Theoretical Mathematics & Applications, vol.3, no.4, 2013, 105-137
ISSN: 1792- 9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Well-Posedness of N.G-KdV Class (3+1) Equation
Under Noncharscteristic Data
K.A. Moustafa 1 and H.A.F. Mahmoud 2
Abstract
In this article, we introduce the well-posedness of N.G-KdV class (3+1) equation
which has an important physically phenomena of the propagation of traveling wave
with all types solitary waves. We prove, first of all, that the general class of N.GKdV class (3+1) equation can be reduced for certain data to a semi-linear system of
first order partial differential equations. We find the characteristics of this system
and show that it is equivalent to a system of ordinary differential equations in
which differentiation is along characteristic direction. These equations can be
integrated to give the solution of the system provided that the data is not specified
on a characteristic. This method of solution is called the generalized characteristics
in three dimensions.
Mathematics Subject Classification: Partial differential equation
1
2
Mathematics Department, Faculty of science, Suez University, Egypt.
E-mail: dr.khaledabdm@gmail.com
Mathematics Department, Faculty of science, Suez University, Egypt.
E-mail:bebamath@yahoo.com
Article Info: Received: August 12, 2013. Revised: October 14, 2013.
Published online : December 16, 2013.
106
Well-Posedness of N.G-KdV class (3+1) equation …
Keywords: N.G-KdV class (3+1) the new generalized Korteweg-de Vries class
Equation in (3+1) dimensions (x , y , z , t ) ; generalized characteristics; initial value
problem
1 Introduction
We establish well- posedness for the nonsingular class by applying the wellknown theorems on uniqueness, existence and continuous dependence on the initial
conditions for semi-linear systems. As regards the singular class, we divide it
further according to the multiplicity of the (essential) characteristic roots of N.GKdV class (3+1) equation which defined as
α1u xxt + α 2u xxx + α 3u xtt + α 4uu xxt + α 5uu xxx + α 6uu xtt + α 7u x u xt + α 8u x u xx + α 9u x utt
+α10ut u xx + α11ut u xt + α12u xxxxx + α13u xxxxt + α14u xxxtt + α15u xtttt + α16u xxttt + α17uttttt
(1.1)
= σ 2u yy + σ 2u zz
We introduce reduction to a semi-linear system of first partial differential
equations.
1.1 Reduction to a semi-linear system of partial differential
equations
Consider the initial value problem which corresponds to the equation (1.1).
Let the initial curve which supports the data be non-characteristic, as shall be
defined in the next section, and without loss of generality let this curve be the usual
one t = 0 , i.e.,
α1u xxt + α 2u xxx + α 3u xtt + α 4uu xxt + α 5uu xxx + α 6uu xtt + α 7u x u xt + α 8u x u xx
+α 9u x utt + α10ut u xx + α11ut u xt + α12u xxxxx + α13u xxxxt + α14u xxxtt + α15u xtttt
+α16u xxttt + α17uttttt =σ u yy + σ u zz
2
2
(1.1.1)
K.A. Moustafa and H.A.F. Mahmoud
107
with the initial value problem
=
u ( x, y, z , 0) f=
( x, y, z ), ut ( x, y, z , 0) g ( x, y, z ),
utt ( x, y, z , 0) = H ( x, y, z ),
(1.1.2)
=
uttt ( x, y, z , 0) ψ=
( x, y, z ), utttt ( x, y, z, 0) φ ( x, y, z )
We introduce now the following.
Lemma 1.1.1 The initial value problem for N.G-KdV class (3+1) equation (1.1.1)
with non -characteristic initial data may be reduced to a non-characteristic initial
value problem for a first order system of partial differential equations.
Proof. Re-writing equation (1.1.1) in the form
F (u , p, q, r , s,τ , v, ω , µ ,υ , γ ,η , M , N , I , h, L, k , E , O, π , ξ , β , ) =
α1β + α 2  + α 3ξ + α 4u β + α 5u + α 6uξ + α 7 qs + α 8 qτ + α 9 qr + α10 pτ
0
(1.1.3)
+ α11 ps + α12 v + α13ω + α14 µ + α15γ + α16υ + α17η − σ 2 h − σ 2 k =
Where
=
p u=
u=
u=
u xt=
,τ u xx=
, v u xxxxx=
, ω u xxxxt=
, µ u xxxtt ,
t,q
x,r
tt , s
=
=
=
=
υ u xxttt
u=
utxxx ,
, γ u xtttt
,η uttttt=
,, M utttt
, N u=
tttx , I
ttxx , h
=
=
L u=
u yy=
u=
u=
uttx=
, E u=
, β utxx
,  u xxx
ttt , k
yt , O
zz , π
zt , ξ
(1.1.4)
We use the initial conditions equation (1.1.2) and differentiating (1.1.3) with
