On nonclassical symmetries of generalized Huxley equations

On nonclass ical symmetries of gen eralized Huxley equations N.M. Iv ano v a † , ‡ and C. Sopho cleous ‡ † Institute of Mathematics of NAS of Ukr aine, 3 T er eshchenkivska Str., 016 01 Kyiv, Ukr aine e-mail: ivanova@imath.kiev.ua ‡ Dep artment of Mathematics and Statistics, University of Cyprus, CY 1678 Nic osia, Cyprus e-mail: christo d@ucy.ac.cy Nonclassical symmetries of a class of generalized Huxley e q uations of fo rm u t = u xx + k ( x ) u 2 (1 − u ) are found. More precis ely , for the c lass under co nsideration we completely classify reduction op erator s with τ = 1 and give a wide num b er of ex amples of equa tions admitting reduction ope rators with τ = 0 . 1 In tro duction Consider reaction-diffusion equation of form u t = u xx + k ( x ) u 2 (1 − u ) , (1) where k ( x ) 6 = 0. This equation mo dels many phenomena that o ccur in d ifferent areas of math- ematical physics and biology . In particular, it can b e used to describ e the spr ead of a recessive adv ant ageous allele through a p opulation in w hic h there are only t w o p ossible alleles at th e lo cus in question. Equation (1) is interesti ng also in the area of n er ve axon p oten tials [18]. Case k = const is the famous Huxley equation. F or more details ab out app lication see [3, 4] and references therein. Bluman and C ole [2] int ro d uced a new metho d for findin g group -in v ariant (called also simi- larit y) solutions of partial differenti al equations. The metho d was called b y the authors “non- classical” to emph asize the d ifference b et ween it and the “classical ” Lie r eduction metho d de- scrib ed, e.g., in [14, 15]. A precise and rigorous d efinition of nonclassical in v ariance was firstly form ulated in [10] as “a generalization of the Lie defi n ition of inv ariance” (see also [21]). Later op erators satisfying the nonclassical in v ariance criterion w ere also called, by differen t authors, nonclassical symm etries, cond itional symmetries, Q -conditional symmetries and red u ction op er- ators [8, 9, 13]. The necessary defin itions, includin g ones of equiv alence of reduction op erato rs, and relev an t statemen ts on th is su b ject are collected in [17, 19]. Bradsha w-Ha jek at al [3, 4] started studying class (1) from the symmetry p oint of view. More precisely , they found some cases of equations (1) admitting Lie and/or nonclassical symmetries. Complete classification of Lie symmetries of class (1 ) is p erformed in [12]. C on d itional s ymme- tries of Huxley and Burgers–Haxley equations ha ving n ontrivial intersectio n with class (1) are in v estigated in [1, 5 – 7, 11]. The present pap er is a step to w ards to the complete classification of nonclassical symmetries of class (1). Reduction op erators of equations (1) h a v e the general form Q = τ ∂ t + ξ ∂ x + η ∂ u , wh ere τ , ξ and η are functions of t , x and u , and ( τ , ξ ) 6 = (0 , 0). W e consider t wo cases: 1. τ 6 = 0. Without loss of generalit y τ = 1. 2. τ = 0 , ξ 6 = 0. Without loss of generalit y ξ = 1. W e present a complete classificat ion for the case 1, while for the ca se 2 w e found sev eral examples. 