W.I. Fushchych Scientific Works 23 Vol. 5 5. 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. 52 58.
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 =
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.
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 2 2 + u 2 2 + 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 2 + 2 2 + 3 2 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 2 2 3 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 2 2 + 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)
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)
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) 2 2 2 u = ± x ] (35) c [x ± x 2 +(x c) 2 2. 2 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)
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. 5 478 p.; Vol. 6 596 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 964 83 с. 4. Фущич В.И. Серов Н.И. Негрупповая симметрия некоторых нелинейных волновых уравнений Докл. АН УССР 99 9 45 49. 5. Fushchych W.I. Serov N.I. Tychynin V.A. Amerov Т.К. On nonlocal symmetries of nonlinear heat equation Докл. АН Украины Сер. A 992 27 33. 6. Фущич В.И. Тычинин В.А. О линеаризации некоторых нелинейных уравнений с помощью нелокальных преобразований Препринт 82.33 Киев Ин-т математики АН УССР 982 53 c. 7. Фущич В.И. Тычинин В.А. Жданов Р.З. Нелокальная линеаризация и точные решения некоторых уравнений Монжа Ампера Дирака Препринт 85.88 Киев Ин-т математики АН УССР 985 28 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 986 54 6.