Hodograph transformations and generating of solutions for nonlinear differential equations

Transcription

1 W.I. Fushchych Scientific Works 23 Vol Hodograph transformations and generating of solutions for nonlinear differential equations W.I. FUSHCHYCH V.A. TYCHYNIN Перетворення годографа однiєї скалярної функцiї в R( ) та R( 3) а також двох скалярних функцiй в R( ) використанi для розмноження розв язкiв нелiнiйних piвнянь; побудованi класи годограф-iнварiантних piвнянь другого порядку. The results of using the hodograph transformations for solution of applied problems are well-known. One can find them for example in [ 2 3]. We note also the paper [4] in which a number of invariants for hodograph transformation as well as hodographinvariant equations were constructed.. Hodograph-invariant and -linearizable equations in R( ). Let us consider the hodograph transformation for one scalar function (M =)of two independent variables x =(x x ) n =2: u(x) =y x = y x = v(y) δ = v = v = v y y =(y y ). Differential prolongations of the transformation generate such expressions for the first and second order derivatives: u = v u = v v u = v 3 v u = v 3 (v v v v ) u = v 3 [v2 v 2v v v + v 2 v ]. It is clear that is an involutory transformation. This allows to write a set of differential expressions of order 2 which are absolutely invariant under the transformation : f (x ) f (x u) f 2 (u u ) f3 (u u u ) f4 (u u 3 u ) f 5 (u u 3 (u u u u )) f 6 (u u 3 [u2 u 2u u u + u 2 u ]). (4) Here f is an arbitrary smooth function f i i = 6 are arbitrary functions symmetric on arguments i.e. f i (x z) =f i (zx). So the second order PDE invariant under the transformation has the form F ({f σ })= {f σ } = {f f...f 6 } σ = 6 (5) F is an arbitrary smooth function. Such well-known equations are contained in the class (5):. u 2 u 2 = the eikonal equation; (6) Доповiдi АН України 993 С

2 6 W.I. Fushchych V.A. Tychynin 2. u u [u 2 u 2u u u + u 2 u ]= the Born Infeld equation; (7) 3. u u u 2 = the Monge Amperé equation; (8) 4. u = f(u )u f(u )=f(u )u 2 the nonlinear heat equation [5]. (9) Particularly such equation as u u u = () is contained in the last class (9). Let u (x x ) be a known solution of Eq. (5). To construct a new solution u (x x ) letuswritethefirstsolutionreplacinginitanargumentx for parameter τ: u (x τ) andsubstituteittothehodographtransformationformula.sowe obtain the solutions generating formula for Eq. (5). u (x x )=τ x = u (x τ). () Let us now describe some class of -linearizable equations. Making use of formulae to transform general linear second order PDE b µν (y)v µν + b µ (y)v µ + b(y)v + c(y) = y =(y y ) µν = (2) we obtain b (x u)u 3 (u2 u 2u u u + u 2 u ) 2b (x u)u 3 (u u u u )+b (x u)u 3 u + + b (x u)u u + b (x u)u b(x u)x c(x u)=. b µν b µ c are arbitrary smooth functions b = b. Summation over repeated indices is understood in the space R( ) with the metric g µν = diag ( ). Therepeated use of this transformation to Eq. (2) turn us again to the Eq. (). For any equation of the class (2) the principle of nonlinear superposition is satisfied u (x x )= u (x τ) (3) u (x x )= u (x x τ) (4) Here (k) u (x x ) k = 2 are known solutions of Eq. (2) u (x x ) isanewsolution of this equation. Parameter τ must be eliminated due to second equality of the system (3). For example such equations important for applications are contained in this class (2): u u 2 u = u u u u = u 2 u 2u u u + u 2 u = u c(x u)u =. Let us consider now an example of constructing new solutions from two known ones by means of solutions superposition formula (3). Example. A nonlinear heat equation u u 2 u =