1 2 Equiv alence transformations and Lie symmetries Since classification of nonclassical symm etries is imp ossible without d etailed kn o wledge of Lie in v ariance pr op erties, we review [12] the equiv alence group and resu lts of the group classification of class (1). Th e complete equ iv alence group G ∼ of class (1) cont ains only scaling and translation transformations of indep en d en t v ariables t and x . More precisely it consists of tr an s formations ˜ t = ε 2 1 t + ε 2 , ˜ x = ε 1 x + ε 3 , ˜ u = u, ˜ k = ε − 2 1 k , where ε i , i = 1 , 2 , 3 are arb itrary constan ts, ε 1 6 = 0. Theorem 1. Ther e exists thr e e G ∼ -ine qui valent c ases of e qu ations f r om class (1) admitting nontrivial Lie invarianc e algebr as (the va lues of k ar e given to gether with the c orr esp onding maximal Lie invarianc e algebr as, c = const ) (se e [12]): 1 : ∀ k , h ∂ t i ; 2 : k = c, h ∂ t , ∂ x i ; 3 : k = cx − 2 , h ∂ t , 2 t∂ t + x ∂ x i . In the follo wing section we search for nonclassical symm etries whic h are not equiv alen t to the ab o v e Lie symmetries. 3 Nonclassical symmetries No w let us r ecall the definition of nonclassical symmetry (or conditional symmetry , or r eduction op erator). Reduction op erators (nonclassical sym m etries, Q -conditional symmetries) of a d ifferen tial equation L of form L ( t, x, u ( r ) ) = 0 ha v e the general form Q = τ ∂ t + ξ ∂ x + η ∂ u , where τ , ξ and η are fun ctions of t , x an d u , and ( τ , ξ ) 6 = (0 , 0). Here u ( r ) denotes the set of all the deriv ativ es of th e fu nction u with resp ect to t and x of order not greater than r , including u as th e d eriv ativ e of order zero. The first-order differenti al fu nction Q [ u ] := η ( t, x, u ) − τ ( t, x, u ) u t − ξ ( t, x, u ) u x is called the char acteristic of the op erator Q . The c haracteristic PDE Q [ u ] = 0 is called also the invariant surfac e c ondition . Denote the manifold defined b y th e set of all the differential consequences of the c haracteristic equation Q [ u ] = 0 in the jet space J ( r ) b y Q ( r ) . Definition 1. T he differentia l equ ation L of form L ( t, x, u ( r ) ) = 0 is called c onditional ly (non- classic aly) invariant with resp ect to the op erator Q if the relation Q ( r ) L ( t, x, u ( r ) )   L∩Q ( r ) = 0 holds, whic h is called th e c onditional invarianc e criterion . T hen Q is called an op erator of c onditional symmetry (or Q -conditional sym m etry , nonclassical symmetry , red u ction op erator etc) of the equ ation L . In Definition 1 the symb ol Q ( r ) stands for the s tandard r -th prolongation of the op erator Q [14, 15]. The c lassical (Lie) symmetries are, in fact, p artial cases of nonclassical sym metries. Th erefore, b elo w w e solv e the problem on fi nding only p ure nonclassica l symmetries whic h are not equiv alent to classical ones. Moreo v er, our approac h is based on application of the notion of equiv alence of nonclassical symmetries with r esp ect to a transform ation group (see, e.g., [17]). F or m ore details, necessary definitions and prop erties of nonclassical symmetries we refer the reader to [16, 17, 19, 21]. 