Báo cáo toán học: "On the Hyperbolicity of Some Systems of Nonlinear " pps

Vietnam Journal of Mathematics 34:1 (2006) 109–128 On the Hyperbolicity of Some Systems of Nonlinear First-Order Partial Differential Equations * Ha Tien Ngoan and Nguyen Thi Nga Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam Received July 6, 2005 Revised September 16, 2005 Abstract. In this paper we study the hyperbolicity of some normal systems of first- order nonlinear partial differential equations, to which some multidimensional Monge- Amp`ere equations have been reduced in [8]. We prove that when the dimension n  5 all these systems are weakly hyperbolic. 1. Introduction We consider the following normal system of 2n + 1 first-order nonlinear partial differential equations with respect to 2n+1 unknown functions X(α),Z(α),P(α) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂X i ∂α n = − n−1  k=1 ∂X i ∂α k + g i (α),i=1, 2, ,n, ∂Z ∂α n = − n−1  k=1 ∂Z ∂α k + n  =1 g  (α)P  (α), ∂P i ∂α n = − n−1  k=1 ∂P i ∂α k − n  =1 a i (X(α),Z(α),P(α))g  (α),i=1, 2, ,n, (1.1) where α ≡ (α 1 ,α 2 , ,α n ) are independent variables, X(α) ≡  X 1 (α),X 2 (α), , ∗ This work was supported in part by t he National Basic Research Program in Natural Science, Vietnam. 110 Ha Tien Ngoan and Nguyen Thi Nga X n (α)  ,P(α) ≡  P 1 (α),P 2 (α), ,,P n (α)  and a ij (X, Z, P) are given smooth functions defined in R 2n+1 , g(α)=(g 1 (α),g 2 (α), ,g n (α)) T = v 1 (α) × v 2 (α) ×···×v n−1 (α) ∈ R n , (1.2) v j (α)= ∂P ∂α j + ∂X ∂α j A(X(α),Z(α),P(α)) =(v j1 (α),v j2 (α), ,v jn (α)) ∈ R n ,j =1, 2, ,n− 1. (1.3) where A(X, Z, P ) ≡ [a ij (X, Z, P )] n×n ,a ij (X, Z, P ) are the same as in (1.1), ∂X ∂α j =( ∂X 1 ∂α j , ∂X 2 ∂α j , , ∂X n ∂α j ) ∈ R n ,j=1, 2, ,n. ∂P ∂α j =  ∂P 1 ∂α j , ∂P 2 ∂α j , , ∂P n ∂α j  ∈ R n ,j=1, 2, ,n v 1 × v 2 ×···×v n−1 =           e 1 e 2 e n−1 e n v 11 v 12 v 1,n−1 v 1,n v 21 v 22 v 2,n−1 v 2,n . . . . . . . . . . . . . . . v n−1,1 v n−2,2 v n−1,n v n−1,n           ∈ R n , (1.4) e 1 ,e 2 , ,e n are unit column-vectors on coordinate axes Ox 1 ,Ox 2 , ,Ox n ,re- spectively. We note from (1.4) that g i (α) will be determined in (2.7) by a determinant of order (n-1), whose elements v jk by (2.8), (2.1) and (2.2) are homogenous polynomials of degree 1 with respect to the same derivatives ∂X(α) ∂α k , ∂P(α) ∂α k ,k = 1, 2, ,n− 1. So all g i (α) are homogenous polynomials of degree (n − 1) with respect to the derivatives ∂X(α) ∂α k , ∂P(α) ∂α k ,k =1, 2, ,n− 1 with coefficients de- pending on a ij (X(α),Z(α),P(α)). Therefore the system (1.1) is normal, because all derivatives of the unknowns X, Z, P with respect