Symmetry, Integrability and Geometry: Methods and Applications Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction ⋆

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.


any of the p
int symmetries admitted by this equation.

Introduction

The construction of group foliations using admitted point symmetry groups for nonlinear partial differential equations (PDEs) is originally due to Lie and Vessiot and was revived in its modern form by Ovsiannikov [1].In general a group foliation converts a given nonlinear PDE into an equivalent first-order PDE system, called the group-resolving equations, whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetry transformations.Each solution of the group-resolving equations geometrically corresponds to an explicit one-parameter family of exact solutions of the original nonlinear PDE, such that the family is closed under the given one-dimensional s mmetry group acting in the solution space of the PDE.

Because a group foliation contains all solutions of the given nonlinear PDE, ansatzes or differential-algebraic constraints must be used to reduce the group-resolving equations into an overdetermined system for the purpose of obtaining explicit solutions.Compared with classical symmetry reduction [2,3], a main difficulty to-date has been how to find effective systematic ansatzes that lead to useful reductions.

An important step toward overcoming this difficulty has been taken in recent work [4,5] on finding exact solutions to semilinear wave equations and heat equations with power nonlineari-ties.Specifically, this work demonstrates that the group-resolving equations for such nonlinear PDEs have solutions arising from a simple separation ansatz in terms of the group-invariant variables.Through this ansatz, many explicit solutions to the nonlinear PDE are easily found, whose form would not be readily obvious just by trying simple direct ansatzes using the original independent and dependent variables in the nonlinear PDE, or by simply writing down the form for classical group-invariant solutions.In particular, some of these solutions are not invariant under any of the point symmetries of the nonlinear PDE and thus fall completely outside of classical symmetry reduction (while others coincide with explicit group-invariant solutions).Most importantly for applications, many of the solutions also have interesting analytical properties related to critical dynamics, blow-u behaviour, asymptotic behaviour and attractors.

We will illustrate this group-foliation reduction method by applying it to obtain explicit exact solutions for − 1)r −1 u r + (p − ku q )u, k = ±1, p = const (1)
for u(t, r), with a nonlinearity power q = 0, −1, where r denotes the radial coordinate in n > 1 dimensions or the half-line coordinate in n = 1 dimension.The symmetry structure of this reaction-diffusion equation is given by [6] time translation
X 1 = ∂/∂t for all n, q, p,(2)
scalin X 2 = 2t∂/∂t + r∂/∂r − (2/q)u∂/∂u only for p = 0,

space translation
X 3 = ∂/∂r only for n = 1,(4)
where X is the infinitesimal generator of a one-parameter group of point transformations acting on (t, r, u).For constructing a group foliation, it is natural to use the time translation generator (2), since this is the only point symmetry admit ed for all cases of the parameters n ≥ 1, q = 0, p.

In Section 2, we first set up the system of group-resolving equations given by the timetranslation symmetry (2) for the reaction-diffusion equation (1), which uses the invariants and differential invariants of the symmetry generator X 1 as the independent and dependent variables in the system.We next state the form required for solutions of the group-resolving system to correspond to group-invariant solutions of the reaction-diffusion equation (1) with respect to he point symmetries generated by X 1 , X 2 , X 3 .

In Section 3, we explain the separation ansatz for directly reducing the system of groupresolving equations.This reduction yields an overdetermined system of differential-algebraic equations which can be readily solved by computer algebra.We present the explicit solutions of these equations and then we derive the resulting exact solutions of the reaction-diffusion equation.These solutions include explicit similarity solutions in the case p = 0, and explicit travelling wave solutions in addition to an ex licit non-invariant solution in the case n = 1.

In Section 4, we show how the success of the reduction ansatz can be understood equivalently as constructing partially-invariant subspaces for a nonlinear operator that arises in a natural way from the structure of the group-resolving equations.This important observation puts our method on a wider mathematical foundation within the general theory of in ariant subspaces developed by Galaktionov [7].

Finally, we mak

some general concluding remarks in Sectio
5.


