EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2562-2573 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations Winter Sinkala1,∗, Molahlehi Charles Kakuli1 1 Department of Mathematical Sciences and Computing, Faculty of Natural Sciences, Walter Sisulu University, South Africa Abstract. The double reduction method for finding invariant solutions of a given partial differ- ential equation (PDE) provides for the reduction of a q-th order PDE that admits a Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order q− 1. In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs. 2020 Mathematics Subject Classifications: 35C05, 58J70, 70H33, 35B06 Key Words and Phrases: Double reduction, Lie symmetry analysis, Conservation law, Invariant solution 1. Introduction The double reduction method, proposed by Sjöberg [19, 20], provides a powerful routine that exploits the relationship between Lie symmetries and conservation laws of a given PDE to find invariant solutions of the PDE. The theory of double reduction relies on the pioneering work by Kara and Mahomed [12, 13] on the relationship between Lie symmetries and conservation laws. For a scalar (1+1)-PDE of order q that admits a Lie symmetry and an associated nontrivial conservation law (in the sense defined in [12]), double reduction of the PDE amounts to a reduction of the equation to an ODE of order q − 1. All the articles we have examined that have used the double reduction method, as proposed by Sjöberg [19, 20], have exploited both the admitted Lie symmetries and the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5318 Email addresses: wsinkala@wsu.ac.za (W. Sinkala), ckakuli@wsu.ac.za (M.C. Kakuli) https://www.ejpam.com 2562 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2563 corresponding nontrivial conservation laws in the reduction algorithm (evident in refer- ences [1–4, 6–9, 11, 14, 15, 17–20]). In this article we demonstrate that it is not essential to utilise an associated conservation law in the reduction routine. Rather, it suffices to know that a given Lie symmetry has an associated nontrivial conservation law. For Euler-Lagrange PDEs, for example, every admitted Noether symmetry has an associated conservation law. Therefore, one could proceed to perform double reduction of an Euler- Lagrange PDE using only an admitted Noether symmetry. The rest of the paper is organised as follows. In Section 2, the theory of double reduction is presented, based on (1 + 1)-scalar PDEs as proposed by Sjöberg [19, 20]. In Section 3, we present the steps involved in the proposed alternative double reduction algorithm and provide illustrative examples based on five PDEs. We give concluding remarks in Section 4. 2. Theory of double reduction of a scalar (1 + 1)-PDE Preliminaries of the double reduction method are extensively covered in numerous articles. To avoid redundancy, these foundational details will not be repeated in this article. Interested readers can consult references [1–4, 6–8, 11, 15, 17–20]), and other related works for comprehensive information. Consider a scalar qth-order (q ≥ 1) PDE with two independent variables (x1, x2) = (t, x) and one dependent variable u = u(t, x), F (t, x, u, . . . , u(q)) = 0, (1) where u(k) denotes the collection {uk} of kth-order partial derivatives. Furthermore, sup- pose that Equation (1) admits a Lie point symmetry with an infinitesimal generator X = ξi(x, u) ∂ ∂xi + η(x, u) ∂ ∂u . (2) A conservation law of (1) DtT t +DxT x = 0 (3) is said to be associated with (2) if X(T i) + T iDk(ξ k)− T kDk(ξ i) = 0, i = 1, 2. (4) In terms of canonical variables r, s and w, i.e., variables under which (2) is transformed into X = ∂ ∂s , the conservation law (3) can be written in canonical variables as [19, 20] DrT r +DsT s = 0, (5) where T r = T tDt(r) + T xDx(r) Dt(r)Dx(s)−Dx(r)Dt(s) (6) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2564 and T s = T tDt(s) + T xDx(s) Dt(r)Dx(s)−Dx(r)Dt(s) . (7) The components T t and T x in (3) depend on (t, x, u, u(1), u(2), . . . , u(q−1)), which means that T r and T s depend on (r, s, w,wr, wrr, . . . , wrq−1) for solutions invariant with respect to X. Therefore, the conservation law in canonical variables (5) becomes ∂T s ∂s +DrT r = 0. (8) From the association of X with T = (T r, T s), it follows that XT r ≡ ∂T r ∂s = 0 and XT s ≡ ∂T s ∂s = 0. (9) This leads to further reduction of the conservation law (8) to DrT r = 0, (10) or, equivalently, T r = k, (11) where k is an arbitrary constant. Equation (11) is an ODE of order q− 1, and its solution can easily be transformed into an invariant solution of (1). 