to the α n are expressed in terms of their derivatives with respect to the rest variables α 1 ,α 2 , , α n−1 . In [1 - 7] the classical hyperbolic Monge-Amp`ere equations (n = 2) has been studied by reducing them to some first-order quasilinear hyperbolic systems (1.1) with 5 equations and 5 unknowns. The Cauchy problem for some hyperbolic or weakly hyperbolic systems had been studied in [11 - 12]. In [8] we have reduced the following multidimensional Monge-Amp`ere equa- tion det [z x i x j + a ij (x, z, p)] n×n =0, (1.5) to the system (1.1), where x =(x 1 ,x 2 , ,x n ) ∈ R n ,z = z(x) is an unknown function, p =(p 1 ,p 2 , ,p n )=(z x 1 ,z x 2 , ,z x n ). The functions a ij (x, z, p)are the same ones as in (1.1). We have shown in [8] that a solution (X(α),Z(α),P(α)) to the system (1.1) with   DX(α) Dα | = 0 gives a solution z(x) to the equation (1.5). On the Hyperbolicity of some Systems of Nonlinear 111 The solvability of the Cauchy problem for the equations (1.5) strongly depends on the hyperbolicity of the system (1.1). So, it is important to study the hyper- bolicity of the system (1.1). In the present paper we study the hyperbolicity for the system (1.1). Our main result is Theorem 2.8 which states that when dimension n  5, the system (1.1) is weakly hyperbolic. Due to a lot of calculations needed, in the case n  6 we get only particular results. The outline of the paper is the following. In Sec. 2 we recall the notions of weak hyperbolicity and hyperbolicity for (1.1). In the following Secs. 3 - 6 we study the hyperbolicity for the dimensions between 2 and 5. We would like to emphasize that the hyperbolicity takes place only in the case n =2, provided that the matrix A(x, z, p)=[a ij (x, z, p)] 2×2 is not symmetric. In the paper we use the Maple 7 for symbolic calculations to calculate the products of matrices, determinants, eigenvalues and to simplify algebraic expres- sions. 2. Hyperbolicity 2.1. Definitions We introduce the following notations. For k =1, 2, ,n− 1, set V k =(V 1k ,V 2k , ,V nk ) ≡ ∂X ∂α k =  ∂X 1 ∂α k , ∂X 2 ∂α k , , ∂X n ∂α k  , (2.1) W k =(W 1k ,W 2k , ,W nk ) ≡ ∂P ∂α k =  ∂P 1 ∂α k , ∂P 2 ∂α k , , ∂P n ∂α k  , (2.2) and U(α)=(X(α),Z(α),P(α)) T = ⎡ ⎣ X T (α) Z(α) P T (α) ⎤ ⎦ F (α)=− n−1  =1 ∂U ∂α  + ⎡ ⎣ g(α) g(α),P(α) −Ag(α) ⎤ ⎦ where ., . stands for the scalar product in R n . We can now write system (1.1) in the matrix form ∂U ∂α n = F. (2.3) For j =1, 2, ,n− 1, we introduce Q j = ∂U ∂α j and 112 Ha Tien Ngoan and Nguyen Thi Nga A j ≡ DF DQ j = ⎡ ⎢ ⎣ Dg DV j − E 0 Dg DW j P Dg DV j −1 P Dg DW j −A Dg DV j 0 −A Dg DW j − E ⎤ ⎥ ⎦ , (2.4) where E is the identity