Group-resolving equations and symmetries

To proceed with setting up the time-translation group foliation for the reaction-diffusion equation (1), we first variants (in terms of t, r, u)
x = r, v = u,(5)
satisfying X 1 x = X 1 v = 0, and the d (in terms of t , tisfying X
(1)
1 G = X (1) 1 H = 0 where X (1)
1 is the first-order prolongation of the generator (2).Here x and v are mutually independent, while G and H are related by equality of mixed n u t and u r , which gives
D r G = D t H,(7)
where D r , D t denote total derivatives with respect to r, t.In addition, v, G, H are related through the reaction-diffusion equation ( 1) by
G − r 1−n D r (r n−1 H) = (p − kv q )v. (8)
Now we put ,(9)G − (n − 1)H/x − H x − HH v = (p − kv q )v,(10)
with independent variables x, v, and dependent variables G, H.These PDEs are called the time-translation-group resolving system for the reaction-diffusion equation ( 1).

The respective solution spaces of equation ( 1) and system ( 9), ( 10) are related by a groupinvariant mapping that is defined thro ifferential invariants (6).

Lemma 1. Solutions (G(x, v), H(x, v)) of the time-translation-group resolving system (9), (10) are in one-to-one correspondence with one-parameter families of solutions u(t, r, c) of the reaction-diffusion equation (1) satisfying the translation-invariance property
u(t + ǫ, r, c) = u(t, r, c(ǫ, c)),
where c(0, c) = c in te d parameter ǫ, such that
u t = G(r, u), u r = H(r, u)(11)
constitutes a consistent pair of par metric first-order ODEs whose integration constant is c.

We now examine the relationship between the symmetry structure of the reaction-diffusion equation ( 1) and the symmetry structure inherited by the time-translation-group resolving system ( 9), (10).

Firstly, through the identifications defined by the variables ( 5), ( 6), the prolongation of any point symmetry generator
X = a 1 X 1 + a 2 X 2 + a 3 X 3 of equation (1) has a natural projection to a point symmetry generator Y = a 2 X (1) 2 + a 3 X (1) 3 modulo X (1)
1 of system ( 9), (10).The time-translation X 1 thus gets annihilate (2/q (1 + 2/q)H∂/∂H when p = 0, (12)
and
Y 3 = ∂/∂x when n = 1.(13)
Secondly, with respect to these inherited symmetries ( 12) and ( 13), the system ( 9), ( 10) has a reduction to ODEs yieldi riant respectively unde scalings x → λx, v → −2/q v when p = 0, and under translations
x → x + ǫ tion-invariant solutions h isfying the ODE system
(h/g) ′ = 0, g − hh /g) ′ = −2/g, g + (2 − n + 2/q)h − (h + 2V /q)h ′ = −kV q+1 .(17)
Integration of the parametric ODEs (11) for such solutions ( 14), ( 15 and ( 16), (17) leads to the following two correspondence results.

Lemma 2. In the case n = 1, there is a one-to-one correspondence between solutions of the translation-group resolving system (9), (10) with the invariant form (14) and one-parameter families of travelling-wave solutions of the reaction-diffusion equation (1) given by the groupinvariant form u = f (ξ) where, modulo time-translations t → t + c, the variable ξ = r − t/a is an invariant of the translation symmetry X = a∂/∂t + ∂/∂r = aX

+ X 3 in
erms of some constant a (determined by ODE system (15)).

Lemma 3. In the case p = 0, there is a one-to-one correspondence between solutions of the translation-group resolving system (9), (10) with the invariant form (16) and one-parameter families of similarity solutions of the reaction-diffusion equation (1) given by the group-invariant form u = r −2/q f (ξ) where, modulo time-translations t → t + c, the variable ξ = t/r 2 is an invaria t of the scaling symmetry X = 2t∂/∂t + r∂/∂r − (2/q)u∂/∂u = X 2 .

Furthermore, in all cases, static solutions u(r) of the reaction-diffusion equation correspond to solutions of the translation-group resolving system with G = 0. Hereafter we will be interested only in solutions with G = 0, corres

nding to dyna
ical solutions of the reaction-diffusion equation.


