Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

1 Introduction

In the analysis of partial differential equations (PDEs), exact solution plays a crucial role, especially in the study of asymptotical behaviour, blow-up, extinction, dispersive, and also in testing numerical solution. This motivates many researchers to propose methods for finding exact solutions, such as Hirota’s bilinear method [1], Darboux transformation [2], Bäcklund transformation [3], inverse scattering transformation [4] and symmetry group method [5], [6], [7]. Among all these methods, the symmetry group method was shown to be effective in dealing with PDEs as evidenced by the number of research papers, books and symbolic manipulation software related to it.

It is well known that Sophus Lie first introduced the theory of continuous group, now referred as Lie point symmetry group or classical symmetry group, which unified and extended various methods for construction of explicit solutions to ordinary differential equations (ODEs). Since then, there were considerable developments related to the symmetry group. On one hand, the symmetry group was found to be of importance in the study of conservation laws [8], [9], Hamiltonian structures [6], fundamental solutions [10], [11], Fourier transformations [12], reconstruction and integrability [13]. On the other hand, the fact remains that some PDEs admit poor point symmetry or symmetry reductions, for many PDEs are unobtainable by applying the classical symmetry method that inspired the creation of several generalisations of Lie point symmetry group, including non-classical symmetry method [14], Lie-Bäcklund symmetry method, potential and nonlocal symmetry method [15] and conditional Lie-Bäcklund symmetry method [16], [17].

Consider wave equation

for u(t, x), where a, b and c are constants, p(≠0) is nonlinearity power and n(>1) is dimension. The analysis of solutions for the Cauchy problem to (1) was discussed in [18], [19]. It can be easily seen that the radial reduction of (1) is given by

where r=|x|, u_r=∂u/∂r and u_rr=∂²u/∂r². The radial wave equation (2) with special values of n arises as a mathematical model to describe various phenomena in physics. For example, when n=2, (2) corresponds to the radial compressible polytropic fluid flow in Lagrangian coordinates [20]. In a recent work [21], point symmetries and conservation laws were fully classified for (2) with restrictions n≠1 and (bp)²+a²(b+c)²≠0. To be one of the extended works of [21], in this paper, we are devoted to deriving explicit group-invariant radial solutions u(t, |x|=r) of (1) by applying the method which was used successfully in the previous works [22], [23] to find the group-invariant solutions to semilinear Schrödinger equations in multi-dimensions and inhomogeneous nonlinear diffusion equation. For this purpose, we, firstly reduce (2) under each point symmetry subgroup in the optimal systems, and then solve the reduced ODEs that have explicit solutions so that the group-invariant radial solutions for (1) can be written out.

The rest of this paper is organised as follows. In Section 2, we present the optimal systems of one-dimensional subalgebras for (2) in terms of the known point symmetries. Section 3 is devoted to solving the reduced ODEs explicitly, which are determined by the reductions under each symmetry in the optimal systems. In Section 4, explicit group-invariant radial solutions of (1) along with their basic analytical features are listed. Finally, Section 5 states the conclusions of this paper.

2 Symmetries and Optimal Systems

For (2), a point symmetry is a one-dimensional Lie group of transformations in (t, r, u)

with a group parameter ε, such that the prolongation prX of infinitesimal generator

satisfies

on the solutions of (2), where prX is the second-order prolongation of generator X (see Ref. [6]). It is straightforward to eliminate u_tt and its derivatives by (2) in (5) and split the resulting equation with respect to u_tr, u_rr, u_t and u_r. Then the system of overdetermined linear PDEs for τ(t, r, u), ζ(t, r, u) and η(t, r, u) reduces to PDEs

which are easily solved to obtain the following result.

Theorem 1:The point symmetries of (2) with n≠1 and (bp)²+a²(b+c)²≠0 are generated by [21]

The corresponding transformation groups acting on solution u=f(t, r) are given by

where group parameters satisfy −∞<ε<∞, 0<λ<∞.

To find all group-invariant solutions, it is sufficient to determine the optimal system of one-dimensional subalgebras. The classification problem is equivalent to the problem of classifying the algebra of system into conjugacy inequivalent subalgebra under the adjoint action of the algebra. It is easily carried out by simplifying a general Lie algebra operator as much as possible in terms of all kinds of adjoint actions [6]. After a systematic calculation, we arrive at the following result.