2 Since (1) is an ev olution equation, th ere exist tw o principally d ifferen t cases of finding Q : 1. τ 6 = 0 and 2. τ = 0. First we consider the case with τ 6 = 0. Here without loss of generalit y we can assume that τ = 1. The results are summarized in the follo wing theorem. Theorem 2. Al l p ossible c ases of e quations (1) admitting nonclassic al symmetries with τ = 1 ar e exhauste d b y the fol lowing ones: 1. k = c tan 2 x : Q = ∂ t − cot x∂ x , 2. k = c tanh 2 x : Q = ∂ t − coth x∂ x , 3. k = c coth 2 x : Q = ∂ t − tanh x∂ x , 4. k = cx 2 : Q = ∂ t − 1 x ∂ x , 5. k = c 2 2 ( c > 0) : Q = ∂ t ± c 2 (3 u − 1) ∂ x − 3 c 2 4 u 2 ( u − 1) ∂ u 6. k = 2 x − 2 : Q = ∂ t + 3 x ( u − 1) ∂ x − 3 x 2 u ( u − 1) 2 ∂ u , wher e c is an arbitr ary c onstant. Pr o of. W e searc h for reduction op erator (op erator of nonclassical ( Q -conditional) symmetry) in form Q = ∂ t + ξ ( t, x, u ) ∂ x + η ( t, x, u ) ∂ u . T hen the system of d etermining equations for the co efficien ts of op erato r Q has the form ξ uu = 0 , 2 ξ ξ u − 2 ξ xu + η uu = 0 , 2 ξ ξ x − 2 η ξ u − 3 k ξ u u 3 + 3 k ξ u u 2 + 2 η xu − ξ xx + ξ t = 0 , − k η u u 2 (1 − u ) + 2 kξ x u 2 (1 − u ) + η xx − 2 ξ x η = η t − k x ξ u 2 (1 − u ) − 2 kη u + 3 kη u 2 . F rom the fi rst equation w e obtain immediately that ξ = φ ( t, x ) u + ψ ( t, x ) . Substituting it to the s econd equation w e deriv e η = − 1 3 φ 2 u 3 − φψ u 2 + φ x u 2 + A ( t, x ) u + B ( t, x ) . Then, s plitting the rest of determin in g equations with resp ect to different p ow ers of u implies the follo w ing system of equations for co efficien ts φ , ψ , A and B . 2 3 φ 3 − 3 k φ = 0 , − 4 φφ x + 2 φ 2 ψ + 3 kφ = 0 , − 2 φ x ψ + φ t − 2 φψ x − 2 φA + 3 φ xx = 0 , 2 ψ ψ x − 2 φB + 2 A x − ψ xx + ψ t = 0 , − 1 3 φ 2 φ x + 1 3 k φ 2 + k φψ = 3 kφ x + k x φ, 2 3 φ 2 ψ x − 2 3 φφ xx − 8 3 φ 2 x − 2 k ψ x − 2 k A + 2 φφ x ψ = − 2 3 φφ t − 2 k φ x − k x φ + k x ψ , − 2 φ x A + φ xxx + 2 φψ ψ x + k A − φ xx ψ − 4 φ x ψ x − φψ xx + 2 k ψ x = φ tx − φ t ψ − φψ t − k x ψ + 3 kB , A xx − 2 ψ x A = A t + 2 φ x B − 2 k B , − 2 ψ x B + B xx = B t . (2) 3 No w from the fir st equation of (2) it is obvious that either (i) φ = 0 or (ii) φ t = 0, k = 2 9 φ 2 . Consider separately these t w o p ossibilities. Case (i ) . φ ( t, x ) = 0. System (2) is read now lik e 2 A x − ψ xx + 2 ψ ψ x + ψ t = 0 , − 2 k ψ x − 2 k A − ψk x = 0 , 2 k ψ x + k A + k x ψ − 3 kB = 0 , A xx − 2 Aψ x + 2 k B − A t = 0 , B xx − 2 ψ x B − B t = 0 . F rom the second and third equations we deduce that A = − 3 B . After substituting this to th e previous system we obtain 2 ψ ψ x − 6 B x − ψ xx + ψ t = 0 , k x ψ + 2 kψ x − 6 k B = 0 , 6 ψ x B − 3 B xx + 3 B t + 2 k B = 0 , B xx − 2 B ψ x − B t = 0 . It follo w s from the last tw o equ ations th at B = 0. Then the r est of th e determinin g equations is r ead lik e 2 ψ ψ x − ψ xx + ψ t = 0 , k x ψ + 2 kψ x = 0 . General solution of this system is k = c, ψ = const , k = c ( ax + b ) 2 , ψ = ax + b 2 ta + m , and k = c ψ 2 , ψ ′ = ψ 2 + a. Nonclassical symmetry op erato r obtained f rom the first tw o branc hes of the solution of the ab o v e system are equiv alent to the usual Lie symm etry . The third branch (up to equiv alence transformations of scaling and tran s lations of x ) giv es cases 1–4 of th e theorem. Case (ii) . φ t = 0, k = 2 9 φ 2 . Substituting this to system (2) we obtain easily that ψ t = A t = B t = 0. Then, the rest of th e system (2 ) has the form − 4 φ x + 2 φψ + 2 3 φ 2 = 0 , − 2 φ x ψ − 2 ψ x φ − 2 φA + 3 φ xx = 0 , 2 ψ ψ x − 2 φB + 2 A x − ψ xx = 0 , 2 9 φ 2 ψ x − 2 3 φφ xx − 8 3 φ 2 x − 4 9 φ 2 A + 14 9 φψ φ x = − 8 9 φ 2 φ x , φ xxx + 4 9 φ 2 ψ x − 2 3 φ 2 B + 2 9 φ 2 A − φψ xx − φ xx ψ − 4 φ x ψ x − 2 φ x A + 2 φψψ x = − 4 9 φφ x ψ − φψ t , A xx − 2 ψ x A = − 4 9 φ 2 B + 2 φ x B + A t , B xx = 2 ψ x B + B t . 