Main results

To find explicit solutions of the group foliation system (9), (10) for (G(x, v), H(x, v)), we will make use of the same general homogeneity features utilized in [4,5].First, the non-derivative terms (p − ku q )u in the reaction-diffusion equation (1) appear only as an inhomogeneous term in equation (10).Second, in both equations ( 9) and ( 10) the linear terms involve no derivatives with respect to v. Third, the nonlinear terms in the homogeneous equation ( 9) have the skewsymmetric form HG v − GH v , while HH v is the only nonlinear term appearing in the nonhomogeneous equation (10).Based on these features, this system c G = g 1 (x)v + g 2 (x)v a , H = h 1 (x)v + h 2 (x)v a , a = 1. (18)
For such a separation ansatz (18), the linear terms G x , G, H/x, H x in equations ( 9) and (10) will contain the same powers v, v a that appear in both G and H, and moreover the nonlinear term HG v −H v G in the homogeneous equation ( 9) will produce only the power v a due to the identities
v a (v) v − (v a ) v v = (a− 1)v a and v(v) v − (v) v v = v a (v a ) v − (v a ) v v a =
0. Thus, equation ( 9) can be satisfied by having the coefficients of v and v a separately vanish.Similarly the nonlinear term HH v in the non-homogeneous equation (10) will only yield the powers v, v a , v 2a−1 .Since we have a = 1 and q = 0, equation ( 10) can be satisfied by again having the coefficients of v and v a separately vanish and by also having the term containing v 2a−1 balance the inhomogeneous term kv q+1 .In this fashion we find that equations ( 9) and (10) reduce to an overdetermined system of 5 algebraic-differential equations for g 1 (x), g 2 (x), h 1 (x), h 2 (x), together with the relation a = 1 + q/2.This system can be solved by a systematic integrability analysis.We have carried out this analysis using the computer algebra program Crack [8], which contains a wide repertoire of modules for reduction of differential order and polynomial degree, splittings with respect to the independent variable(s), eliminations, substitutions, factorizations, integrations, and length-shortening of equat (−p/2) 1/2 tan (−p/2) 1/2 (x + c 1 ) v , n = 1, q = 2, p < 0. (23G = 3 2 pv ± (pk) 1/2 coth (p/2) 1/2 (x + c 1 ) v 2 , H = ±(k/2) 1/2 v 2

(p/2) 1/2
oth (p/2) 1/2 (x + c 1 ) v, n = 1, q = 2, p > 0,
and
G = 3 2 pv ± (−pk) 1/2 cot (−p/2) 1/2 (x + c 1 ) v 2 , H = ± (k/2) 1/2 v 2 + (−p/2) 1/2 cot (−p/2) 1/2 (x + c 1 ) v , n = 1, q = 2, p < 0,
as obtained through the trig i=1 h 1+i (x)v a i ,(24)
where
a i = 1, a i − a j to a considerably more complicated analysis compared to the previous two-term ansatz (with N = 1 −1 in e the non-homogeneous equation (10) contains the further powers v 2a i −1 , and v q+1 .To determine the exponents in these powers, a systematic examination of the possible balances (which rapidly increase in number with N ) is necessary.We carry out this analysis by computer algebra, again using the program Crack [8].

In particular, for any fixed N ≥ 1, Crack can be run automatically by setting up a priority list of modules to be tried recursively with case splittings given a high priority, in order to split equations ( 9) and ( 10) under the ansatz (24), (25) and i each case solve the resulting overdetermined system of algebraic-differential equations for g i and h i , i = 1, 2, . . ., N .Importantly, the modules in Crack are able to organize the case distinctions in an efficient way that lessens a combinatorial explosion by using algebraic conditions (equalities and inequalities) arising in the steps of solving the equations to keep track of whether a possible balance of two powers of v generates a new case (or subcase) which has to be solved or not.Running Crack, we find that the system (9), ( 10), ( 24), (25) has solutions when N = 2 but not when N = 3. (Based on this outcome we have not tried to investigate the system when N > 3.) Proposition 2. For q = 0, −1, and N ≤ 3, the separation ansatz (24) yields two additional solutions of the translation-group resolving system (9), (10) with G = 0:
G = 3 pv + (k 2 p) 1/3 + (kp 2 ) 1/3 v 1/2 , H = −(2/p) 1/2 pv + (k 2 p) 1/3 + (kp 2 ) 1 3 v 1/2 , n = 1, q = −3/2, p > 0;
Remark 2. Solutions (31) and (32) are related by shifts of the arbitrary constants c 1 → c 1 ± iπ/(2p) 1/2 and c → c ∓ iπ/(3p) thr ns, solution (26) is a similarity solution, since it has the form hown in Lemma 3.This solution has been obtained in our previous work [5] using a two-term separation ansatz to solve the group-resolving equations given by the scaling symmetry (3) for the reactiondiffusion equation (1) in the case p = 0.