3 Hodograph transformations and generating of solutions 7 is reduced to the linear equation v v = (5) Therefore the formula (3) is true for (4). The functions u = x u = x 2x (6) are both partial solutions of Eq. (4). We construct a new solution u of this Eq. (4) via u and u.ithastheform u (x x )= 2 ± 4 + x 2x (7) 2. Hodograph-invariant and -linearizable equations in R( 3). The hodograph transformation of a scalar function u(x) of four independent variables x = (x x x 2 x 3 ) has the form v(x) =y x = v(y) x θ = y θ θ = 2 3. (8) Prolongation formulae for (8) are obtained via calculations [6 7]: u = v u θ = v v θ u = v 3 v u θ = v 3 (v v θ v θ v ) v θθ = v 3 (v2 v θθ 2v θ v v θ + vθ 2v ) u θγ = v 3 [v (v v θγ v γ v θ ) v θ (v v γ v γ v )]. Here θ γ = 2 3 θ γ. Making use of involutivity of the transformation (8) we list for it a such set of absolute differential invariant expressions of order 2: f (x x 2 x 3 ) f (x u) f 2 (u u ) f3 (u θ u u θ) f 4 (u u 3 u ) f 5 (u θ u 3 (u u θ u θ u )) f 6 (u θθ u 3 (u2 u θθ 2u u θ u θ + u 2 θ u )). f 7 (u θγ u 3 [u (u u γθ u γ u θ ) u θ (u u γ u γ u )]). There is no summation over θ here as before f is an arbitrary smooth function f j j = 7 are arbitrary symmetric. An equation invariant under transformation (8) has the form (9) (2) F ({f λ })= (λ = 7). (2) The solutions generating formula has the same form as () u (x x x 2 x 3 )=τ x = u (x τx 2 x 3 ). (22) Here u (x) is a known solution of Eq. (2) u (x) is its new solution. The following well-known equations are contained in this class (2):. u 2 u a u a = a = 3 the eikonal equation; 2. ( u ν u ν ) u u µ u ν u µν = µν = 3 the Born Infeld equation [8]; 3. det(u µν )= the Monge Amperé equation.

4 8 W.I. Fushchych V.A. Tychynin Here summation over repeated indices is understood in the space R( 3) with the metric g µν = diag ( ). u = µ µ u = u u u 22 u 33 is the d Alembert operator u a u a = u 2 + u u u 2 3 =( u) 2. The class of hodograph-linearizable equations in R( 3) is constructed analogously as above. Making use of transformation (8) for linear equation () written in R( 3) weget b (x δ u)u 3 u + b θθ (x δ u)u 3 (u2 u θθ 2u u θ u + u 2 θ u )+ + b γθ (x δ u)u 3 [u (u u γθ u γ u ) u θ (u u γ u γ u )] + + b (x δ u)u u θ b(x δ u)x c(x δ u)= x δ =(x x 2 x 3 ). (23) Here δ θ = 2 3 and summation over θ is understood in the space R( 2) with metric g θγ = diag ( ). Note that multidimensional nonlinear heat equation u u 2 ( + u2 2 + u 2 3)u u 22 u 33 +2u (u 2u 2 + u 3 u 3 )= (24) reduces due to transformation (8) to linear equation v = v where is the Laplace operator. So the solutions superposition formula for the equations (23) and (24) is u (x x x 2 x 3 )= u (x τx 2 x 3 ) (25) u (x τx 2 x 3 )= u (x x τx 2 x 3 ). (26) Example 2. Let partial solutions of Eq. (24) u = x x 2 x 3 ln x c 2 c [ ] 9 u = 4 c2 3(x c 4 ) 2 x 2 2 x be initial for generating a new solution u. Then this new solution of Eq. (24) is determined via (25) (26) by the equality u 2 (x)+x x 2 3 = c 3 [ x c 2 c exp{x x 2 x 3 u (x)} ] 2 c 3 = 9 4 c2 3 c 2 = c 4 + c 2. (27) Thus the formula (27) gives us a new solution of Eq. (24) in the implicite form. 3. Hodograph-invariant and -linearizable systems of PDE in R( ). Let us consider two functions u µ (x x ) µ = of independent variables x x.the hodograph transformation in this case as is known [2] has the form u (x x )=y u (x x )=y x = v (y y ) x = v (y y ) δ = u u u u δ = vv vv. (28)