matrix of order n and Dg DV k ≡  ∂g i ∂V jk  n×n = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂g 1 ∂V 1k ∂g 1 ∂V 2k ∂g 1 ∂V nk ∂g 2 ∂V 1k ∂g 2 ∂V 2k ∂g 2 ∂V nk . . . . . . . . . . . . ∂g n ∂V 1k ∂g n ∂V 2k ∂g n ∂V nk ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , Dg DW k ≡  ∂g i ∂W jk  n×n = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂g 1 ∂W 1k ∂g 1 ∂W 2k ∂g 1 ∂W nk ∂g 2 ∂W 1k ∂g 2 ∂W 2k ∂g 2 ∂W nk . . . . . . . . . . . . ∂g n ∂W 1k ∂g n ∂W 2k ∂g n ∂W nk ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . Note that each of the matrices A j depends on X(α),Z(α),P(α), ∂X ∂α k , ∂P ∂α k ,k = 1, 2, ,n− 1. We recall now the notion of hyperbolicity for the system (2.3). Definition 2.1. [9, 10] 1) System (2.3) is said to be weakly hyperbolic if for any given (X(α),Z(α),P(α)) ∈ C 1 and for any ξ  =(ξ 1 , , ξ n−1 ) ∈ R n−1 , all eigenvalues of the matrix A = n−1  i=1 ξ i A i (2.5) are real. 2) System (2.3) is said to be hyperbolic if it is weakly hyperbolic and if for any given (X(α),Z(α),P(α)) ∈ C 1 and for any ξ  =(ξ 1 , , ξ n−1 ) ∈ R n−1 , there exists a basis in R 2n+1 , consisting of its corresponding smooth left eigenvectors of the matrix A. Proposition 2.2. For each k =1, 2, ··· ,n − 1 the matrix Dg DW k is anti- symmetrix, i.e.  Dg DW k  T = − Dg DW k . (2.6) Proof. From (1.2), (1.4) we have On the Hyperbolicity of some Systems of Nonlinear 113 g i =(−1) 1+i ×                 v 11 v 1,i−1 v 1,i+1 v 1n . . . . . . . . . . . . . . . v k−1,1 v k−1,i−1 v k−1,i+1 v k−1,n v k1 v k,i−1 v k,i+1 v k,n v k+1,1 v k+1,i−1 v k+1,i+1 v k+1,n . . . . . . . . . . . . . . . . . . v n−1,1 v n−1,i−1 v n−1,i+1 v n−1,n                 (2.7) From (1.3), (2.1) and (2.2) it follows that v jm = W mj + n  h=1 a hm V hj ,j =1, ,n− 1,m=1, ,n. (2.8) We note that W ik ,k =1, 2, ,n− 1 do not appear in the expression of each g i . Therefore, ∂g i ∂W ik =0,i=1, ··· ,n, k =1, ··· ,n− 1. (2.9) If j<i,then (2.7) yields ∂g i ∂W jk =(−1) 1+i ×                  v 11 v 1,j–1 0 v 1,j+1 v 1,i–1 v 1,i+1 v 1n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v k–1,1 v k–1,j–1 0 v k–1,j+1 v k–1,i–1 v k–1,i+1 v k–1,n v k,1 . . . v k,j–1 1 v k,j+1 . . . v k,i–1 v k,i . . . v k,n v k+1,1 v k+1,j–1 0 v k+1,j+1 v k+1,i–1 v k+1,i+1 v k+1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . v n–1,1 v n–1,j–1 0 v n–1,j+1 v n–1,i–1 v n–1,i+1 v n–1,n                  (2.10) On the other hand, if j<i,then we can rewrite (2.7) as follows (2.11) So from (2.11) we have 114 Ha Tien Ngoan and Nguyen Thi Nga ∂g j ∂W ik =(−1) 1+j ×                  v 11 v 1,j−1 v 1,j+1 v 1,i−1 0 v 1,i+1 v 