respect to t yields, we obtain the following equation.
∂F ∂F ∂u ∂F ∂p ∂F ∂q ∂F ∂r ∂F ∂s ∂F ∂τ ∂F ∂v ∂F ∂ω
=
+
+
+
+
+
+
+
∂t ∂u ∂t ∂p ∂t ∂q ∂t ∂r ∂t ∂s ∂t ∂τ ∂t ∂v ∂t ∂ω ∂t
∂F ∂µ ∂F ∂υ ∂F ∂γ ∂F ∂η ∂F ∂M ∂F ∂N ∂F ∂I ∂F ∂h
+
+
+
+
+
+
+
+
∂µ ∂t ∂υ ∂t ∂γ ∂t ∂η ∂t ∂M ∂t ∂N ∂t ∂I ∂t ∂h ∂t
∂F ∂L ∂F ∂k ∂F ∂E ∂F ∂O ∂F ∂π ∂F ∂ξ ∂F ∂β ∂F ∂
+
+
+
+
+
+
+
+
∂L ∂t ∂k ∂t ∂E ∂t ∂O ∂t ∂π ∂t ∂ξ ∂t ∂β ∂t ∂ ∂t
= α1βt + α 2  t + α 3ξt + α 4 [ut β + u βt ] + α 5 [ut  + u  t ] + α 6 [utξ + uξt ]
+ α 7 [qt s + qst ] + α 8 [qtτ + qτ t ] + α 9 [qt r + qrt ] + α10 [ ptτ + pτ t ] + α11[ pt s + pst ]
+ α12 vt + α13ωt + α14 µt + α15γ t + α16υt + α17ηt − σ 2 ht − σ 2 kt =
0,
but
(1.1.5)
108
Well-Posedness of N.G-KdV class (3+1) equation …
ut p=
h t =
st ξ
=
vt ω x
υt γ=
ξt N =
=
x
qt s =
τ t β=
ωt υ x
=
(1.1.6)
rt L =
βt I=
γ t ηz
=
By inserting equation (1.1.6) into equation (1.1.5), we have
∂F
= (α1 + (α 4 + α 5 )u ) I + α 2 h + (α 3 + α 6u ) N
∂t
+ [α 4 β + α 5  + (α 6 + α11 )ξ + α10 β ] p + s (α 7 s + α 8τ + α 9 r )
+ (α10τ + α 9 s )r + q (α 9 L + α 8 β ) + α12ω x + α13 µ x + α14υ x
(1.1.7)
0
+ α15η x + α16γ x + α17ηt − σ kt − σ Ot =
2
2
Thus equations (1.1.6) and (1.1.5) can be combined to form the following
system:
(α1 + (α 4 + α 5 )u ) I + α 2 h + (α 3 + α 6u ) N + [α 4 β + α 5  + (α 6 + α11 )ξ + α10 β ] p
+ s (α 7 s + α 8τ + α 9 r ) + (α10τ + α 9 s )r + q (α 9 L + α 8 β ) + α12ω x + α13 µ x + α14υ x
+ α15η x + α16γ x + α17ηt − σ 2 E y − σ 2π z =
0
=
ut p =
h =
st ξ
t
=
υt γ x =
ξt N=
vt ω x
=
τt β
=
βt I
=
qt s=
ωt υ x
=
rt L
=
γ t η=
kt E y =
Ot π z
z
(1.1.8)
This is a system of first order partial differential equation in the dependent
variable u , p , q , r , s ,τ ,v , ω , µ ,υ , γ ,η , M , N , I , h , L , k , E ,O , π , ξ , β
and  . The
initial conditions may be obtained from equations (1.1.2) and amount to the
specification of u , p , q , r , s ,τ ,v , ω , µ ,υ , γ ,η , M , N , I , h , L , k , E ,O , π , ξ , β and  .
However, η is not known explicitly, but since the initial conditions are assumed
specified on non-characteristic curve, then η may always be determined. Thus
initial condition t = 0 become
K.A. Moustafa and H.A.F. Mahmoud
( x, y , z )
( x, y , z )
u ( x, y, z , 0) f =
p( x, y, z , 0) g=
q( x, y, z, 0)
=
s ( x, y, z, 0) g=
r ( x, y, z, 0)
τ ( x, y, z, 0) f=
=
xx ( x, y , z )
x ( x, y , z )
v( x, y, z , 0) f =
g=
µ ( x, y, z, 0)
=
xxxxx ( x, y , z ) ω ( x, y , z , 0)
xxxx ( x, y , z )
L( x, y, z , 0) = R(=
( x, y, z ) υ ( x, y, z, 0)
x, y , z )
M ( x, y, z , 0) K=
109
f x ( x, y , z )
H ( x, y , z )
H xxx ( x, y, z )
Rxx ( x, y, z )
( x, y , z )
N ( x, y, z, 0) R=
I ( x, y, z, 0) H xx ( x, y, z )
γ ( x, y, z, 0) K x=
=
x ( x, y , z )
( x, y , z )
( x, y , z )
h( x, y, z, 0) g xxx
k ( x, y, z, 0) C=
E ( x, y, z, 0) g y ( x, y, z )
=
=
( x, y , z )
O( x, y, z, 0) θ=
π ( x, y, z, 0) g z ( x, y, z ) ξ ( x, y, z, 0) = H x ( x, y, z )
=
β ( x, y, z, 0) g=
=
( x, y, z, 0) f xxx ( x, y, z )
xx ( x, y , z )
η ( x, y, z, 0) = G ( f , g , f x , f xx , g x , H , f xxxxx , g xxxx , H xxx , R, K , Rxx , K x , Rx , H xx , g xxx , C , g y ,
θ , g z , H x , g xx , f xxx )
(1.1.9)
The system (1.1.8) can be expressed in the matrix form:
U t + A24×24U x + B24×24U y + D24×24U z + C24×24 =
0
(1.1.10)
U (x , y , z , 0) = w (x , y , z ) .
where U , A24×24 , B24×24 , D24×24 and C24×24 are
U T = [u, p, q, r , s,τ , v, ω , µ ,υ , γ ,η , M , N , I , h, L, k , E , O, π , ξ , β , ]
(1.1.11)
C25T ×25 =−
[ p, −r , − s, − L, −ξ , − β ,0,0,0,0,0, J , −η , −γ , −υ , − µ , − M ,0,0,0,0, − N , − I , −h]
With
J
=
1
α17
[(α1 + (α 4 + α 5 )u ) I + α 2 h + (α 3 + α 6u ) N + [α 4 β + α 5  + (α 6 + α11 )ξ
+ α10 β ] p + s (α 7 s + α 8τ + α 9 r ) + (α10τ + α 9 s )r + q (α 9 L + α 8 β )],
α17 ≠ 0,
110
A24×24
Well-Posedness of N.G-KdV class (3+1) equation …
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K.A. Moustafa and H.A.F. Mahmoud
B24×24
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112
Well-Posedness of N.G-KdV class (3+1) equation …
D24×24
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
α17
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−σ 2
0
0 
0