4 It follo ws then that B = 9(2 ψ x A − A xx ) 2(2 φ 2 − 9 φ x ) , ψ = 6 φ x − φ 2 3 φ , A = 4 φφ x − 3 φ xx 6 φ . Substituting this v alues to the ab ov e sys tem we obtain a system of 4 differen tial equ ations for one f u nction φ only that has fir s t order d ifferen tial consequence of form 3 φ 2 x + φ 2 φ x = 0 . It is not difficult to s ho w th at general solution of th is constraint φ = 3 x + c , φ = c satisfies the whole system for φ . These tw o v alues of φ (tak en u p to equiv alence tr ansformations) giv e r esp ectiv ely cases 6 and 5 of the theorem. Note 1. Cases 1, 2 and 4 with c > 0 we re kn o wn in [3, 4], constan t co effici ent case 5 with c = 2 can b e f ound in , e.g., [11], while 1,2 and 4 with c < 0, 3 and 6 are new. No w we turn in to the case 2. T hat is, we consider nonclassical symmetry op erator with τ = 0. Without loss of generalit y we assume it to b e of the form Q = ∂ x + η ( t, x, u ) ∂ u . An y op erator nonclassical symmetry of the ab ov e form satisfies th e follo wing equation − η xx − 2 η η xu − η 2 η uu + k η u u 2 − k η u u 3 + η t + k x u 3 − k x u 2 − 2 k ηu + 3 k η u 2 = 0 . (3) The corresp onding inv arian t su rface condition is u x = η . Eliminating u x and u xx , equation (1) reads u t = η η u + η x + k ( x ) u 2 (1 − u ) . (4) Using a solution of (3), we can fin d u by inte grating inv ariance s urface condition and then substituting in (4) to deriv e a s olution of (1). No w as it w as shown in [16,20] for more general ca se of (1+ n )-dimensional ev olution equations, in tegration of equation (3) is, in some sense, equiv alen t to int egration of th e initial equation (1). Ho w ev er, since it con tains bigger num b er of u nknown v ariables it is p ossib le to construct certain partial solutions. Th us, for example, we h a v e su cceeded to find all ( G ∼ -inequiv alen t) partial solutions of equ ation (3) of the form η ( x, t, u ) = n X p = − m φ p ( x, t ) u p , where m and n are p ositiv e integers and φ p ( x, t ) unkn o wn functions. W e find the follo wing results: 1. k = 2 B 2 , η = B ( x ) u 2 − tan xu , wh ere − 4 B B ′ + 4 B ′ tan x − B ′′ + 2 B + 2 B 2 tan x = 0 . F or example, this equation has a solution of form B = tan x . 2. k = 2 B 2 , η = B ( x ) u 2 + tanh xu , where 4 B B ′ + 4 B ′ tanh x + B ′′ + 2 B + 2 B 2 tanh x = 0 . In particular, this equation has a solution of form B = − tanh x . 5 3. k = 2 B 2 , η = B ( x ) u 2 + coth x u , wh ere 4 B B ′ + 4 B ′ coth x + B ′′ + 2 B + 2 B 2 coth x = 0 . In particular, this equation has a solution of form B = − coth x . 4. k = 2 B 2 , η = B ( x ) u 2 + u x , wh ere 4 xB B ′ + 4 B ′ + x B ′′ + 2 B 2 = 0 . A solution of th is equation is B = − 1 x . 5. k = 2 x 2 , η = 1 x ( u 2 − 1). 6. k = 1 2 x 2 , η = 1 2 x u 2 . 7. k = 2 tan 2 2 x , η = − u 2 tan 2 x . 8. k = 2 tanh 2 2 x , η = u 2 tanh 2 x . More detaile d inv estigat ion of conditional symmetries and constr u ction of asso cia te similarit y solutions of equ ations fr om class (1) will b e th e s ub ject of a forthcoming pap er. 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