1n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v k−1,1 v k−1,j−1 v k−1,j+1 v k−1,i−1 0 v k−1,i+1 v k−1,n v k1 . . . v k,j−1 v k,j+1 . . . v k,i−1 1 v k,i+1 . . . v kn v k+1,1 v k+1,j−1 v k+1,j+1 v k+1,i−1 0 v k+1,i+1 v k+1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v n−1,1 v n−1,j−1 v n−1,j+1 v n−1,i−1 0 v n−1,i+1 v n−1,n                  (2.12) From (2.10) and (2.12) we see that the formula (2.10) is true for i = j. Moreover, from (2.12), (2.10) it follows that ∂g j ∂W ik =(−1) 1+j (−1) i−j−1 ×                  v 11 v 1,j−1 0 v 1,j+1 v 1,i−1 v 1,i+1 v 1n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v k−1,1 v k−1,j−1 0 v k−1,j+1 v k−1,i−1 v k−1,i+1 v k−1,n v k,1 . . . v k,j−1 1 v k,j+1 . . . v k,i−1 v k,i . . . v k,n v k+1,1 v k+1,j−1 0 v k+1,j+1 v k+1,i−1 v j+1,i+1 v k+1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . v n−1,1 v n−1,j−1 0 v n−1,j+1 v n−1,i−1 v n−1,i+1 v n−1,n                  = − ∂g i ∂W jk . The proposition is proved.  Proposition 2.3. For k =1, 2, ,n− 1 we have Dg DV k = Dg DW k A T , (2.13) where A =[a ij ] n×n . Proof. From (2.7) - (2.10) it follows that On the Hyperbolicity of some Systems of Nonlinear 115 ∂g i ∂V jk =(−1) 1+i ×                  v 11 v 1,i−1 v 1,i+1 v 1n . . . . . . . . . . . . . . . v k−1,1 v k−1,i−1 v k−1,i+1 v k−1,n a j,1 . . . a j,i−1 a j,i+1 . . . a j,n v k+1,1 v k+1,i−1 v k+1,i+1 v k+1,n . . . . . . . . . . . . . . . v n−1,1 v n−1,i−1 v n−1,i+1 v n−1,n                  = n  h=1 ∂g i ∂W hk a jh . (2.14) The proposition is proved.  Set M ≡ n  k=1 ξ k Dg DW k =[m ij ] n×n , (2.15) M i =                 v 11 v 1,i−1 v 1,i+1 v 1n . . . . . . . . . . . . . . . v k−1,1 v k−1,i−1 v k−1,i+1 v k−1,n v k1 v k,i−1 v k,i+1 v k,n v k+1,1 v k+1,i−1 v k+1,i+1 v k+1,n . . . . . . . . . . . . . . . . . . v n−1,1 v n−1,i−1 v n−1,i+1 v n−1,n                 , (2.16) and for i<jdenote by M ij the matrix obtained from the matrix M i by replacing its (j − 1)-column by the column [ξ 1 ξ 2 ξ n−1 ] T . Proposition 2.4. For i<j we have m ij =(−1) 1+i det M ij . Proof. From (2.15), m ij = n−1  k=1 ξ k ∂g i ∂W jk . (2.17) From (2.7) we get 116 Ha Tien Ngoan and Nguyen Thi Nga ∂g i ∂W jk =(−1) 1+i ×                  v 11 v 1,i−1 v 1,i+1 v 1,j−1 0 v 1,j+1 v 1n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v k−1,1 v k−1,i−1 v k−1,i+1 v k−1,j−1 0 v k−1,j+1 v k−1,n v k1 . . . v k,i−1 v k,i+1 . . . v k,j−1 1 v k,i+1 . . . v kn v k+1,1 v k+1,i−1 v k+1,i+1 v k+1,j−1 0 v k+1,j+1 v k+1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v n−1,1 v n−1,i−1 v n−1,i+1 v n−1,j−1 0 v n−1,j+1 v n−1,n                  (2.18) The proposition follows from (2.17) and (2.18).  