Solutions ( 27) to (30) are travelling-wave solutions with the form shown in Lemma 2 (where the constant a is given by the wave speed).The two solutions ( 27) and (28) were first obtained in [11] through standard symmetry reduction.To the best knowledge of the authors, the other solutions ( 29) and (30) are new.

In contrast, solutions (31) to (34) are non-invar efines a nonlinear operator acting on the pair of variables (G, H).Then we view the ansatz (18) as defining a linear space of functions with a separable power form
G H = g 1 h 1 v + g 2 h 2 v a , a = 1(36)
where the coefficients depend only on x.This linear space (36) is not invariant under Φ because
Φ g 1 h 1 v + g 2 h 2 /x − h 2 1 − p v + (a − 1)(h 2 g 1 − h 1 g 2 ) g 2 + (1 − n)h 2 /x − (a + 1)h 1 h 2 v a + 0 −ah 2 2 v 2a−1 + 0 k v q+1
produces terms that have additional powers v 2a−1 , v q+1 .However, the operator Φ will preserve a set of functions contained in the linear space (36) if the power a and the coefficients g 1 , g 2 , h 1 , h 2 satisfy the conditions 2a − 1 = q + 1, ah 2 2 = k, so that the terms containing v 2a−1 and v q+1 cancel out.In this sense the linear space will then be partially-invariant [7] under Φ.As a consequence of this invariance property, the evolution equation ( 35) for (G, H) reduces to a overdetermined system of differential-algebraic equations which we can solve for g 1 , g 2 , h 1 , as discussed in Section 2.

The separable power ansatz (18) thus can be viewed as explicitly constructing a partiallyinvariant linear subspace for the group-foliation operator Φ.A similar discussion applies to the multi-term ansatz (24).

It is worth pointing out that apart from the mathematical interpretation given to the ansatzes (18) and (24), the theory of (partially) invariant linear subspaces does not provide any general constructive method or algorithm for finding a successful ansatz.So the main aspect of the present work is to demonstrate the effectiv

ess of a general se
aration ansatz for solving group-foliation equations.We also emphasize that this approach is much more effective than if an analogous separation method were to be applied either directly for sol ing the given reactiondiffusion PDE or instead for seeking a (partially) invariant linear subspace for that PDE itself.


Concluding remarks

In general the method of group foliation reduction using a separation ansatz as illustrated in this paper is able to yield exact solutions to nonlinear 2nd order PDEs with power nonlinearities.

This method works with any admitted group of point (or contact) symmetries and gives a systematic reduction of the group foliation equations into an overdetermined system consisting of algebraic equations and 1st order differential equations that can be derived and in most cases solved by means of computer algebra (e.g. using t e program Crack [8]).In particular, for a given nonlinear 2nd order PDE having two independent variables, solutions are produced in an explicit form, whereas standard symmetry reduction only gives a 2nd order ODE that still has to be solved to find group-invariant solutions explicitly and in general this step can be quite difficult.

Moreover, because the group foliation equations contain all solutions of the given nonlinear PDE, the method can yield solutions that are not invariant under any of the point (or contact) symmetries admitted by the nonlinear PDE.

It is straightforward to extend this method to higher-order PDEs with power nonlinearities.As well, it should be possible to apply the same method to PDEs having more general forms of nonlinearities by utilizing a general eparation of variables ansatz with respect to all of the independent variables in the group foliation equations.In general, the success of such an ansatz can be interpreted mathematically [7] as constructing a (partially) invariant linear subspace for a nonlinear operator coming from the structure of the group foliation equations.

Related work using a similar group-foliation method applied to nonlinear diffusion equations appears in [12].Group foliat