5 Hodograph transformations and generating of solutions 9 Thefirstandsecondorderderevativesarechangingas u = δ v u = δ v u = δ v u = δ v (29) u = δ 3 [(v) 2 (vv vv )+(v ) 2 (vv vv ) 2vv (u v vv )] u = δ 3 [(v) 2 (vv vv +(v) 2 (vv vv ) 2vv (v v vv )] u = δ 3 [vv (v v vv )+v v (v v vv ) (vv vv )(v v + vv )] u = δ 3 [(v) 2 (vv vv )+(v ) 2 (vv vv ) 2vv (v v vv )] u = δ 3 [(v) 2 (vv vv )+(v ) 2 (vv vv ) 2vv (v v vv )] u = δ 3 [vv (v v vv )+v v (v v vv ) (vv vv )(v v + vv )]. (3) Let us now construct the absolute differential invariants with respect to (28) (3) of order 2. Making use of involutivity of this transformation we get f (x µ u µ ) µ = f 2 (u µ µδu ν ν) µ ν µ ν = there is no summation over repeated indices here f 3 (u µ ν δ u µ ν ) µ ν µ ν = ; f 4 (u δ 3 [(u ) 2 (u v u u )+(u ) 2 (u u u u ) 2u u (u u u u )]) f 5 (u δ 3 [(u ) 2 (u u u u )+(u ) 2 (u u u u ) 2u u (u v u u )]) f 6 (u δ 3 [u u (u u u u )+u u (u u u u (u u u u )(u u + u u )]) f 7 (u δ 3 [(u ) 2 (u u u u )+(u ) 2 (u u u u ) 2u u (u u u u )]) f 8 (u δ 3 [(u ) 2 (u u u u )+(u ) 2 (u u u u ) 2u u (u u u u )]) f 9 (u δ 3 [u u (u u u u )+u u (u u u u ) (u u u u )(u u + u u )]). (3) All functions f k k = 9 are arbitrary smooth and symmetric. Sowenowareabletoconstructthehodograph-invariantsystemofsecondorder PDEs F σ ({f k })= k = 9 σ = 2...N. (32)

6 W.I. Fushchych V.A. Tychynin We construct a new solution u = ( u u ) of system (32) via known solution u =( u u ) according to the formula u (x) =τ x = u (τ). (33) Here x =(x x ) τ =(τ τ ) τ µ are parameters to be eliminated out of system (33). Example 3. Let us consider the simplest hodograph-invariant system of first order PDE u u = u u =. (34) It is easily to verify that pair of functions u =2x x + c u x 2 + x 2 is the solution of system (34). Making use of formula (33) one obtain the new solution of this system u = ± ] [x ± x 2 +(x c) u = ± x ] (35) c [x ± x 2 +(x c) Let us consider the linear system of first order PDEs b σν µ (y)vµ ν + b σν (y)v ν + c σ (y) =. (36) Here b σν µ b σν c σ are arbitrary smooth functions of y =(y y ) summation over repeated indices is understood in the space with metric gµν = diag ( ). This system (36) under transformation (28) reduces into system of nonlinear PDEs b σ (u)δ u b σ (u)δ u b σ (u)δ u + + b σ (u)δ u + b σ (u)x + b σ (u)x + c σ (u) =. The solutions superposition formula for the system (37) has the form (37) u (x x )= u (τ τ ) u (x x )= u (τ τ ) u (τ τ )= u (x τ x τ ) u (τ τ )= u (x τ x τ ). (38) Making use of designations u =(u u ) x =(x x ) τ =(τ τ )onecanrewrite the formula (38) in another way: u (x) = u (τ) u (τ) = u (x τ). Example 4. It is obviously that two pairs of functions (38a) u = 2 x u = x ρ =(2λ) 4 x2 x ] [ 2 c + x ρ =(2λx ) c (39)

7 Hodograph transformations and generating of solutions give two partial solutions of the system u + uu +4λ 2 ρρ = ρ + u ρ + uρ =. (4) Let us apply the formula (38) to construct a new solution u ρ via (39). Finally we get u 2 (x x ) c 2 2(x 2 u (x x )) 2 x u (x x )+x + 2 c = [ ρ (x x )=(2λ) x u (x x ) u 2 (x x ) x ] 2 c 2.. Forsyth A.R. Theory of differential equations New York Dover Publication 959 Vol p.; Vol p. 2. Ames W.F. Nonlinear partial differential equations in engineering New York Academic Press 965 Vol. 5 p.; 972 Vol. 2 3 p. 3. Курант Р. Уравнения в частных производных М. Мир с. 4. Фущич В.И. Серов Н.И. Негрупповая симметрия некоторых нелинейных волновых уравнений Докл. АН УССР Fushchych W.I. Serov N.I. Tychynin V.A. Amerov Т.К. On nonlocal symmetries of nonlinear heat equation Докл. АН Украины Сер. A Фущич В.И. Тычинин В.А. О линеаризации некоторых нелинейных уравнений с помощью нелокальных преобразований Препринт Киев Ин-т математики АН УССР c. 7. Фущич В.И. Тычинин В.А. Жданов Р.З. Нелокальная линеаризация и точные решения некоторых уравнений Монжа Ампера Дирака Препринт Киев Ин-т математики АН УССР c. 8. Тычинин В.А. Нелокальная линеаризация и точные решения уравнения Борна Инфельда и некоторых его обобщений в сб. Теоретико-групповые исследования уравнений математической физики Киев Ин-т математики АН УССР