0
0

0
0

0
0

0

0

0

0

0
0

0

0
0

0
0

0
0

0

0
□
This completes the proof of lemma.
Lemma 1.1.2 The system of equation (1.1.10) under the non singular linear
transformation
α ∗ β ∗ γ ∗
 ∗
δ µ∗ ω∗

S= ∗
 φ η∗ ψ ∗

0
0
 0
0

0
0

1 
This reduces to the system U t + A1U ζ + C 24×24 =
0.
Proof. Let
x → ζ ∗ = α ∗ x + β ∗ y + γ ∗ z,
∗
y → ε ∗ = δ ∗ x + µ y + ω ∗ z,
z → ξ ∗ = φ ∗ x + η ∗ y +ψ ∗ z,
K.A. Moustafa and H.A.F. Mahmoud
113
and t → t ′ =
t . Using this transformation, we have
Ux =
∂U ∂ζ ∗ ∂U ∂ε ∗ ∂U ∂ξ ∗
+
+
= U ζ ∗α ∗ + U ε ∗ δ ∗ + U ξ ∗φ ∗ ,
∂ζ ∗ ∂x ∂ε ∗ ∂x ∂ξ ∗ ∂x
Uy =
∂U ∂ζ ∗ ∂U ∂ε ∗ ∂U ∂ξ ∗
+
+
= U ζ ∗ β ∗ + U ε ∗ µ ∗ + U ξ ∗η ∗ ,
∂ζ ∗ ∂y ∂ε ∗ ∂y ∂ξ ∗ ∂y
Uz =
∂U ∂ζ ∗ ∂U ∂ε ∗ ∂U ∂ξ ∗
+
+
= U ζ ∗ γ ∗ + U ε ∗ ω ∗ + U ξ ∗ψ ∗ .
∂ζ ∗ ∂z ∂ε ∗ ∂z ∂ξ ∗ ∂z
Substituting these values in the system of equation (1.1.10) we have
U t + AU
0,
1 ζ ∗ + A2U ε ∗ + A3U ξ ∗ + C24×24 =
where A1 = α ∗ A24×24 + β ∗ B24×24 + γ ∗ D24×24 ,
(1.1.12)
A2 = δ ∗ A24×24 + µ ∗ B24×24 + ω ∗ D24×24 and
A3 = φ ∗ A24×24 + η ∗ B24×24 +ψ ∗ D24×24 .
Since any solution of this equation depends on the matrices, then the space of
solution
is
9-dimensional
space
span
by
the
matrices
α ∗ , β ∗ , γ ∗ , δ ∗ , µ ∗ , ω ∗ , φ ∗ ,η ∗ ,ψ ∗ . Also the properties of eigenvalues and eigenvectors
under linear transformations denote by [1]. So we shall study the solution on a
subspace of the space of all solutions. This subspace satisfies the condition,
α ∗ , β ∗ , γ ∗ = I and A 2 , A 3 = 0 . Then the equation is reduced to
U t + AU
0,
1 ζ ∗ + C24×24 =
where
A1 = A24×24 + B24×24 + D24×24 and defined the form
(1.1.13)
114
0 0
0 0

0 0

0 0
0 0

0 0
0 0

0 0
0 0

0 0

0 0

0 0
A24×24 + B24×24 + D24×24 =

0 0

0 0
0 0

0 0

0 0
0 0

0 0
0 0

0 0
0 0

0 0

0 0
Well-Posedness of N.G-KdV class (3+1) equation …
0 0 0 0 0
0
0
0
0
0
0 0 0 0 0 0
0
0
0
0 0 0 0 0
0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
0 0 0 0 0
0
0
0
0
0
0 0 0 0 0 0
0
0
0
0 0 0 0 0 −1
0 0 0 0 0 0
0
−1
0
0
0
0 0 0 0 0 0
0
0
0
0
0
0
0 0 0 0 0 0
0
0
0
0 0 0 0 0
0 0 0 0 0
0
0
0
0
−1
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
−1
0
−1 0 0 0 0 0 0
0
0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
α12 α13 α14 α16 α15
α17 α17 α17 α17 α17
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
−σ 2
−σ 2
0 0 0 0 0 0
0
α17
α17
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
−1
0
0
0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
−1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0
0
0
0
0 0 0
0 0 0
0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0
0 0 0

0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0

0 0 0
(1.1.14)
Definition 1.1.3 The system of equation (1.1.13) is called quasi-linear if A1 depend
on U . If A1 independent of U , and C depends on U , but not linearly, the
system is called Semi-linear. If C is also a linear function of U , the system is
called linear.
Theorem 1.1.4 The initial value problem of equation (1.1.1) for the general class
of N.G-KdV class (3+1) equation with non- characteristic data can be reduced to a
non-characteristic initial value problem for a first order semi-linear system of
partial differential equations.
Proof . Making use of the definition (1.1.3) of the semi-linear system and the result
of lemma (1.1.2) the theorem is proved.
□
K.A. Moustafa and H.A.F. Mahmoud
115
2 Characteristics of the system
Definition 2.1 A characteristic of the system (1.1.13) is a curve along which the
values of U , combined with the equation (1.1.13) is insufficient to determine the
derivatives of the normal this curve.
The problem of determining the derivatives of U normal to our data is easily
resolved by considering the effect on system (1.1.13) of a change of coordinates
t →t
and ζ ∗ → Φ (ζ ∗ , t ) =constant.
(2.1)
Then the system (1.1.13) reduces under equation (2.1) to
 ∂U ∂Φ 
 ∂U ∂U ∂Φ 
+
+
+ C24×24 (U ) =
A
0
1

∗
 ∂t ∂Φ ∂t 
 ∂Φ ∂ζ 
 ∂Φ
∂Φ   ∂U   ∂U 
i.e., Ι
0
+ A1 ∗  
+
+ C24×24 (U ) =
∂ζ   ∂Φ   ∂t 
 ∂t
(2.2)
 ∂U 
where 
is the normal derivative of U to, this normal derivative is determined
 ∂Φ 
 ∂Φ
∂Φ 
if det Ι
+ A1 ∗  ≠ 0.
∂ζ 
 ∂t
Combining this result with definition (2.1), then characteristic of the system
(1.1.13) is given by the equation
 ∂Φ
∂Φ 
+ A1 ∗  =
det Ι
0.
∂ζ 
 ∂t
(2.3)
∂Φ
−
∂ζ ∗
∂t
λ =
=
Putting
, then equation (2.3) can be written as
∂Φ
∂t
∂ζ ∗
det( A1 − λΙ) =0
(2.4)
Equation (2.4) is called the characteristic equation of the system (1.1.13), where λ
is now an eigenvalue of the matrix A1 .
The above analysis leads to the following.
116
Well-Posedness of N.G-KdV class (3+1) equation …
Theorem 2.2 The characteristic of the system (1.1.13) which corresponds to the
N.G-KdV class (3+1) equation (1.1.1) is given by the roots of the equation
λ 19 (α12 − α13λ + α14 λ 2 − α16 λ 3 + α15λ 4 − α17 λ 5 ) =
0
 ∂ζ
where λ = 
 ∂t
(2.5)

.