3. An alternative double reduction algorithm It follows from the double reduction routine that canonical variables, while reducing the conservation law (3) into the ODE (10), must necessarily transform the PDE (1) into an ODE equivalent to (10), i.e., of the same order as the original PDE. The reduction of order by one of this “equivalent” ODE is subsequently achieved through routine calculations. The steps for implementing the alternative double reduction routine are presented below: (i) Identify a symmetry X = ξi(x, u)∂xi + η(x, u)∂u of the PDE (1) for which there exists an associated nontrivial conservation law. (ii) Find similarity variables r, s and w, i.e., variables under which the symmetry X is transformed to its canonical form Y = ∂ ∂s . The similarity variables can be determined from the conditions X(r) = 0, X(s) = 1, X(w) = 0. These variables establish an invertible mapping r = R(t, x), s = S(t, x), w =W (t, x, u), ∂w ∂u ̸= 0 from the space (t, x, u, u(1), . . . , u(q)) to the space (r, s, w,wr, . . . , wrq). (iii) Express all partial derivatives of u in (1) in terms of r, s, w and derivatives of w. W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2565 (iv) Write the PDE (1) in canonical variables in the form Drψ(r, w,wr, wrr, . . . , wrq−1) = 0, (12) for some function ψ, where w = w(r). It follows from (12) that ψ(r, w,wr, wrr, . . . , wrq−1) = k, (13) where k is an arbitrary constant. Equation (13) is the desired ODE of order q − 1. In the illustrative examples that follow, we perform double reduction of five PDEs following the algorithm outlined above. Although we have provided associated nontrivial conservation laws for each of the symmetries used in the examples, the conservation laws are not used in the double reduction procedure. Example 1. The PDE [7] utt − utxx − 3u2uxx + uxxxx − 6uu2x − 1 x5 = 0 (14) admits a Lie point symmetry with infinitesimal generator X = 2t∂t + x∂x − u∂u (15) associated with the nontrivial conservation law Dt(ϕ t) +Dx(ϕ x) = 0, where ϕt = (ut − uxx)t− t2 2x2 , ϕx = tuxxx − 3tu2ux + ux. (16) To find canonical variables, we solve characteristic equations dx x = dt 2t = du −u (17) corresponding to (15). We obtain canonical variables r = x√ t , s = ln t 2 , w = u √ t or w = xu, (18) where w = w(r). In (18) we essentially have two sets of canonical variables, one corre- sponding to w = u √ t and the other to w = xu. Case 1.1: Reduction using canonical variables with w = u √ t The inverse canonical variables of (18) in this case are given by t = e2s, x = res, u = e−sw, (19) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2566 and the partial derivatives that appear in (14), expressed in terms of the canonical variables using the routine outlined in [10], are: ux = e−2swr, uxx = e−3swrr utt = 1 4e −5s ( r2wrr + 5rwr + 3w ) uxxt = −1 2e −5s(rwrrr + 3wrr), uxxxx = e−5swrrrr. (20) Substituting (19) and (20) into (14), we obtain an ODE of the same order as (14), namely r7wrr + 5r6wr + 2r6wrrr − 12r5w2wrr − 24r5ww2 r +3r5w + 6r5wrr + 4r5wrrrr − 4 = 0. (21) To obtain a lower order ODE, we find functions h and ψ so that (21) can be written in the form h(r, w)Drψ(r, w,wr, wrr, wrrr) = 0. (22) Comparison of equations (21) and (22) leads to a system of determining equations which we solve to obtain h = r5, and ψ = 1 r4 − 12w2wr + 3rw + r2wr + 4wr + 2rwrr + 4wrrr. (23) Case 1.2: Reduction using canonical variables with w = xu The inverse canonical