2.2. Transformation of the Matrix A Set B = A T − A, C = BM, where M is given by (2.15) ν = n−1  i=1 ξ i . From (2.4) and Proposition 2.3 we have A = n−1  i=1 ξ i A i = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ n−1  i=1 ξ i Dg DW i A T −  n−1  i=1 ξ i  E 0 n−1  i=1 ξ i Dg DW i P n−1  i=1 ξ i Dg DW i A T − n−1  i=1 ξ i P n−1  i=1 ξ i Dg DW i −A n−1  i=1 ξ i Dg DW i A T 0 −A n−1  i=1 ξ i Dg DW i − E n−1  i=1 ξ i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎣ MA T − νE 0 M PMA T −νPM −AM A T 0 −AM − νE ⎤ ⎦ . (2.19) Theorem 2.5. The matrix A is similar to the following one On the Hyperbolicity of some Systems of Nonlinear 117 ˜ A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −E n−1  i=1 ξ i 0 n−1  n=1 ξ i Dg DW i 0 − n−1  i=1 ξ i P n−1  i=1 ξ i Dg DW i 00(A T − A) n−1  i=1 ξ i Dg DW i − E n−1  n=1 ξ i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.20) = ⎡ ⎣ −Eν 0 M 0 −νPM 00C − Eν ⎤ ⎦ Proof. Setting the block matrix D = ⎡ ⎣ E 00 010 −A T 0 E ⎤ ⎦ , we have D −1 = ⎡ ⎣ E 00 010 A T 0 E ⎤ ⎦ . It is easy to see that ˜ A = D −1 AD. The theorem is proved.  Corollary 2.6. If A T = A, (i.e.B =0)then the system (1.1) is we akly hyper- bolic. Corollary 2.7. If all eigenvalues of the matrix C = BM are real, then the system (1.1) is weakly hyperbolic. We formulate now the main result of the paper. Theorem 2.8. For n =2, if A T = A T , i.e. if a 12 = a 21 , then the system (1.1) is hyperbolic. For n =3, 4, 5, it is weakly hyperbolic. All the following sections are devoted to the proof of the theorem when n =2,n=3,n=4andn =5. For the last three cases we will prove that all the eigenvalues of the matrices C are real, and therefore, the systems (1.1) in this cases are weakly hyperbolic. 3. Proof of the Theorem 2.8 for the Case n =2 Suppose that A T = A, i.e. a 12 = a 21 . We prove that the system (1.1) is hyper- bolic. 118 Ha Tien Ngoan and Nguyen Thi Nga 1 c irc ) First we prove that all eigenvalues of the matrix ˜ A are real. From (2.20) we have | ˜ A−λE| = −(ξ 1 + λ) 3    (A T − A)ξ 1 Dg DW 1 − (ξ 1 + λ)E    (3.1) where Dg DW 1 =  ∂g 1 ∂W 11 ∂g 1 ∂W 21 ∂g 2 ∂W 11 ∂g 2 ∂W 21  =  01 −10  (3.2) (A T − A)=  0 a 21 − a 12 a 12 − a 21 0  . (A T − A)ξ 1 Dg DW 1 =  ξ 1 (a 12 − a 21 )0 0 ξ 1 (a 12 − a 21 )  = ξ 1 (a 12 − a 21 )E.(3.3) From (3.1), (3.2) we obtain λ 1 = λ 2 = λ 3 = −ξ 1 , λ 4 = λ 5 = ξ 1 (a 12 − a 21 − 1). This means that in the case n = 2 the system (1.1) is always weakly hyper- bolic. 