Proof. By using the expression of
A1
as in (1.1.14) and expanding
det(A1 − λΙ) =0 , then obviously equation (2.5) follows and the theorem is proved.□
Definition 2.3
(1) If all the roots of equation (2.4) are real and distinct the system of equation
(1.1.13) is called "totally hyperbolic".
(2) If some of the roots of equation (2.4) are complex, the system is called "ultrahyperbolic".
(3) If all the roots of equation (2.4) are complex, the system (1.1.13) is "elliptic".
(4) The system (1.1.13) is hyperbolic if equation (2.4) has at least one real root.
In the following we shall concentrate only on the case in which the system is
hyperbolic.
2.1 Normal form of the first order system
In the previous section we have demonstrated that the characteristic of
equation (1.1.13) are given by the eigenvalues of eigenvalue problem
A1X = λ X .
It is now convenient to transform the system (1.1.13) to a simple form in which the
differentiation should be in one direction only, i.e., directed along a characteristic
of the system. This new system is called the normal form of equation (1.1.13).
For doing this let the eigenvectors corresponding to the eigenvalues λi of
K.A. Moustafa and H.A.F. Mahmoud
117
A1 span E 24 and let T be the matrix in which each column is one of these
eigenvectors, then T is nonsingular. Suppose that
U = TV
(2.1.1)
Inserting this transformation into (1.1.13) then
(TV )t + A1 (TV )ζ + C=
0, TV (=
ζ , 0) H (ζ ).
24×24
(2.1.2)
Hence
+ ATV
TVt + TV
0.
ζ + AT
t
1
1 ζ V + C24×24 =
Multiplying both sides of equation by T
(2.1.3)
−1
 0,
Vt + T −1 ATV
ζ +C =
1
(2.1.4)
such that
−1
C =
T −1C + T −1 AT
t .
1 ζ V + T TV
(2.1.5)
Since A1 is a matrix of constant coefficients, equation (1.1.13), and then the
eigenvalues of A1 don’t depend on ζ ,t and U consequently T doesn’t depend on
ζ ,t and U this implies that T t = 0= T ζ .
Since T −1A1T = D is diagonal, then equation (2.1.4) can be written as
Vt + DVζ + =
C 0, =
D diag (λ1 , λ2 ,..., λ24 ),
(2.1.6)
with the initial condition
−1
=
V (ζ , 0) T=
U (ζ , t ) ψ (ζ ).
(2.1.7)
Finally, equation (2.1.7) can be written in terms of components and the i-th
component, which corresponds to the i-th characteristic, has the form
i
i i
i
=
V
0, =
V i (ζ , 0) ψ i (ζ ).
t + λ Vζ + C
(2.1.8)
From the theory of a single first order partial differential equation, it follows
that on the characteristic traces for the equation, the equation reduces to an ordinary
118
Well-Posedness of N.G-KdV class (3+1) equation …
differential equation. Hence, V i + λ iV ζi is a directional derivative in the direction
λ i . Thus, every equation in the form (2.1.6) and (2.1.7) contains a differentiation
in one direction only which is the characteristic direction. The form of equation
(2.1.6) and (2.1.7) is the normal form of the system (1.1.13).
Example 2.1.1 Consider the initial value problem
u=
u xx + u yy + u zz
tt
−∞ < x < ∞
, t ≥ 0.
−∞ < y < ∞
−∞ < z < ∞
=
u ( x, y, z , 0) f=
( x, y , z )
q( x, y, z , 0) g ( x, y, z )
=
p ( x, y, z , 0) f=
r ( x, y, z , 0) f xx ( x, y, z )
x ( x, y , z )
(2.1.9)
=
s ( x, y, z , 0) g=
k ( x, y, z , 0) f yy ( x, y, z )
x ( x, y , z )
=
o( x, y, z , 0) g=
h( x, y, z , 0) f zz ( x, y, z )
y ( x, y , z )
=
A( x, y, z , 0) g=
τ ( x, y, z, 0) G ( f , g , f x , f xx , g x , f yy , g y , f zz , g z )
z ( x, y , z )
To find the solution of this equation by using the method of characteristics,
we firstly reduce it to a system of first order partial differential equations. Thus let
F (u , p, q, r , s,τ , k , o, h, A) = τ − r − k − h = 0,
(2.1.10)
where
,
=
q u=
p u=
r u xx=
s u xt ,
t,
x,
,
=
k u=
o u=
h u zz=
A u zt .
τ u=
tt ,
yy ,
yt ,
(2.1.11)
Differentiating equation (2.1.9) with respect to t
∂F ∂F ∂u ∂F ∂p ∂F ∂q ∂F ∂r ∂F ∂s ∂F ∂τ
=
+
+
+
+
+
∂t ∂u ∂t ∂p ∂t ∂q ∂t ∂r ∂t ∂s ∂t ∂τ ∂t
∂F ∂k ∂F ∂o ∂F ∂h ∂F ∂A
+
+
+
+
∂k ∂t ∂o ∂t ∂h ∂t ∂A ∂t
(2.1.12)
Using equation (2.1.11), then equation (2.1.9) reduces to the form
τ t − rt − kt − ht =
0,
where
(2.1.13)
K.A. Moustafa and H.A.F. Mahmoud
119
=
q ut ,=
pt s=
,
st τ x=
,
rt sx ,
(2.1.14)
=
st τ=
kt o=
ot τ=
ht A=
At τ z .
x,
y,
y,
z,
Combining equations (2.1.13) and (2.1.14), then the original equation (2.1.9)
reduces to the system
,
,
qt τ ,=
pt s=
st τ x=
rt sx ,
=
(2.1.15)
st τ=
kt o=
ot τ=
ht A=
At τ z ,
=
x,
y,
y,
z,
with the initial data
=
u ( x, y, z , 0) f=
q( x, y, z , 0) g ( x, y, z )
( x, y , z )
=
p ( x, y, z , 0) f=
r ( x, y, z , 0) f xx ( x, y, z )
x ( x, y , z )
=
s ( x, y, z , 0) g=
k ( x, y, z , 0) f yy ( x, y, z )
x ( x, y , z )
(2.1.16)
=
o( x, y, z , 0) g=
h( x, y, z , 0) f zz ( x, y, z )
y ( x, y , z )
=
A( x, y, z , 0) g=
τ ( x, y, z, 0) G ( f , g , f x , f xx , g x , f yy , g y , f zz , g z )
z ( x, y , z )
Clearly, the system of equations (2.1.15) and (2.1.16) can be written in the matrix
form
U t = AU x + BU y + DU z + CU ,
U ( x, y, z , 0) = H ( x, y, z ),
(2.1.17)
where
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0


0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
A =
, B
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 
0
0

0

0
0

0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
0

0
0

0
0

0
0

0 
120
C
Well-Posedness of N.G-KdV class (3+1) equation …
0 0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 0 


0 0 0 0 1 0 0 0 0 0 


0 0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0
=
, D
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 
0
0

0

0
0

0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
0

0
0
,
1
0

0
1

0 
and U T = [u , p, q, r , s,τ , k , o, h, A] . Thus the characteristic roots are
(0, 0, 0, 0, 0, 0, 0, 0,1, −1) .
Then, the eigenvectors can be written in the form of the
matrix T
1

0
0

0
0
T =
0
0

0
0

0

0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0 0
0 0
0 0
0 −1
0 1
0 −1
0 0
1 0
0 1
0 0
Clearly, the inverse of this matrix exists and has the form
0

0
0

1
1

1
0

0
0 
1 
K.A. Moustafa and H.A.F. Mahmoud
1

0
0

0
0
T −1 = 
0
0

0
0

0

0
1
0
0
0
0
0
0
0
0
121
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0 0 0

