Group properties of a class of semilinear hyperbolic equations

Inr., .Von-Lincur Medwcs, Printed tn Great Bruin OOZO-7462.86, S3.M) + ,443 Pergamon Journals Ltd. Vol. 21. No. L PP. 117-155. 1986 zyxwvutsrqpo GROUP PROPERTIES OF A CLASS OF SEMILINEAR HYPERBOLIC EQUATIONS zyxwvutsrqponmlkjihgfedc EDVIGEPuccr and M. CESARINASALVATORI Department of Mathematics, Perugia University, via Vanvitelli 1, 06100 Perugia, Italy (Received 30 April 1984; revised 2 July 1985; received/or publication 1 October 1985) Abstract-We present the results of a systematical investigation of invariance properties of a semilinear hyperbolic equation u,, = J(u), under a one-parameter Lie group of transformations, for arbitrary f(u). The intinitesimais, resulting generators of the Lie algebra, and ordinary differential equations determining invariant solutions, are determined. 1. INTRODUCTION In the last years, the group theoretical analysis has been intensively applied to non-linear evolution equations [l-3]; in this paper the group properties of the semilinear hyperbolic equation in two variables: us= f(u) (1.1) are developed, using the Lie theory of continuous transformation groups [4-6].Equation (1.1) arises in a variety of theoretical physics and applied mathematics situations;f(u) is any reasonable non-linear function which is chosen as a derivative ofa potential energy [7]. This is especially true in the casef(u) = sin u, which is known as the Sine-Gordon equation, it is a useful model for a large class of mechanical questions (propagation of a crystal dislocation. magnetic-flux, propagation in a large Josephson junction, propagation of ultrashort optical pulses, propagation in ferromagnetic materials of waves carrying rotations of direction of magnetization etc.) and so for geometrical problems (study of geometry of surfaces with gaussian curvature zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA k = - 1) [8]. Furthermore, equation (1.1) has been discussed [9] with a cubic non-linearity: f(u) = u - u3 as a model in non-linear meson theory of nuclear forces (phi-four equation). Other particular cases are obtained from investigation of non-linear theory of elementary particles as well [lo]. We are able to reduce equation (1.1) to an ordinary differential equation through its group analysis with the characterization of: (a) point transformations which leave this equation invariant; (b) the respective infinitesimals; (c)the related similarity variables. For the essential steps of this procedure see e.g. [3]. The solutions of this equation depend on free constants which enter into the infinitesimals; this class of exact invariant solutions is the larger all the more the algebra is extended. The above solutions play an important role, either because they are stationary with respect to the transformation groups or because they appear in problems of “source” type and in asymptotic limits of other solutions. In this class of functions, there are no free functions included; nevertheless it is possible to take into account these solutions even for initial and boundary problems; this occurs when the further conditions are themselves invariant under the transformation group. Let us consider, first of all, the infinitesimals of the “universal” group of a one-parameter group transformation, that is to say, those infinitesimals which leave (1.1) invariant for each f(u). Later we obtain all f(u) for which its relative “particular” group is larger than the universal one. In connection to the various infinitesimals we study the invariant properties and determine, if possible, the relative Lie algebra and the finite transformations. In this way, we find again some well-known results for the two expressions off(u) pointed out before; one finds that the two f(u) are not included in that class whose functions have some more extended particular groups; the invariant group for both these functions is the universal one VI. 147 E. Pucn 148 2. INVARIANCE and M. C. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM SALV AT ORI AND INFINITESIMALS section we proceed through the characterization of the infinitesimals of the oneparameter group transformation. Let a one-parameter E group transformation of the variables X, t, u be taken as: In this t* = T(x, t, u, E), x*= X(x, 6 4 For a function 8(x, t) under every transformation defined by: E), u*= U(x, t, u, &). (2-l) we can have a function 3*(~*, t*) which is 3: cwx, t,wf,0, E),7-k [, qx,t), 43 = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH U(x, t, 8(x, t), 6). The transfo~ation group (2.1) leaves equation (1.1) invariant if, for every solution 9(x, t) of (l.l), it associates a function 9*(x*, t*) solution of the equation: u,* ., 1 = (2.2) f(u”). Among the solutions of (1.1) we point out those which are invariant under the transformation group (2.1); these satisfy the one-parameter functional equation: 9 [X(x, t, 9, E),7-(x, t, 9, E)I = Uk 4 9, d* In the usual notation p = a,, 4 = u,, r = uXr, s = y,, u = uxr = u,, (1.1) becomes zyxwvutsrqponmlkjih F = u -f(u) = 0, As is known [5,6], the invariance condition under the transformation infinitesimals: (2.3) group involves the and becomes : lEFf+* =O (2.4) with 3Ethe operator: where CC, $, y, 6, < are the first and second order extensions of the transformation defined by (2.3) the invariance condition (2.4) (with u = f(u)) is: [5]. For F df where f’ = ;II. Since this relation has to be identically zero for a11values of p, q, r, s, the terms in brackets must vanish; we get a system of linear homogeneous partial differential equations from which we can derive the infinitesimals t, g, q. 149 Group properties of a class of semilinear hyperbolic equations We have: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 5 = T(T), q = au + b(x, t) 5 = 5(x), (2.6) zyxwvutsrq uf’(w)a + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f’(u)b -f(u)@ - 5’ - T’) - b, = zyxwvutsrqponmlkjihgfedcbaZY 0 (2.7) dt T’ = g. where C’= z, 3. INFINITESIM ALS FOR THE “UNIV ERSA L” AND “PARTICULAR” GROUPS We can observe that (2.7) is the only equation containingj(u); it shows howf(u) enters in determining the infinitesimals of the group. Equation (2.7) is a linear equation in uf’(u), f’(u), f(u). For f( u1 is arbitrary, uf’, f’, f are free variables; (2.7) is verified if and only if a = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC b(x, t) = 0, 5’ + T’ = 0; these conditions and (2.6) imply: Theorem 1. The infinitesimals of the universal group (valid for every f(u)) are: Go:r=It+p, <= v=o --It+v, with J., p, v arbitrary parameters. Iff(u) is a common solution of all the differential equations obtained for every (x, t) from (2.7), we can define a set of infinitesimals larger than Go. For fixed (x, t) the corresponding differential equation has the following solutions: f(u) = -b,,/(a - 5’ - r’); f(u) = c exp[ - 45 + r')/b] + b,,/(<' + t’) f(u) = d + b,,u/b for a = 0,5’ + r’ z 0; for a = 0, <’ + T’ = 0; f(u) = b,,/(C’ + T’ - a) + g(au + b)@-C'-6)'o f(u) = baa- ’ (log@ + (3.1) b) + h fora#O,a-c’-r’#O; foraZO,a-<‘-r’=O. (3.2) (3.3) (3.4) (3.5) As (x, 1)varies,f(u) must remain the same; this implies that the above expression must define somef(u) independent of (x, t), eventually obtained through pertinent values of the constants c, d, g, h. This last condition leaves out the expression (3.5) because its independence from (x, t) implies b = constant as well. For (3.4) we have an analogous simplification and finally we state: Theorem 2. A larger particular set of infinitesimals is given for the following assumptions: (0 f(u) = constant = k (ii) f(u) = hu + k (iii) f(u) = (hu + k)les (iv) f(u) = k + xexp@u) h# O s#O,l;h#O X # 0;p # 0. (M(u) = k Equation (2.7) becomes: k(a - <’ - T’) + b, = 0 and so: b(x,t) = k[tC(x) + XT(~) - axt] + al(x) + al(t) where a1 eaz are arbitrary functions of x and t respectively. E. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Puccr and M . C. SALVATORI 150 Therefore we have: Theorem 3. For f(u) = k the infinitesimals G, of the particular group are: G1 : T = t(t), q = au - kuxr + k(rt(.x) + xr(t)) it = T(x), + cx,(xf + q(t). (ii)f(u) = hu + k Equation (2.7) becomes: u({’ c t’) + hb - k(u - 5’ - T’) - zyxwvutsrqponmlkjihgfedcbaZYXWVUT b, = 0. This relation is verified (identically in u) if and only if: 4’ + hb - Tf = 0, ka - b,, = 0 and these, in addition (2.6), imply: Theorem 4. Forf(tr) = hu + k the infinitesimals Gz of the particular group are: G2:7 5 = --IlxfV, = It + p, v = au + ka/h + b(x,t) where 5(x, t) is a solution of: h7j - Ii,, = 0. (3.6) (iii)f(u) = (hu - t k)‘- ’ Equation (2.7) becomes: [bh(l - s) - k(a - g’ - t’)] i- uh(<’ + T’ - as) - b,,(hu + k)” = 0 and this last, in addition (2.6), gives the following: Theorem 5. For f(u) = (hu $ k)’ - s the infinitesimals G3 of the particular group are: zyxwvutsrqpon G 3:7 = (A + as,‘2)t f p, 