variables of (18) in this case are given by t = e2s, x = res, u = e−sw r , (24) and partial derivatives that appear in (14), expressed in terms of the canonical variables are: ux = e−2s ( wr r − w r2 ) uxx = e−3s ( 2w r3 − 2wr r2 + wrr r ) , utt = 1 4e −5s(rwrr + 3wr) uxxt = −1 2e −5swrrr uxxxx = e−5s ( 24w r5 − 24wr r4 + 12wrr r3 − 4wrrr r2 + wrrrr r ) . (25) Substituting (24) and (25) into (14), we obtain 4r4wrrrr + 2r3 ( r2 − 8 ) wrrr + ( r6 − 12r2w2 + 48r2 ) wrr − 24r2ww2 r +3r ( r4 + 24w2 − 32 ) wr − 48w3 + 96w − 4 = 2r5Drψ = 0, (26) where ψ = 1 2r4 + 6w3 r4 + ( 1− 12 r4 ) w + ( r 2 + 12 r3 − 6w2 r3 ) wr + ( 1− 6 r2 ) wrr + 2wrrr r . (27) Example 2. The PDE [7] uxxxx − utxx − nenuuxx + utt − n2enuu2x − 1 x4 = 0 (28) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2567 admits a Lie point symmetry with infinitesimal generator X = t∂t + x 2 ∂x − 1 n ∂u. (29) The symmetry (29) has an associated nontrivial conservation law Dt(ϕ t) + Dx(ϕ x) = 0, where ϕt = x(ut − uxx), ϕ x = −nxenuux + enu − uxx + xuxxx + 1 2x2 . (30) Canonical variables arising from (29) are r = x√ t , s = ln t, w = ln t n + u or w = 2 lnx n + u, (31) where w = w(r). Case 2.1: Reduction using canonical variables with w = n−1 ln t + u The inverse canonical variables of (31) in this case are given by t = es, x = res/2, u = nw − s n , (32) and the following partial derivatives of u in terms of the canonical variables are obtained: ux = e− s 2wr, uxx = e−swrr, utt = e−2s 4n ( nr2wrr + 3nrwr + 4 ) uxxt = −1 2e −2s(rwrrr + 2wrr), uxxxx = e−2swrrrr. (33) Substituting (32) and (33) into (28), we obtain wrrrr + rwrrr 2 + r2wrr 4 + wrr − nenwwrr + 3rwr 4 − n2enww2 r − 1 r4 + 1 n = r−1Drψ = 0, (34) where ψ = rwrrr + r2wrr 2 − wrr + r3wr 4 − nrenwwr + enw + r2 2n + 1 2r2 . (35) Case 2.2: Reduction using canonical variables with w = 2n−1 lnx + u The inverse canonical variables of (31) in this case are given by t = es, x = res/2, u = w − 2 ln r + s n . (36) Therefore, the partial derivatives of u expressed in terms of the canonical variables are: ux = e− s 2 ( wr − 2 nr ) uxx = e−s ( 2 nr2 + wrr ) , utt = 1 4re −2s(rwrr + 3wr) uxxt = −1 2e −2s(rwrrr + 2wrr), uxxxx = e−2s ( 12 nr4 + wrrrr ) . (37) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2568 Upon substituting (36) and (37) into (28), we obtain 4nr4wrrrr + 2nr5wrrr − nr2 ( 4nenw − r4 − 4r2 ) wrr − 4n3r2w2 re nw +nr ( 16nenw + 3r4 ) wr − 24nenw − 4n+ 48 = 4nr3Drψ = 0, (38) where ψ = rwrrr + ( r2 2 − 1 ) wrr + ( r3 4 − nenw r ) wr + 3enw r2 +− 6 nr2 + 1 2r2 + 1 n . (39) Example 3. The BBM equation [19] utxx − ut + uux = 0 (40) admits Lie point symmetries with infinitesimal generators X1 = ∂x and X2 = ∂t. (41) These symmetries have an associated nontrivial conservation law Dt(ϕ t) + Dx(ϕ x) = 0, where ϕt = u3 3 , ϕx = u2t − u2tx − u2utx − u4 4 . (42) Let X = αX1 +X2, where α is an arbitrary constant. Canonical variables derived from X are r = x− αt, s = t, w = u, (43) where w = w(r). From (43), we obtain the inverse canonical variables t = s, x = αs+ r, u = w, (44) and the following partial derivatives of u expressed in terms of the canonical variables: ux = wr, ut = −αwr, uxxt = −αwrrr. (45) Substituting (44) and (45) into (40), we obtain αwr − αwrrr + wwr = Drψ = 0, (46) where ψ = αw − αwrr + w2 2 . (47) Example 4. The PDE [5] uxxx + ut + u2ux + uux = 0 (48) admits Lie point symmetries with infinitesimal generators X1 = ∂x and X2 = ∂t. (49) W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2569 The nontrivial conservation law of (48) Dt(ϕ t) +Dx(ϕ x) = 0, where ϕt = u2 2 + u 2 + 1 2 , ϕx = u4 4 + u3 2 + u2 4 − u2x 2 + ( u− 1 2 ) uxx (50) is associated with the symmetries in (49). Let X = αX1 + X2, where α is an arbitrary constant. Canonical variables derived from X are r = x− αt, s = t, w = u, (51) where w = w(r). From (51), we obtain inverse canonical variables t = s, x = αs+ r, u = w, (52) and the partial derivatives: ux = wr, ut = −αwr, uxxx = wrrr. (53) Substituting (52) and (53) into (48), we obtain wrrr + w2wr + wwr − αwr = Drψ = 