0 0 0
0 0 0

0 1 −1
0 −1 −1
,
0 1 −1
0 0 0

1 0 0
0 1 0 
0 0 1 
0
0
0
0
0
0
1
0
0
0
then we have
and
0
0

0

0
0
T −1 AT = 
0
0

0
0

0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0
0

0

0
0
T −1 BT = 
0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
0 0

1 1
−1 1 
,
1 1
0 0

0 0
0 0

0 0 
0
0
0 0
0 0
0 0
0 0
0 0
1 0
1 0
0 −1
0 0
0 0
0
0 
0

0
0
,
0
0

1
0

0 
122
Well-Posedness of N.G-KdV class (3+1) equation …
0
0

0

0
0
T −1CT = 
0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0

0

0
0
T −1 DT = 
0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 −1
0 −1
0 −1
0 0
0 0
0 0
0 1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 1
0 −1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
1 
1

0
0
,
0
0

0
0

0 
0 0 0
0 0 0 
0 0 0

0 1 0
0 1 −2 
,
0 1 1
0 0 0

0 0 0
0 0 1

0 −1 1 
then using the lemma (1.1.2). The equation (2.1.17) reduce to the form
=
U t AU
1 ξ ∗ + CU ,
Now, let U = TV , then the system of equation (2.1.17) reduces to
=
(TV )t A1 (TV )ξ ∗ + C (TV ).
(2.1.18)
Since T doesn’t depend on x , y , z and t , then equation (2.1.18) implies that
=
Vt
(T
−1
−1
AT
1 ) Vξ ∗ + ( T CT ) V ,
(2.1.19)
then using the Lemma 1.1.2. The equation (2.1.19) reduce to the form
=
Vt
(T
−1
−1
AT
1 ) Vζ ∗ + ( T CT ) V ,
Such that A1 = A + B + D , i.e.,
V (ζ ∗ , 0) = H (ζ ∗ ).
(2.1.20)
K.A. Moustafa and H.A.F. Mahmoud
 V1 
V 
 2
 V3 
 
 V4 
 V5 
=
 
 V6 
V 
 7
 V8 
V 
 9
V10  t
123
0
V3

 





0

 V5 + V9 + V10 

 V6 − V9 + V10 
0

 

0
 V5 − V6 + 2V9 + V10  





−V10
0

+
,
0
V5 − V6 + V8 + 2V9 + 2V10  


 

V8
0

 

V6 − V9 + V10
0

 





V10
0

 


 
V6 − V9 + V10

0
with the initial data V (ζ ∗ , 0) = T −1U (ζ ∗ , 0) , i.e.,
=
V [ f , f x , g , − g z + f zz + f xx , − g z − f zz + g x , − g z + f zz , f yy , g y ]T .
(2.1.21)
Next, the first component yields, we find that
∂V1
g (ζ ∗ ).
= V=
3
∂t
(2.1.22)
So, by integrating equation (2.1.22), we get that
x + y + z +t
t
1
=
V1 ∫=
g (ζ )d ζ
g (ζ ∗ )d ζ ∗ .
∫
2
x + y + z −t
0
∗
∗
(2.1.23)
Since V 1=
(x , y , z , 0) u=
(x , y , z , 0) f (x , y , z ) , equation (2.1.23) implies that
t
x + y + z +t
1
( x, y, z , t ) ∫ g=
(ζ )d ζ
=
V1 P u=
g (ζ ∗ )d ζ ∗ .
∫
2 x + y + z −t
0
∗
∗
(2.1.24)
3 Well-posedness of N.G-KdV class (3+1) equation
This section is devoted to the proof of the well-posedness of N.G-KdV class
(3+1) equation (1.1.1), under characteristic data, by using the method of
characteristics. For this purpose, we first establish an integral formula for the
solution of this equation.
124
Well-Posedness of N.G-KdV class (3+1) equation …
3.1 Integral formula of the solution
It has been proved in subsection (1.1), that the equation (1.1.1) of N.G-KdV
class equation can be reduced to the semi-linear system of first order partial
differential equations.
U (ζ ∗ , 0) H (ζ ∗ ) ,
=
0,
Ut + =
AU
1 ζ ∗ + C24×24
and it has been shown that the latter reduces to the normal form
=
Vt + MVζ * + C 0, =
V (ζ ∗ , 0) ψ (ζ ∗ ),
where M = diag (λ1 , λ2 ,..., λ24 ) and C is defined in (2.1.5).
Thus the ith component is
Vt i +=
λ iVζ * C i ,
V i=
(ζ ∗ , 0) ψ i (ζ ∗ ) =
(i 1, 2,...),
(3.1.1)
Such that C = −C . Along the characteristics, equations (3.1.1) are ordinary
differential equations, since the differentiation is now in one direction only. This is
the root of establishing the integral formula.
Definition 3.1.1 (Domain of determinacy) Consider the linear or semi-linear
system U t + AU
0. The domain of determinacy for this system is defined
1 ξ∗ + C =
as the set of all points p (ζ ∗ , t ) which can be connected to the initial interval by
characteristic trajectories.
Now, if p (ζ ∗ , t ) is any point in the domain of determinacy of the system
(3.1.1), then integrating along the characteristic pq i , we have
p

V=
( p ) V (qi ) + ∫ C i (V i )dη ,
i
i
(3.1.2)
qi
where q are those points the initial interval, connected to p by the i-th
characteristic for all i = 1, 2,... , then equation (3.1.2) converted to
K.A. Moustafa and H.A.F. Mahmoud
125
p

V=
( p ) ψ (q ) + ∫ C i (V i )dη ,
i
i
(3.1.3)
qi
This is the integral formula of the underlying system.
3.1.2 Uniqueness
To prove the uniqueness of the solution of the system
=
U (ζ ∗ , 0) H (ζ ∗ ) ,
Ut + =
AU
0,
1 ζ ∗ + C24×24
where A1 and C 25×25 are given by equation (1.1.13), it is important to note that it
can be rewritten in the form
0,=
U=
U (ζ ∗ , 0) H (ζ ∗ ),
t + AU
1 ζ ∗ + Ε 24×24U
(3.1.4)
since by the expression of C 24×24 , it can be shown that C 24×24 = Ε 24×24U , where
U is given by equation (1.1.11) and
−1
0
0
0
0
0
−
0
0
1
0

0
−1
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

 (α 4 β + α 5 + (α 6 + α11 )ξ + α10 β ) (α 9 L + α8 β ) (α10τ + α 9 s) (α 7 s + α8τ + α 9 r )
0
α17
α17
α17
α17
Ε24×24 =

0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(α 3 + α 6u ) (α1 + (α 4 + α 5 )u ) α 2
0 0 0 0 0 0 0 0
α17
α17
α17
0
0
0
−1
0
0
0
0
0
0
0
0 0 0 0 0 0 −1 0
0 0 0 0 0 −1 0 0
0
0
0
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 −1 0 0 0
0 0 0 −1 0 0 0 0
0
0
0
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 −1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0
0
−1
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
−1 0 0 0 0 0 0 0
0
0
0

