ON NEW EXACT SOLUTIONS OF NONLINEAR WAVE EQUATIONS. Abstract

Transcription

1 UDK 517.9: Ivan YURYK ON NEW EXACT SOLUTIONS OF NONLINEAR WAVE EQUATIONS Abstract A new simple method for constructing solutions of multidimensional nonlinear dalembert equations is proposed Let us consider a nonlinear Poincare-invariant d'alembert equation u + F(u) = 0, (1) qu gu gu dxl дх'( dx 2 n' F(u) is an arbitrary smooth function. Papers [1, 5, 6, 7] are devoted to the construction of exact solutions to equation (1) for different restrictions on the function F(x). Majority of these solutions is invariant with respect to a subgroup of the invariance group of equation (1), i.e., they are Lie solutions. One of the methods for constructing solutions is the method of symmetry reduction of equation (1) to ordinary differential equations. The essence of this method for equation (1) consists in the following. Equation (1) is invariant under the Poincare algebra AP( 1, n) with the basis elements Joa = %fida + xad0, Jab = xbda - xadb, P0 = до- Pa = da (a, b = 1, 2,..., n). Let L be an arbitrary rank n subalgebra of the algebra AP{l,n). The subalgebra L has two main invariants и, и = w(xq, x\,..., xn). The ansatz и = ц>(и) corresponding to the subalgebra L reduces equation (1) to the ordinary differential equation (Vu;) 2 + фпи + F(<p) = 0, Such a reduction is called the symmetry reduction, and the ansatz is called the symmetry ansatz. There exist eight types of nonequivalent rank n subalgebras of the algebra AP( 1, n) [5]. In Table 1, we write out these subalgebras, their invariants and values of (Vw) 2, Do; for each invariant. 1

2 ()UJ\ duj'2 dio i dio 2 0(jJ\ du>2 Vwj Vw2 =. dxo dx0 dxi dxi dxn dxn Let us impose the condition on equation (4), under which equation (4) coincides with the reduced equation (3). Under such assumption, equation (4) decomposes into two equations k fn + <Pi + F(v>) = 0, Wi 2^12(Vwx Vw2) + V22(Vu;2) 2 + <p i2nu>2 = 0. (5) Equation (5) will be fulfilled for an arbitrary function ip if we impose the conditions w2 = 0, (Vw2) 2 = 0, (6) Vu>i-Vu;2 = 0 (7) on the variable u>2. Therefore, if we choose the variable oj2 such that conditions (6), (7) are satisfied, then the multidimensional equation (1) is reduced to the ordinary differential equation (3) and solutions of the latter equation give us solutions of equation (1). So, the problem of reduction is reduced to the construction of general or partial solutions to system (6), (7). The overdetermined system (6) is studied in detail in papers [10, 11]. A wide class of solutions to system (6) is constructed in papers [10, 11]. These solutions are constructed in the following way. Let us consider a linear algebraic equation in variables x0, xi,...,xn with coefficients depending on the unknown lo2: 00(^2)^0 - ai(w2)a;i an(u>2)xn - b(uj2) = 0. (8) Let the coefficients of this equation represent analytic functions of uj2 satisfying the condition [a0(w2)] 2 - [ai(w2)] 2 K(w2)] 2 = 0. Suppose that equation (8) is solvable for w2 and let a solution of this equation represent some real or complex function ^2(^0,..., xn). (9) Then function (9) is a solution to system (6). Single out those solutions (9), that possess the additional property VuvVo>2 = 0. It is obvious that du>2 a0 duj2 «1 du>2 an dx0 6' ' dxi 6' ' " ' ' dxn 6' ' S(u2) = a0(uj2)x0 - ai(u2)xi an(co2)xn - b(cj2) and 5' is the derivative of S with respect to io2. Since du>i _ x0 dull _ xi du>i xn dxo u\ ' dxi u}\ ' ' dxn ui ' 3

