implies that it must always be strictly positive. Hence, on the real interval and the function is therefore bounded for all real . Choosing the constants as values of the initial constant for which solution (2.10) is bounded then guarantees the boundedness of functions and in solution (2.16). Therefore, solution (2.16) is bounded everywhere, even when the Riemann invariants admit the gradient catastrophe. A similar analysis can be applied to every solution presented in Table 3 to show that they are bounded. This can be accomplished by the same procedure as presented above through the use of the theorem from [6].

This solution is physically interesting since it remains bounded for every value of the Riemann invariants . Thus, it represents a bounded solution with periodic flow velocities (see figure 1). Similarly, it is possible to submit the arbitrary functions of the differential equations no listed in (2.4) to obtain other types of bounded solutions. Table 3 presents these various types of solutions with their corresponding Riemann invariants. They are all bounded solutions of periodic, bump or kink type. Note that these solutions of (1.1) admit gradient catastrophes at some finite time. Hence, some discontinuities can occur like shock waves [5, 13]. Note also that the solutions remain bounded even when the first derivatives of