0
0

0
0

0
0

0

0

0

0

0
0

0
0
0

0
0

0
0

0
0
126
Well-Posedness of N.G-KdV class (3+1) equation …
Lemma 3.1.3 [2] If W (ξ ∗ , t ) is a solution of the linear system
Wt + AW
0,=
W (ξ ∗ , 0) 0,
=
1 ξ ∗ + Ε 24×24W
(3.1.5)
where A1 is a symmetric matrix, then W = 0 .
Using the result of Lemma 3.1.3, then the uniqueness of the solution of the original
system 3.1.4 can be proved.
Theorem 3.1.4 If U is a solution of the semi-linear system (3.1.4), then U is
unique.
Proof. The semi-linear system (3.1.4) can be reduced, by nonsingular linear
transformation to the normal form:
=
+ C (V ) 0,
Vt + MV
ξ∗
=
V (ζ ∗ , 0) ψ (ζ ∗ ),
(3.1.6)
ϒ −1E 24×24 ϒ and λi for every
where U = ΚV , M = diag (λ1 , λ2 ,..., λ24 ) , C =
i = 1, 2,..., 24 are the eigenvalues of the matrix A1 . Hence, to prove the uniqueness
of the system (3.1.4) it suffices, without loss of generality, to prove that the
solution of equation (3.1.6) is unique.
Let V 1 and V 2 be two solutions of equation (3.1.6) and W= V 1 −V 2 , then
W satisfies
Wt + MWξ ∗ + C (V1 ) −=
C (V2 ) 0,
W=
(ξ ∗ , 0) 0
Using the mean-value theorem, we have
C (V1 ) − =
C (V2 ) k (V1 , V2 )(V1=
− V2 ) k (V1 , V2 )W .
Then equation (3.1.7) reduce to
Wt + MW=
+ k (V1 , V2 )W 0,
ξ∗
=
W (ξ ∗ , 0) 0,
(3.1.7)
then the previous equation (3.1.7) with A1 diagonal and k doesn’t depend on W .
Now since W (ξ ∗ , 0) = 0 then by using lemma (3.1.3) W (ξ ∗ , t ) = 0 , i.e., V 1 =V 2 .
Consequently the solution of the equation (3.1.4) is unique.
□
K.A. Moustafa and H.A.F. Mahmoud
Example 3.1.5
127
Consider the initial value problem
0,
u xxt + u xtt + uu xxt + ut u xt + u xtttt + uttttt − u yy − u zz =
(3.1.8)
=
u ( x, y, z , 0) f=
ut ( x, y, z , 0) g ( x, y, z )
( x, y, z ),
=
utt ( x, y, z , 0) H=
uttt ( x, y, z , 0) ϕ ( x, y, z )
( x, y, z ),
utttt ( x, y, z , 0) = Ω( x, y, z ),
(3.1.9)
It is clear that the equation (3.1.8) belongs to the N.G-KdV class (3+1) equation
(1.1). To prove that the solution of this problem is unique, we reduce the problem
into a system of first order partial differential equations. Thus let
F (u , p, q, r , s,τ , ω , µ ,υ , γ ,η , M , N , I , h, L, k , E , O, π , ξ , β ) = 0,
(3.1.10)
=
=
τ u=
ω u xxxxt
p u=
q u=
r u=
s u xt=
,
, µ u xxxtt ,
t,
x,
tt ,
xx ,
=
=
=
υ u=
γ u=
uttttt
N u=
I uttxx
, M u=
, h utxxx , (3.1.11)
xxttt ,
xtttt , η
tttt ,
xttt ,
=
π u zt=
ξ uttx=
β utxx ,
L u=
k u=
E u=
O u=
,
,
ttt ,
yy ,
yt ,
zz ,
then by using equation (3.1.11) into equation (3.1.8)
ξ + β (1 + u ) + ps + γ + η − k − O =
0.
(3.1.12)
Differentiating equation (3.1.10) with respect to t and using equation (3.1.11), we
get
∂F ∂F
∂F
∂F
∂F
∂F
∂F
∂F
∂F
∂F
ut +
pt +
qt +
rt +
st +
τt +
ωt +
µ t + υt
=
∂t ∂u
∂p
∂q
∂r
∂s
∂τ
∂ω
∂µ
∂υ
∂F
∂F
∂F
∂F
∂F
∂F
∂F
∂F
Mt +
Nt +
It +
Lt +
kt +
Et
γ t + ηt +
+
∂γ
∂η
∂M
∂N
∂I
∂L
∂k
∂E
∂F
∂F
∂F
∂F
Ot +
πt +
ξt +
βt ,
+
∂O
∂π
∂ξ
∂β
then equation (3.1.10) reduce to the form
ξt + β ut + βt (1 + u ) + pst + pt s + γ t + ηt − kt − Ot =
0,
(3.1.13)
But
=
p ut=
, s qt ,=
β τt ,=
βt I ,=
ξt N ,=
pt r=
, st ξ ,
=
γ t η=
kt E y=
, Ot π =
ωt µ =
µt υ =
π t r=
Et ry .
x,
z,
x,
x,
z,
Combining equation (3.1.13) and (3.1.14) then, we get
(3.1.14)
128
Well-Posedness of N.G-KdV class (3+1) equation …
0,
N + β p + I (1 + u ) + pξ + rs + η x + ηt − E y − π z =
(3.1.15)
with the initial data
=
u ( x, y, z , 0) f=
( x, y , z )
p( x, y, z , 0) g ( x, y, z )
=
q ( x, y, z , 0) f=
r ( x, y, z , 0) H ( x, y, z )
x ( x, y , z )
=
s ( x, y, z , 0) g=
τ ( x, y, z, 0) f xx ( x, y, z )
x ( x, y , z )
=
L( x, y, z , 0) ϕ=
( x, y, z ) ξ ( x, y, z , 0) H x ( x, y, z )
β ( x, y, z , 0) = g xx ( x, y, z ) M ( x, y, z, 0) = Ω( x, y, z )
=
N ( x, y, z , 0) ϕ=
I ( x, y, z , 0) H xx ( x, y, z )
x ( x, y , z )
γ ( x, y, z , 0) =
Ωx ( x, y, z ) k ( x, y, z , 0) =
f yy ( x, y, z )
=
E ( x, y, z , 0) g=
O( x, y, z , 0) f zz ( x, y, z )
y ( x, y , z )
=
π ( x, y, z , 0) g=
h( x, y, z , 0) g xxx ( x, y, z )
z ( x, y , z )
η ( x, y, z , 0)= G= [ f , g , f x , h, g x , f xx , ϕ , hx , g xx , g xxx , Ω, ϕ x , hxx , Ωx , f yy , g y , f zz , g z ]
The system (3.1.14) and (3.1.15) can be written in the matrix form
U t + AU x + BU y + DU z + CU =
0,
(3.1.16)
then by using the Lemma 1.1.2. The equation (3.1.16) reduce to the form
U t +=
AU
0,
1 ξ ∗ + CU
=
U (ξ ∗ , 0) H (ξ ∗ ),
(3.1.17)
where
U T = [u , p, q, r , s,τ , ω , µ ,υ , γ ,η , M , N , I , h, L, k , E , O, π , ξ , β ],
then
(CU )T =[− p, −r , − s, − L, −ξ , − β , 0, 0, 0, 0, N + β p + I (1 + u ) + pξ + rs, −η ,
− γ , −υ , 0, − M , 0, 0, 0, 0, − N , − I ]
where A1 = A + B + D ,
(3.1.18)
K.A. Moustafa and H.A.F. Mahmoud
0
0