Theorem 2:The optimal systems of one-dimensional subalgebras admitted by (2) are generated by (i) X₁, X₂in the general case; (ii) X₁, X₃+a₁X₁, X₂+a₂X₃when cp(p−1)=0, b≠0; (iii) X₄, X₅, X₁+a₁X₅, X₂+a₂X₄when p=1 and b=0; (iv) X₃, X₁+a₁X₃, X₂+a₂X₃, X₁+X₆+a₃X₃when p=−4 and c=0; (v) X₁, X₂, X₃+a₁X₁, X₂+a₂X₃, X₇+a₃X₁with a₂≠0 when cp(p−1)=0, (a+b+c(1−p))(n−3)=bp(n/2−1) and n≠2; (vi) X₁, X₂+a₁X₈, X₃+a₂X₁, X₈+a₃X₁when cp(p−1)=0, a+b+c(1−p)=0, n=2 and b≠0; (vii) X₂, X₃, X₇, X₁+a₁X₃, X₁+a₂X₇, X₂+a₃X₃, X₁+X₆+a₄X₃, X₁+X₆+a₅X₇with a₁a₃a₄≠0 when p=−4, c=0, (a+b)(n−3)=bp(n/2−1) and n≠2, (viii) X₇, X₁+a₁X₅, X₂+a₂X₄, X₄+a₃X₇, X₅+a₄X₇, X₁+X₇+a₅X₅when p=1, n=3 and b=0; (ix) X₃, X₈, X₁+a₁X₃, X₁+a₂X₈, X₂+a₃X₈, X₁+X₆+a₄X₃, X₁+X₆+a₅X₈with a₂a₅≠0 when p=−4, c=0, a+b=0 and n=2 where −∞<a_i<∞, (i=1, …, 5) are parameters.

3 Reduced ODEs and Solutions

In this section, we focus on solving reduced ODEs formulated in terms of invariants [7] which are determined by point symmetry groups in optimal systems. For convenience, we replace the parameters a₁, …, a₅ in Theorem 2 by constant α. Firstly, it is readily seen that under symmetry (18), (2) is reduced to vʺ=0 possessing solution v=C₁ξ+C₂, where and hereafter C_i (i=1, 2, …) denote arbitrary constants. The reduced ODE obtained under symmetry X₁+X₆+αX₃ is not easy to handle for finding the general solution, but we can obtain its invariant solution v=((2bn−a−5b)/a)^1/4. Then let us turn to solve the other reduced ODEs in detail.

Example 1: Consider equation

which arises from the reduction of (2) under time-translation symmetry (12). The corresponding group-invariant solution is u=v(ξ), where ξ=r. To get explicit solutions, we consider the following two cases.

Case 1:p=0, b+c≠0 and a+b+c≠0

In this case, (30) has an integrating factor 1/v′, which yields a first-order ODE

possessing explicit solutions

Thus solutions (32) and (33) provide two solutions for (30) with p=0 and (b+c)(a+b+c)≠0.

Case 2:p=0, b+c≠0 and a+b+c=0

In this case, the integrating factor 1/v′ reduces (30) to a first-order ODE

which can be solved explicitly

Here we obtain two solutions of (30) with p=0 and a=−(b+c)≠0, given by (35) and (36).

Example 2: Consider equation

which arises from the reduction of (2) with c(p−1)=0 and bp≠0 under scaling symmetry (14). The corresponding group-invariant solution is u=r^2/pv(ξ)−cp/b, where ξ=t. It is easily seen that (37) is a non-linear oscillator equation. By utilising integrating factor v′, (37) can be integrated to quadratures

where bp≠0. Note that quadrature (38) presents an implicit solution of (37) for p=−2 and b≠0, whereas quadrature (39) can be evaluated to get explicit solutions for v in terms of elementary functions for special values of parameters. To proceed, we shall make use of a change of variable

so that the respective expression is

Taking into account the value of C₁, we arrive at the following two cases.