3 we have 1. Vui-Vu>2 = 7j( a o x o - iixi - anxn). u>i o' Hence, with regard for (8), the equality Vu>i-Vti;2 = 0 is fulfilled if and only if b(co2) = 0. Therefore, we have constructed the wide class of ansatzes reducing the d'alembert equation to ordinary differential equations. The arbitrariness in choosing the function lo2 may be used to satisfy some additional conditions (initial, boundary and so on). b) The symmetry ansatz u = ip(ui), = (xj-\ \-xf) 1/2, 1 < I < n-1, is generalized in the following way. Let u>2 be an arbitrary solution to the system of equations d 2 u> d 2 uj d 2 u> _ dx"l 9xf+1 dx 2 ' ^ The ansatz u = <p(ui,w2) reduces equation (1) to the equation d 2 ip k - 1 dtp + F(<p) = 0. du>l u> i du 1 If I = n - 1, then the ansatz u = <p(u i,cj2), symmetry ansatz u = <p{u 1). u>2 = x0 - xn is a generalization of the Ansatzes corresponding to subalgebras 2, 6 and 8 in Table 1, are particular cases of the ansatz constructed above. Doing in a similar way, one can obtain wide classes of ansatzes reducing equation (1) to two-dimensional, three-dimensional and so on equations. Let us present some of them. c) The ansatz u = <p{ui,...,w/,ui+i), u>i = xi,...,ui = xi, is an arbitary solution of system (10), / < n 1, is a generalization of the symmetry ansatz u tp{oji,...,u>1) and reduces equation (1) to the equation ^ ^ ^ + = 0. dio\ dj\ duf d) The ansatz u = <p(wi,.,cos,u;s+1), <Ji = [xq x\ x 2 ) 1^2, u>2 = xi+i,..,,u)s = xi+s-1, I > 2, / + s - 1 < n, ws+i is an arbitrary solution of the system w,+i = 0, (Vcjs+i) 2 = 0, V«t-VwH.i = 0, i 1, 2,..., s, (11) is a generalization of the symmetry ansatz u = the equation..., ws) and reduces equation (1) to <fu ~ Vl -<P22 Vss + F(<p) = 0. Let us construct in the way described above some classes of exact solutions of the equation u + = k^l. (12) 4

4 The following solution of equation (12) is obtained in paper [7]: u.1 -k r (M)(»i+ + *?). (13) Solution (13) defines a multiparameter solution set u 1 -** = a{k, Z)[( i + Ci) 2 + +(«/ + Ci)\ Ci,...,Ci are arbitrary constants. Hence, according to c), we obtain the following set of solutions to equation (12) for I < n 1: u l-k = <r(k, l)[(xi + hx (w)) (xi + hi(u))% k ± I I-2' u> is an arbitrary solution of system (10) and h\ (w),..., hi(u) are arbitrary twice differentiable functions of a;. In particular, if n = 3 and / = 1, then equation (12) possesses in the space Rit3 the solution set Next, let us consider the following solution of equation (1) [7]: u 1 k = a(k,s)(x 2 -x 2 x 2 ), s = 2,...,ra, (14) (U A(l-&) 2,, s + l a^ S ) -'2(s-ks + k + iy s-l Solution (14) defines the multiparameter solution set u l ~ k = <r{k, s)[x 2 0-xl xf - (xi+1 + Ci+1) 2 (xa + C,) 2 ], Ci-)_i,..., Cs are arbitrary constants. According to d) we obtain the following solution set for I > 2 u l ~ k = a(k, s)[z 2 - x 2 x] - (xl+1 + hl+l{u)) 2 (xs + hs(u)) 2 ], lj is an arbitrary solution of system (11), and hi+i(w),..., hs(u>) are arbitrary twice differentiable functions. In particular, if I = 2 and s = 3, then equation (1) possesses in the space Rii3 the following solution set u l ~ k = J ^ - l x l * l - ( * 3 - M«)fl, k + 2. The equation Ou + 6u 2 =0 (15) 5