0

0
0

0
0

0
0

0

0
A1 = 
0

0
0

0
0

0

0
0

0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0
0 −1 0 0 0
0 0 −1 0 0
0 0 0 −1 0
0 0 0 0 −1
0 0
0 0
0
0
0 0 0
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 −1 0 0 0
0 0 0 0 0 0
0 0 −1 0 0 0
0
0
0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0
0 0 0
0
0
0
0
0
0
0
0
129
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0 0
0 0
0
0
0 0 0 −1 0 −1 0
0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0
0
0
0 0 0
0 0 0
0 0 0
0
0
0
0 0 0
0 −1 0
0 0 0
0
0
0
0
0
0
0 0
0 0
0
0
0 0 0
0 0 0
0
0
0
0
0 0
0 0
0 −1
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 −1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
0

0
0

0
0

0
0

0

0
0

0
0

0
0

0

0
0

0
0

0 
The eigenvalues of A1 given by det(A1 − λ I ) =
0 , thus
λi = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,1],
(3.1.19)
the eigenvectors correspond these eigenvalues (3.1.19) are the solution of
A1X = λ X , then T and T
−1
can be easily obtained and yields
T −1A1T = diag (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,1) .
Now, applying the nonsingular transformationU = TV , to the system of equation
(3.1.10) this leads to the characteristic form, i.e.,
Vt + GVξ ∗ + C 0, =
V (ξ ∗ , 0) φ (ξ ∗ ),
=
(3.1.20)
where G = diag (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,1) and
[ p, −r , − s, − L, −ξ , − β , I , − I , I , − I , J + I , −η , −γ , −υ , 0, − M , 0, 0, 0, 0, − N , − I ]T .
C =−
130
Well-Posedness of N.G-KdV class (3+1) equation …
Hence to prove the uniqueness of the system (3.1.14) and (3.1.15) it is
sufficient to prove the uniqueness of (3.1.20). Thus, let V 1 and V 2 be two
solutions of equation (3.1.20).
Let W= V 1 −V 2 . Then W satisfies the initial value problem
Wt + GWξ ∗ + C (V1 ) =
− C (V2 ) 0,
W=
(ξ ∗ , 0) 0.
(3.1.21)
By using the differentiation of C from equation (3.1.18) with the relations
J + I = N + β p + I (1 + u ) + pξ + rs + I
β1 p1 − β 2 p2 = ( β1 − β 2 ) p1 + ( p1 − p2 ) β 2 = β p1 + pβ 2 ,
and
p1ξ1 − p2ξ 2 = ( p1 − p2 )ξ1 + (ξ1 − ξ 2 ) p2 = pξ1 + ξ p2 ,
r1s1 − r2 s2 = (r1 − r2 ) s1 + ( s1 − s2 )r2 = rs1 + sr2 ,
I1u1 − I 2u2 = ( I1 − I 2 )u1 + (u1 − u2 ) I 2 = Iu1 + uI 2 ,
where=
p1 u1=
u=
W t …etc, . Then
t , p1
2t , p
C (V 1) − C (V2 ) =
[− p, −r , − s, − L, −ξ , − β , I , − I , I , − I , N + β p1 + p β 2 + Iu1 + uI 2 + pξ + rs1 + sr2 + 2 I ,
− η , −γ , −υ , 0, − M , 0, 0, 0, 0, − N , − I ].
 , where
MW
Hence C (V 1 ) − C (V 2 ) =
K.A. Moustafa and H.A.F. Mahmoud
−1
0
0
0

0
0

0
0
0
0

0
0
0
0

0
0
0
0

0
0

u β 2 + ξ1
M =  2
0
0

0
0
0
0

0
0
0
0

0
0

0
0
0
0

0
0
0
0

0
 0
0 0
0 −1
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
131
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
−1
0 0
0 0
0 0
0 −1
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0
0
0
0
0
0
1
0 0
0 s1
0 0
0 0
0 0
0 0
0 0
0 0
0
r2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 −1
0 0
0 0
0 0
0
0
0
−1
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
1
0
0
0
0
0
0
−1
2 + u1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Wt + GWξ ∗ + K (V1 =
, V2 )W 0,
0 0 0
0 0 0 
0 0 0

0 0 0
0 −1 0 

0 0 −1
0 0 0

0 0 0
0 0 0

0 0 0

0 p2 p1 
0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0

0 0 0
0 0 0

0 0 0
0 0 0

0 0 0 
W=
(ξ ∗ , 0) 0.
In the case of this system is linear in W and the matrix M is symmetric and
W (ξ ∗ , 0) = 0 , then the hypotheses of lemma (3.1.3) has at most one solution.
Consequently, the equation (3.1.8) has at most one solution also.
4 Existence
The existence theory for hyperbolic system of quasi-linear partial differential
equations
U t + AU x + BU
=
0, =
U ( x, y, z , 0) H ( x, y, z )
y + DU z + C
has been studied in one dimension by many [3], [4], [5], for the analytic problem,
132
Well-Posedness of N.G-KdV class (3+1) equation …
now we introduce this initial value problem in (3+1) dimensions, i.e., when
A , B , D and C are analytic in x , y , z , t and H is analytic in x , y and z ,
then the solution exists and depends continuously on data in the small (i.e., for
suitably narrow neighborhood of
x , y , z = 0 and t = 0 ) by the Caushy-
Kowalewsky theorem [2]. This result was extended by Lax [6] who was able to
show firstly that for analytic data the solution exists not only in the small but it can
be continued analytically until it reaches the boundary of the domain of analyticity.
Secondly, by approximating a non- analytic problem by a sequence of analytic
problems and using the above results, the solution of a non-analytic initial value
problem which is now a generalized solution is shown to get Lax proved that if all
the matrices A , B , D , C and T (where T is the matrix of eigenvectors of A )
have continuous first derivatives and the first derivative of H (x , y , z ) is almost
everywhere continuous at all regular points of the system, i.e., points that don’t lie
on characteristics through points of discontinuity of the initial data.
Now we will study the existence theory for hyperbolic system of semilinear of equation (3.1.4) which has the normal form
=
Vt + MVξ * + C 0 =
V (ξ * , 0) ψ (ξ * ),
where M = diag (λ1 , λ2 ,..., λ24 ) ,
(4.1)
and ξ * = ξ * ( x, y, z ) .
Now, we introduce the next lemma and then prove the existence for equation (4.1).
Lemma 4.2 The system of differential equations (4.1) can be replaced equivalently
by a system of nonlinear integral equations.
Proof. Let M =
k
∂
∂
in the k-th component of equation (4.1) then M k
+ λk
∂t
∂ξ *
can be regarded as differentiation along the characteristic C k . Thus, by similar
arguments as were used to derive the integral formula (4.1), the system of equation
(3.1.3) corresponds to the nonlinear integral equations.
V = LV
(4.2)
K.A. Moustafa and H.A.F. Mahmoud
133
p