Case 1:C₁=0

In this case, integral (39) together with (40) and (41) becomes

which yields one explicit solution of (37) for p≠0, −2 and b≠0, given by

Case 2:C₁≠0

In this case, the required conditions arising from (41) are

By solving conditions (44) and (45) for p and s, we have

Subcase 2.1: Quadrature for p=−1

For case (46), the integral (39) together with (40) and (41) becomes

which yields one explicit solution of (37) for p=−1 and b≠0

Subcase 2.2: Quadrature for p=−4

For case (47), the integral (39) together with (40) and (41) becomes

which yields one explicit solution of (37) for p=−4 and b≠0

Example 3: Consider equation

which arises from the reduction of (2) with c(p−1)=0, a+b=0, n=2 and ap≠0 under symmetry (19). The corresponding group-invariant solution is u=(r ln r)^2/pv(ξ)−cp/b, where ξ=t. Equation (52) is also a non-linear oscillator equation. Thus, analogous to the analysis in Example 2, we arrive at two quadratures

with ap≠0, where quadrature (53) determines an implicit solution of (52) for p=−2 and a>0. In terms of the change of variable (40) with (41), two cases are needed to be considered.

Case 1:C₁=0

In this case, quadrature (54) can be evaluated to get one explicit solution of (52) for p≠0, −2 and a≠0

Case 2:C₁≠0

To construct explicit solutions in terms of elementary functions, conditions (44) and (45) are required, which lead to case (46) and case (47).

Subcase 2.1: Quadrature for p=−1

When p=−1 and s=1, the integral (54) is transformed to

which leads to one explicit solution of (52) for p=−1

Subcase 2.2: Quadrature for p=−4

When p=−4 and s=2, the integral (54) is transformed to

which leads to one explicit solution of (52) for p=−4

Example 4: Consider equation

which is derived from the reduction of (2) with p=0 and bα≠0 under optimal symmetry X₃+αX₁. The corresponding group-invariant solution is u=exp(2bt/α)v(ξ), where ξ=r. To solve (60) completely, we consider the following three cases.

Case 1:b+c=0

In this case, (60) becomes a first-order ODE

where bα≠0, whose solution is given by

Case 2:a+b+c=0, a≠0

In terms of integrating factor ξⁿ⁻¹v⁻¹, (60) can be reduced to a first-order ODE

where bα≠0. By solving (63), we get three explicit solutions for (60) with a+b+c=0 and abα≠0

Case 3:a+b+c≠0, b+c≠0

In this case, we use a change of variables s=(2b−(a+b+c)/(α(b+c)))ξ and w=s^−1+n/2v^{(a+b+c)/(b+c)} to transform (60) into Bessel’s equation

whose general solution is

where J_μ(s) and Y_μ(s) are Bessel functions. Thus the explicit solution of (60) with αb(b+c)(a+b+c)≠0 is given by

Example 5: Consider equation

which is obtained from the reduction of (2) with p=1 and b=0 under optimal symmetry X₁+αX₅. The corresponding group-invariant solution is u=v(ξ)+αt²/2, where ξ=r. The case of α=0 is the reduction discussed in Example 1, so here we assume α≠0. To solve (70) completely, we distinguish the following two cases.

Case 1:c=0

In this case, (70) becomes to av′² – α=0, with general solution

Case 2:c≠0

In this case, with transformations s=−aαξ/c and w=s^−1+n/2exp((a/c)v), (70) gets transformed to (67) with general solution (68). Thus, the general solution of (70) for c≠0 is given by

Example 6: Consider equation

which arises from the reduction of (2) with (a+b+c(1−p))(n−3)=bp(n/2−1), cp(p−1)=0, n≠2, b+c(1−p)≠0 and a+b+c(1−p)≠0 under optimal symmetry X₇+αX₁. The corresponding group-invariant solution is u=r−mv(ξ)−cpb+c(1−p), where ξ=t−α(n−2)(a+b+c(1−p))rn−2 and m=(n−2)(b+c(1−p))a+b+c(1−p). Here constant α is also assumed to be non-zero. The condition cp(p−1)=0 leads to the following three cases.

Case 1:c=0

When c=0, (73) has an integrating factor ((a+b)²−bα²v^p)^−1+a/(bp), which yields a quadrature, viewed as an implicit solution of (73) with c=0, (a+b)(n−3)=bp(n/2−1) and αb(a+b)(n−2)≠0,

Case 2:p=0

In this case, (73) becomes

where α(b+c)(a+b+c)≠0, which has three solutions

Solutions (76)–(78) thereby provide three solutions of (73) with p=0 and α(b+c)(a+b+c )≠0.