5 possesses the solution u = V(xs + C2), V(x3 + C2) is an elliptic Weierstrass function with the invariants g2 = 0 and #3 = C\. Therefore, according to c) we get the following set of solutions of equation (15): u = V(x3 + h(u)), uj is an arbitrary solution to system (10) and h(u) is an arbitrary twice differentiable function of U). Next consider the Liouville equation u + Aexp«= 0. (16) The symmetry ansatz u = ui\ = x3l reduces equation (16) to the equation d^p = Aexpv(wi). Integrating this equation, we obtain that <p coincides with one of the following functions: In {(-^sec 2 ^/ z CT (wi + C2) (Cj < 0, A > 0, C2 G 2C1C2exp(v / CTa;i) In A[l-C2exp(VCTo;1)] 2 m (J^+c). (Cl > 0, AC2 > 0); Hence, according to c) we get the following solutions set for equation (16): v / z Mw) (u 1 + h2(u)) (hi(u) < 0, A > 0);, ( 2hi[u)h2(u) exp(\/hi(u)ui) I..,,,, u= - In I \j-u 1 + h(u) hi(u>), /12(^)5 h(u>) are arbitrary twice differentaible-functions; to is an arbitrary solution to system (10). Using, for example, the solution to the Liouville equation (16) [7] 2(s - 2) u = In s ^ 2, \[xq - x\ - xl] we obtain the wide class of solutions to the Liouville equation u = In 2(s 2) 2i ' X[xl -x\ xf - {xl+1 + h,+1(0;)) (xs + hs(u>)y] 6

6 lo is an arbitrary solution to system (11), and hi+i (w),..., hs(u) are arbitrary twice differentaible functions. If s = 3, then equation (16) possesses in the space R13 the following solution set 2 \[xl-x\-xl- (*3 + M")) 2 ]' Let us consider now the sine-gordon equation m + sin u = 0. Doing in an analogous way, we get the following solutions: u = 4arctan/ii(w) e (1 - E)TT, e0 = ±l, = ±1; u = 2 arccos[dn(z3 + h^u)), to] + ^(1 + e)tt, 0 < to < 1; u 2 arccos x3 + hi(u) cn TO 1,, TO + -(1 + 0 < to < 1, hi(u}) is an arbitrary twice differentiable function, to is is an arbitrary solution to system (10). [1] Fushchych W.I., Shtelen V.M. and Serov N.I., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht, [2] Fushchych W.I., Nikitin A.G., Symmetries of Maxwell's Equations, Dordrecht, Reidel, [3] Fushchych W.I. and Tsyfra I.M., On a reduction and solutions of nonlinear wave equations with broken symmetry, J. Phys. A: Math. Gen., 1987, V.20, N 2, L45-L48. [4] Levi D. and Winternitz P., J. Phys. A, V.22, [5] Grundland A.M., Harnad J. and Winternitz P., Symmetry reduction for nonlinear relativistically invariant equations, J. Math. Phys., 1984, V.25, N 4, [6] Cieciura G. and Grundland A., A certain class of solutions of the nonlinear wave equations, J. Math. Phys., 1984, V.25, N 12, [7] Fushchych W.I., Barannyk L.F. and Barannyk A.F., Subgroup Analysis of Galilei and Poincare Groups, and Reduction of Nonlinear Equations, Naukova Dumka, Kyiv, 1991 (in Russian). [8] Collins C.B., Complex potential equations. I. A technique for solutions, Proc. Cambridge Phil. Soc., 1976, N 9, [9] Fushchych W.I. and Zhdanov R.Z., Nonlinear Spinor Equations: Symmetry and Exact Solutions, Kyiv, Naukova Dumka, 1992 (in Russian). [10] Smirnov V.I. and Sobolev S.L., New method for solving a plane problem of elastic oscillations, Proc. of Seismological Institute of Acad. Sci. USSR, 1932, V.20, [11] Smirnov V.I. and Sobolev S.L., On application of a new method to the study of elastic oscillations in a space with the axial symmetry, Proc. of Seismological Institute of Acad. Sci. USSR, 1933, V.29,