5=(-A-t-af/2)x+v, q = au + ak/h. (iv)f(u) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA au f x exp(pu) Equation (2.7) becomes: ~~~~exp(~u~ + [bm - ~(a - r’ - r’)]exp(pu) + k(lj’ + *t’ - a) - b, = 0 from which we obtain: a= 0, 0, k(<’ + 7’) - b,, = 0. + 7YP? k(c’ f t’) = 0 bp + r + T’ = These conditions and (2.6) impiy : l.Z= 0, b= - (i? and therefore the following: Theorem 6. For f(u) = k + x exp(pu) there exists a particular group if and only if k = 0. The infinitesimals G4 of this group are: G.,: 7 = t(t), 5 = 5’(x), 4. SIM ILARITY q = -(<’ + 7’)lP. SOLUTIONS We have defined the infinitesimals t, 4, q of the various groups for all the possibie assumptions off(u). Each of these i~~tesimals generates a one-parameter transformation group and the invariability of the solutions of (1.1) under this group is given by: 151 Group properties of a class of semilinear hyperbolic equations <(x)u, + T(C)& = au + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ b(x,t). (4.1) The general solution of (4.1) is implicitly defined by: rpk 4 u) = 9[z(x,Ol (4.2) where S(z) is an arbitrary function, z and cp are a couple of first integrals of system (4.2) named similarity variable and dependent variable respectively [5]. Determining the invariant solutions is the same as finding the 9 functions which define solutions of (l.l)using (4.2); an ordinary differential equation of second order in 9(z) arises. For every 9, solution of this equation, a similarity solution is defined by solving (4.2): t4 = @-‘{X,t,.F[z(X,~)l}. Let us determine the similarity and dependent variables and the relative ordinary equation in correspondence with the infinitesimals Gi i = 0, 1,. . . ,4. G,;f(u) arbitrary. For the infinitesimals G,, is cp=u (4.3) and z = vt - px for A = 0 and z = (t + p/1)(x - v/E.) for 1 # 0. (4.4) By substituting in (1.1) we have the differential equations: .F”Vp = -f(F) Fz + 9’ = f(F) (4.5) (4.6) for rl.= 0 and L # 0 respectively. The general solution of (4.5) is and defines the similarity solutions when f(u) is prescribed. These solutions are given as u = 9(vt - p) and are travelling-wave form The general solution of (4.6) is not generally calculated; we may analyse this equation, characterizing the possible movablecritical points, that is to say, equation (4.6) may or may not be a Painlevl: equation [ 111; for f(u) = sin u, it was found, that its resultant ordinary differential equation falls within the V” Painleve class. For f(u) = u - u3 (4.6) does not fall within any Painlevt class [2]. G,;f(u) = k. Equation (1.1) is in this case: u, = k. (4.7) The full class of solution of (4.7) is obtained as a class of similarity solutions. It is enough to consider the subclass of G, infinitesimals with a = O,al(x) = O,az(t) = O.Then thesimilarity 152 E. Puca and M. C.SALVATOIU variable, the dependent variable and the ordinary equation are: j(x)-‘dx, cp = u- kxr, .P = 0. (4.8) We can see that the similarity solutions are: u =fi(x) +fiw + kxr (4.9) withf, ,fz arbitrary functions because 7 and < are arbitrary as well. Therefore, in this case, we find that all the solutions of (4.7) are the same ones (4.9) which are invariant with respect to the transformations defined by the above subclass of infmitesimals. G2;f(u) = hu + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ k. Equation (1.1) is now linear: U xt = hu + (4.10) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM k through the infinitesimals Gz we find the same similarity variables as in the case of the universal group, that means (4.4). For 1 = 0 the similarity solutions are defined by: u = exp(ar/lr)jp- rj [6(x, t) + ku/h]exp( -at/p) dr + 9(z)} where x = (vr - z)/p, @x,r) an arbitrary solution of (3.6) and g(z) as a solution of: .?Ft"V,U2 -9'+h=O is given by: 9(z) = c,exp@,z) + czexp(bzz). While for 1 # 0 the similarity solutions are: u = (k + p,)lljA j [6(x, r) + ka/h](Ar + p)-“ ‘ ” -‘ dr + F(z) where x = 1-l (v + z/(,lr -t p)), 6(x, r) as above and P(z) solution of 9”i2Z + 6’(A2 + al) - .Fh = 0, is given by: 9(z) = (z/n2)-.‘2,~-,,,(2J~). G3; f(u) = (hu + k)’ -I. Taking kl = 1 + as/2 and k2 = -1 + as/2 from the infinitesimals G3 the similarity variable and the similarity solutions are: z = (k,x + v)(k,r + p)-lrt’ lr2; u = -k/h + .F(k,r + p)@“ ’ where Y(z) satisfies the following: d” k:z + Fk,(a -t k,) + (.? W I)‘-~ = 0 GA;f(u) = 1 w(w). 