Case 3:p=1

In this case, (73) becomes

where (a+b)(n−3)=b(n/2−1), αb(a+b)(n−2)≠0. In terms of transformation w=(a+b)²−bα²v, (79) is converted to

whose general solution is

We then can find the general solution of (79)

Hence, solution (82) provides a solution of (73) with p=1, (a+b)(n−3)=b(n/2−1) and αb(a+b)(n−2)≠0.

Example 7: Consider equation

which arises from the reduction of (2) with b=c(p−1), bp=0, n=3 and a≠0 under optimal symmetry X₇+αX₁. The corresponding group-invariant solution is u=v(ξ)−(cp/α)ln r, where ξ=t−(α/a)r. Assuming α≠0. For further analysis, we distinguish three cases.

Case 1:c=0

When c=0, (83) reads as

where aα≠0, whose general solution can be expressed by quadrature

Thus (85) provides an implicit solution for (83) with c=0 and apα≠0.

Case 2:p=0

When p=0, (83) reads as

where aα≠0 and c=−b, which has two solutions

Solutions (87) and (88) thus are solutions of (83) with p=0, c=−b and aα≠0.

Case 3:p=1

In this case, (83) reads as

where aα≠0. Solving (89), we obtain solutions

which are two solutions of (83) with p=1 and aα≠0.

Example 8: Consider equation

which arises from the reduction of (2) with a+b+c=0, p=0, n=2, bα≠0 under optimal symmetry X₈+αX₁. The corresponding group-invariant solution is u=r^2bt/α exp(2bt/α) v(ξ), where ξ=r. Note that when a=0, (92) only has trivial solution v=0. Alternatively, when a≠0, two integrating factors v⁻¹ and 1 of (92) produce its general solution

Example 9: Consider equation

which is obtained from the reduction of (2) with p=1, n=3, b=0 and aα≠0 under optimal symmetry X₄+αX₇. The corresponding group-invariant solution is u=v(ξ)+ r/(aα)−(c/a) ln r, where ξ=t. It is easy to know that the general solution of (94) is given by

Example 10: Consider equation

which is obtained from the reduction of (2) with p=1, n=3, b=0 and aα≠0 under optimal symmetry X₅+αX₇. The corresponding group-invariant solution is u=v(ξ)+ tr/(aα)−(c/a)ln r, where ξ=t. Solving (96), we get

Example 11: Consider equation

which is derived from the reduction of (2) with p=1, n=3, b=0, a≠0 under optimal symmetry X₁+X₇+αX₅. The corresponding group-invariant solution is u=v(ξ)− αr²/(2a²)+αtr/a−(c/a) ln r, where ξ=t−r/a. To solve (98) completely, we need to discuss two possibilities: a²=c and a²≠c.

Case 1:a²=c

In this case, (98) thereby becomes to a first-order ODE

with a≠0, which has general solution

Then we find one solution of (98) with a²=c, given by (100).

Case 2:a²≠c

In this case, (98) gets converted to a second-order linear ODE

by use of transformation v=(2aaαξ+aαξ2+(c−a2)ln(w2(ξ)/(16a3α)))/(2a), (a≠0). The general solution of (101)

determines the general solution of (98) with a≠0, which is of form

Example 12: Consider equation

which is derived from the reduction of (2) with a+b=0, c=0, p=−4, n=2 under optimal symmetry X₁+X₆+αX₈. The corresponding group-invariant solution is u=v(ξ)/rlnr/(1+t2), where ξ=2α arctan(t)−ln ln r. Equation (104) is not tractable to solve in general by standard integrating method. Nevertheless, it is still available to construct explicit solution of (104) with α=0. For the case of α=0, a change of variables θ=exp(−ξ)/2 and V=v exp(ξ/2) converts (104) into

which has solutions

Hence, (104) with b=−a and α=0 has three explicit solutions

Additionally, an invariant solution of (104) can be derived, given by

4 Exact Solutions of Radial Wave Equation

From the explicit solutions of reduced ODEs shown in Section 3, group-invariant solutions u=f(t, r) to (2) will now be written out in this section.

Theorem 3:The radial wave equation(2) withn≠1 and (bp)²+a²(b+c)²≠0 has the following group-invariant solutions