2 ◦ ) Suppose that ξ 1 =0. Since the martix A is simmilar to ˜ A,toprove the theorem we have to show that there exists a basis of R 5 generated by left eigenvectors  1 , 2 ,  5 of the matrix ˜ A. Lemma 3.1. Let X 1 be the spa ce of left eigenvectors of the matrix ˜ A corre- sponding to the eigenvalue λ = −ξ 1 . Then dimX 1 =3. Proof. From (2.20) with n =2andλ = −ξ 1 we have ˜ A−λE = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 00 ξ 1 Dg DW 1 00 Pξ 1 Dg DW 1 00(A T − A)ξ 1 Dg DW 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Because det  ξ 1 Dg DW  = ξ 2 1 = 0 we have rank( ˜ A−λE)=2. Therefore, dimX 1 = 5 − 2=3.  Lemma 3.2. Let X 2 be the spa ce of left eigenvectors of the matrix ˜ A corre- sponding to the eigenvalue λ = −ξ 1 (a 12 − a 21 − 1). Then dim X 2 =2. Proof. From (2.20) with n =2andλ = −ξ 1 (a 12 − a 21 − 1) we have [...]... − μE| = 0 2 if and only if −μ(μ + b12 m12 + b23 m23 + b13 m13 ) = 0 So eigenvalues of the matrix C are the following μ1 = 0, μ2 = μ3 = −b12 m12 − b23 m23 − b13 m13 Theorem 2.8 in the case n = 3 follows from the Corollary 2.7 5 Proof of the Theorem 3.8 for the Case n = 4 We put On the Hyperbolicity of some Systems of Nonlinear 121 ⎡ ⎤ b13 b14 0 b12 0 b23 b24 ⎥ ⎢ −b B = (AT − A) = [bij ] = ⎣ 12 ⎦ −b13... ξ1 v33 − v31 v13 ξ2 + v31 ξ1 v23 (5.7) From (5.2) - (5.7) we obtain m12 m34 + m14 m23 − m13 m24 = 0 So we have proved Δ ≥ 0 The Theorem 2.8 in the case n = 4 follows from Corollary 2.7 6 Proof of Theorem 2.8 for the Case n = 5 We put On the Hyperbolicity of some Systems of Nonlinear ⎡ 0 ⎢ −b12 ⎢ B = (AT − A) = [bij ] = ⎢ −b13 ⎣ −b14 −b15 ⎡ 0 4 ⎢ −m12 Dg ⎢ ξi = [mij ] = ⎢ −m13 M= ⎣ DWi i=1 −m14 −m15...On the Hyperbolicity of some Systems of Nonlinear ˜ A − λE ⎡ −ξ (a − a21 )E ⎢ 1 12 ⎢ ⎢ =⎢ 0 ⎢ ⎣ 0 119 ⎤ Dg ξ ⎥ DW1 ⎥ Dg ⎥ Pξ ⎥ ⎥ DW1 ⎦ Dg T (A − A)ξ1 − ξ1 (a12 − a21 )E DW1 (3.4) 0 −ξ1 (a12 − a21 ) 0 From (3.2), (3.3) and (3.4) we have ⎡ ⎤ ξ1 0 ⎥ ⎥ P1 ξ1 ⎥ ⎦ 0 0 (3.5) ˜ = a21 , then rank (A−λE) = 3 Therefore, dimX2 = 5−3 = −ξ1 (a12 − a21 ) 0 0 −ξ1 (a12... 4 = 0 Proof Since 1 ∈ X1 , 1 0 0 0 −ξ1 −ξ1 (a12 − a21 ) −P2 ξ1 0 0 0 0 4 ∈ X1 , ∈ X2 If 1 A = λ1 1 + 4 = 0, then (3.6) Analogously, 4 4 ∈ X2 ⇒ A = λ4 4 (3.7) On the other hand, λ4 4 4 = λ1 + λ4 4 − λ1 4 = λ1 4 + (λ4 − λ1 ) 4 (3.8) From (3.6), (3.7), (3.8) we get ( From (3.9) and 1 + 1 4 + 4 )A = λ1 ( = 0 we have 1 4 + 4 ) + (λ4 − λ1 ) 4 = 0 and 1 (3.9) = 0 Continuation of the proof of Theorem 2.8... 0 ⎡ ⎤ m13 m14 0 m12 3 Dg 0 m23 m24 ⎥ ⎢ −m12 ξi = [mij ] = ⎣ M= ⎦ −m13 −m23 0 m34 DWi i=1 −m14 −m24 −m34 0 C = B × M = C1 + C2 , where We prove that all eigenvalues of the matrix C are real With the aid of the Maple 7, the eigenvalues of the matrix C are calculated as following 1 1 1 b14 m14 − b13 m13 − b23 m23 − 2 2 2 1 1 1 1 − b12 m12 − b24 m24 + Δ 2 , 2 2 2 1 1 1 μ3 = μ4 = − b14 m14 − b13 m13 − b23... m34 m25 b23 b45 + 4 m35 m24 b23 b45 − 2 m23 m45 b23 b45 − 4 m23 m45 b34 b25 − 4 m12 m45 b15 b24 + 2 b13 m13 b15 m15 + b25 2 m25 2 We prove that Δ ≥ 0 To this end we write Δ in the form On the Hyperbolicity of some Systems of Nonlinear 125 Δ = –4 b34 m34 b15 m15 –4 b24 m24 b35 m35 –4 b24 m24 b15 m15 –4 b14 m14 b35 m35 − 4 b14 m14 b25 m25 − 4 b34 m34 b25 m25 + 4 b34 m35 b25 m24 + 4 b34 m35 b15 m14 +... Hermann, Paris, 1932 e 4 M Tsuji, Formation of singularities for Monge-Amp`re equations, Bull Sci e Math (1995) 433–457 5 M Tsuji and H T Ngoan, Integration of hyperbolic Monge-Amp`re equations, e In: Proceedings of the Fifth Vietnamese Mathematical Conference, Publishing House of Sci & Tech., Hanoi, 1997, pp 205–212 6 M Tsuji and N D Thai Son, Geometric solutions of nonlinear second order hyperbolic equations,... + v31 v23 ξ1 v45 − v31 v23 v15 ξ4 − v31 v43 ξ1 v25 + v31 v43 v15 ξ2 + v41 v13 ξ2 v35 − v41 v13 v25 ξ3 − v41 v23 ξ1 v35 + v41 v23 v15 ξ3 + v41 v33 ξ1 v25 − v41 v33 v15 ξ2 , (6.12) On the Hyperbolicity of some Systems of Nonlinear 127 m15 = v12 v23 v34 ξ4 − ξ12 ξ23 ξ3 v44 − v12 v33 v24 ξ4 + v12 v33 ξ2 v44 + v12 v43 v24 ξ3 − v12 v43 ξ2 v34 − v22 v13 v34 ξ4 + v22 v13 ξ3 v44 + v22 v33 v14 ξ4 − v22 v33 ξ1... c3 5 =0 4 + c5 5 ) = 0 3 =0 Ha Tien Ngoan and Nguyen Thi Nga 120 Hence, c1 = c2 = c3 = 0 c4 = c5 = 0 So the vectors 1 , 2 , 3 , 4 , 5 form a basis of the space R5 Therefore Theorem 2.8 is proved in the case n = 2 4 Proof of Theorem 2.8 for the Case n = 3 Put ⎡ ⎡ ⎤ 0 b12 b13 0 b23 ⎦ (AT − A) = B = [bij ] = ⎣ −b12 −b13 −b23 0 ⎡ ⎤ 2 0 m12 m13 Dg M= ξi = ⎣ −m12 0 m23 ⎦ DWi i=1 −m13 −m23 0 −b12 m12 − b13... m14 + m45 m12 ) = 0, (m14 m23 − m13 m24 + m34 m12 ) = 0 From (6.6) and (6.17) we have Δ ≥ 0 (6.17) 128 Ha Tien Ngoan and Nguyen Thi Nga Therefore, we obtained that all eigenvalues of the matrix C are real Theorem 2.8 in the case n = 5 follows from Corollary 2.6 Thus Theorem 2.8 is completely proved References 1 G Darboux, Le¸ons sur la th´orie g´n´rale des surfaces, tome 3, Gauthier-Villars, c e e e . equation (1.5). On the Hyperbolicity of some Systems of Nonlinear 111 The solvability of the Cauchy problem for the equations (1.5) strongly depends on the hyperbolicity of the system (1.1). So,. we have proved Δ ≥ 0. The Theorem 2.8 in the case n = 4 follows from Corollary 2.7. 6. Proof of Theorem 2.8 for the Case n =5 We put On the Hyperbolicity of some Systems of Nonlinear 123 B =(A T −. are devoted to the proof of the theorem when n =2,n=3,n=4andn =5. For the last three cases we will prove that all the eigenvalues of the matrices C are real, and therefore, the systems (1.1) in