153 Group properties of a class of semilinear hyperbolic equations For the infinitesimals G4 the similarity variable is defined again by (4.8) while the similarity solutions are : s -5 ’ u=9-p-’ + 7 ’ dt 7 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO with Y(z) solution of: - Xexp(pS) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP 0 9” and so implicitly defined by: Z= 5. INFINITESIM AL J(- :exp(p9) OPERATORS + C,)-‘I2 d9. AND FINITE TRANSFORM ATIONS We define now some group properties with respect to the infinitesimals determined before [6]. The infinitesimals GOof the universal group depend on three constants 1, ~1,v; the linear space L of the infinitesimals is here finite dimensional: dim zyxwvutsrqponmlkjihgfedcbaZYXWVUT L = 3; so the relative linear space of the differential operators is three dimensional as well. Let us denote this space with L3. The basis in this space can be given as three operators below: (XI and X1 are operators of translation, I3 is an operator of dilatation). From the table of commutators it follows that L3 is a Lie algebra. The finite transformations are solutions of Cauchy problem: dx* dE=-Ax*+v, dr* -=Ac*+/I, ds x*(o) = x, du* z=O u*(o) = u, t*(O) = t, and therefore : t* = -p/j. + (t + p/L/l)exp(-&A), x* = v/A + (x - v/A)exp( --El), u* = U. Let us consider besides the three operators (5.1) the following: St a sx a 14=ydr+yjy+u a ka xl;;+hau we obtain a basis for the linear space L4 of the differential operators that are related to the particular infinitesimals Ga which are admitted whenf(u) = (hu + k)l- ‘. This space is a Lie algebra as well; the finite transformations are in this case: t* = (A + as/2)-‘{ [(A + as/2)t + p]exp[e(A + as/2)] - Ic), x* = (-1 + OS/~)-‘{[(-I + as/2)x + v]exp[e(-A + as/21 -v}, u* = -h/k + (u + k/h)exp(as). L is finite dimensional in any other case, that is for the particular infinitesimals G, , G2, G, because they depend on some arbitrary functions. E. Puca and M. C, SALVATORI I.54 When = k we can easily see that G t related a space of operators L = L’ @ L” which is a direct sum of L” and L”; L* is a one dimensional space whose basis is and L” is an infinite dimensional space to which corresponds ind~~ndent operators of the form: The group of finite transfo~at~o~s t” = t* an infinite set of linearly related to Lr is: XJ = x, u* = kxt -f f@- ~~~X~~~~~ related to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Lm are defined by the Cauchy problem: while the finite transformations dt* = r(Ph de x*(o) = x, t*(O) = t, u*(o) = U” For J(u) = ttu f k, with the particular in~nites~mals Gzt the space of the operators is L = L4 @ Lm where L4 is a four dimensional space with a basis Xt , X2 ‘IX3 given by (5.I) and X.$= @i-t- Wh;; and La is the infinite ~~e~sio~~ space corresponding to the operators of the form 6(x, r)i with f;(x, t)a solution of (3.6). The finite transfo~ations generated by L4 and L” are: r* =5 --#A + (t c ~~~~x~~&~); u* = -k/h x* = v/J. i- (x - v/2.)exp( -en); + (u + k/h)exp(ae) and t* = t* x’ = x, u* = 6{x9 Q& f tl respectively. Finally forf(u) = xexp(pu) with particular infinitesimals C4 it is L = L” ; L” is defined by the following infinite set of linear i~de~nde~t operators: For consequence the finite transformations dt* = s(P), d& dx* x = @P), are given by the Cauchy problem: - fUx*) + T’~~~))~~, Group properties of a class of semilinear hyperbolic equations 155 3. R. Chand, D. T. Davy and W. F. Ames, On the similarity solutions of wave propagation for a general dass of non-linear dissipative materials. Inc. J. Non-linear Mech. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM 11, 191-205 (1976). 4. W. F. Ames, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Non- linear Partial Differential Equations in Engineering. Academic press, New York (1982). 5. G. W. Bluman and J. D. Cole. Similarity M ethods for Differential Equations. Springer, Berlin (1974). 6. L. V. Ovsiannikov, Group A~ly sic ofDifirentia1 Equations (Edited by W. F. Ames). Academic press, New York (1982). 7. G. B. Whitham, Linear and Non- linear W aves. John Wiley, New York (1974). 8. A. Barone, F. Esposito, C. J. Magee and A. C. Scott, Theory and application of Sine-Gordon equation. Rivista Ntwoo Cimento 1 (2). 227-267 (1971). 9. L. I. Shiff, Non-linear Meson theory of nudear forces. Phys. Reu. 84, 1-11 (1951). 10. J. K. Perring and T. H. R. Skyrme, A model unit’ied field equation. Nucl. Phys. 31, 550-555 (1962). 11. E. L. Ince, Ordinary Difirential Equations. Dover, New York (1956).