0, (54) where ψ = wrr + w3 3 + w2 2 − αw. (55) Example 5. The PDE [14] (see also [16]) ut − uux − uxxx = 0 (56) admits a Lie point symmetry with infinitesimal generator X = x∂x + 3t∂t − 2u∂u. (57) Furthermore, the nontrivial conservation law of (56) Dt(ϕ t) +Dx(ϕ x) = 0, where ϕt = tu2 2 + ux, ϕx = − tu 3 3 − tuuxx + tu2x 2 − xu2 2 + ux − xuxx. (58) is associated with (57). Canonical variables derived from (57) are r = x3 t , s = ln t 3 , w = ut2/3 or w = ux2, (59) where w = w(r). W. Sinkala, M.C. Kakuli / Eur. J. Pure Appl. Math, 17 (4) (2024), 2562-2573 2570 Case 5.1: Reduction using canonical variables with w = ut2/3 From (59), the inverse canonical variables in this case are given by t = e3s, x = r1/3es, u = e−2sw. (60) Therefore, the following partial derivatives of u expressed in terms of the canonical vari- ables are obtained from (60): ux = 3r2/3e−3swr, ur = −1 3e −5s(3rwr + 2w) uxxx = e−5s ( 27r2wrrr + 54rwrr + 6wr ) . (61) Substituting (60) and (61) into (56), we obtain 81r2wrrr + 162rwrr + 3 ( 3r2/3w + r + 6 ) wr + 2w = 3r2/3 r1/3 + w Drψ = 0, (62) where ψ = r2/3w + 2r1/3w2 + w3 + 9 ( r2/3 + 2r1/3w ) wr +27 [( r4/3w + r5/3 ) wrr − 1 2 r4/3w2 r ] . (63) Case 5.2: Reduction using canonical variables with w = x2u We obtain from (59), in this case, inverse canonical variables t = e3s, x = r1/3es, u = e−2sw r2/3 , (64) and the following partial derivatives of u expressed in terms of the canonical variables: ux = e−3s(3rwr−2w) r , ut = −r1/3e−5swr uxxx = e−5s ( 27r4/3wrrr + 24wr r2/3 − 24w r5/3 ) . } (65) Substituting (64) and (65) into (56), we obtain 6w2 + w(72− 9rwr)− 3(r + 24)rwr − 81r3wrrr = 3r3 r + w Drψ = 0, (66) where ψ = −w 3 r2 − 2(r + 6)w2 r2 − (r + 24)w r + 27w2 r 2 + 27wr − 27(r + w)wrr. (67) REFERENCES 2571 4. Concluding remarks In the standard double reduction method as proposed by Sjöberg [19, 20], both a Lie symmetry of a PDE and an associated nontrivial conservation law are used to find invariant solutions of the PDE. The algorithm involves writing the conservation law in terms of canonical variables derived from the associated Lie symmetry. The association of the conservation law with the symmetry ensures that, in canonical variables, the conservation law is reduced to an ODE of order one less than that of the PDE. In this article, we have demonstrated that double reduction can be realised by simply transforming the PDE, in stead of the conservation law, using the constructed canonical variables. This means that any symmetry of the PDE known to have an associated nontrivial conservation law may be used to perform double reduction of the PDE, without explicitly using the conservation law in the algorithm. We have provided examples involving five (1 + 1)-PDEs. For each of the PDEs in the illustrative examples, we started by identifying an ad- mitted Lie point symmetry that has an associated nontrivial conservation law. We then found canonical variables r, s and w, i.e., variables under which the Lie symmetry is transformed into X = ∂/∂s. The canonical variables constitute an invertible mapping from the (t, x, u, u(1), . . . , u(q))-space to the (r, s, w,wr, . . . , wrq)-space. In the latter space, the PDE is reduced to an ODE of the same order, but one which could be written in the form Dr(·) = 0 to complete the reduction. Conflict of interest The authors declare that they have no competing interests. Author contributions WS conceived the presented idea. WS and MCK performed the computations, dis- cussed the results, and contributed to the final manuscript. Both authors approved the submitted version. Acknowledgements We would like to thank the Directorate of Research Development and Innovation of Walter Sisulu University for continued support. We also thank the anonymous reviewers for their careful reading of our manuscript and their useful comments. References [1] Stephen C Anco and Maria Luz Gandarias. Symmetry multi-reduction method for partial differential equations with conservation laws. 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