LV =
( ρ ,τ ) ψ (ξ ) + ∫ C k (ξ * ,η , )dη ,
k
i
*
k
(4.3)
0
□
This proves the lemma.
Now, we define the region in which the existence proof is valid. Let H be a
closed domain in x , y , z and t space in which all the characteristics C i followed
from a point p in H backwards in t meet a given section J of the initial data
line t = 0 in the points p i , as in the following figure
P
Ai
Pi
Ak
Pk
Let S be the set of all functions V
with domain H having continuous
derivatives and equal to ψ (x , y , z ) on t = 0 . Finally, we define the norm of
elements of S to be the largest value of the functions attained in the closed
domain H . However, if we choose
ψ (ξ * ) = N * and restrict admissible
functions by choosing V < 2N * then there exists a common upper bound  > 0
such that [7]:
CVk < , Cξk* <  , and
C tk < ,
(4.4)
where CVk is a functional gradient of C k with respect to V . Note that
k
k
C=
C=
0 for the equation (3.1.4).
t
ξ*
Now, we introduce the following theorem:
134
Well-Posedness of N.G-KdV class (3+1) equation …
Theorem 4.3 Let ψ (ξ * ) , C have continuous first derivatives, then the system
=
Vt + MVξ * + C 0 =
V (ξ * , 0) ψ (ξ * ),
(4.5)
possesses a solution which has the same differentiability as ψ (ξ * ) .
Proof.
If we choose α sufficiently small, then equation (4.3) implies that
V
k
≤ ψ (ξ * ) + α = N * + α  ≤ 2N * .
The system of equation (4.3) lends itself immediately to a process of solution by
iteration and for a suitably narrow strip H α the desired fixed element will be
constructed as the uniform limit, as derivatives with respect to ξ * , since the
t − derivatives follows from the known directional derivatives.
Now, the existence and continuity of V in the characteristic direction follows
directly system of equation (4.3) and from the continuity of the solution obtained.
∂V
we observe, first of all,
∂ξ *
To prove the existence and continuity of derivatives
that the assumed continuous differentiability of ψ (ξ * ) and C implies that all the
approximations constructed in proving the existence of a solution, have continuous
derivative with respect to ξ * . Differentiating the (n + 1) th approximation,
τ
V n +1 (ξ ,τ ) ψ (ξ (0,τ , ε )) + ∫ C (ξ * , t ,V n )d η
=
*
*
0
By differentiate with respect to ε . Thus
τ
 ∂C ∂Vn ∂ξ * ∂C ∂ξ * 
∂Vn +1
∂ε
= ψ ′(ξ * (0,τ , ε )) * + ∫ 
+
dη
∂ε
∂ξ
∂Vn ∂ξ * ∂ε ∂ξ * ∂ε 
0
τ
∂ε
=ψ ′ * + ∫ CV Vξ * + Cξ * ξε*dη .
∂ξ
0
(
)
(4.6)
Similar to the assumption for the system of equation (4.3) we can prove the
 ∂V
uniform convergence of the sequence  *n
 ∂ξ
 *
 { ξ instead of ε }, n = 1, 2,... , by

using the same method which we used to prove the convergence of (V n ) . This
K.A. Moustafa and H.A.F. Mahmoud
135
∂V n ∂V
= * , which suffices to prove the existence of the solution of
n →∞ ∂ξ *
∂ξ
gives us lim
the characteristic system (4.4) locally. We must be sure that the solution exists
globally. i.e., in a larger region, we use the line t = α as new initial line and solve
the problem by the same procedures, as above, in the strip α < t < 2α . We continue
stepwise in this way which implies the existence of the solution in an arbitrary
large t so long as the assumption of the continuity and bounded remains satisfied
the existence of the original system
U=
U (ξ * , 0) G (ξ * ).
0 =
t + AU
1 ξ* + C
□
Theorem 4.4 Let U (x , y , z , t ) and W (x , y , z , t ) be two solutions of equation
(4.1), such that U (x , y , z , 0) = ψ (x , y , z ) and ϕ −ψ < δ . Then W −U < ε and
ε → 0 as δ → 0 (continuous dependence of the solution on the initial data).
Proof. Let
ϕ ( x, y , z ) − ψ ( x, y , z ) =
α ( x, y, z ),
where α (x , y , z ) < δ , and
U (x , y , z , t ) −W (x , y , z , t ) =
Ω( x , y , z , t ) .
Then, as in Theorem 4.3, and Ω satisfies the integral equation
τ
Ω( x, y, z , t ) = δ ( x, y, z ) + ∫ CV ( x, y, z ,η , V )(U − W )dη
0
τ
=
δ ( x, y, z ) + ∫ CV ( x, y, z ,η , V )Ω( x, y, z ,η )dη ,
0
where V
is intermediate value.
Let max Ω(x , y , z , t ) =
ε , then by estimates analogous to that used in the
x , y , z ,t ∈S
existence proof
ε < δ + ετ ,
( C < ),
(4.7)
replacing Ω in integral equation (4.6) by the right hand side of equation (4.7) and
repeating, we obtain
136
Well-Posedness of N.G-KdV class (3+1) equation …
ε < δ (1 + τ ) + ε
 2τ 2
.
2
Repeating this operation n times we have

 2τ 2
 n −1τ n −1
 nτ n 
ε < δ 1 + τ +
+ ... +
+ε
2
(n − 1)
n 

Now, as n → ∞ , we get ε < δ e t . Thus if t is bounded, then δ → 0 implies
ε → 0 , which proves the theorem.
□
5 Conclusion
In this article, the well-posedness of N.G KdV class (3+1) equation was
investigated. For this investigation it was convenient to reduce N.G KdV class
(3+1) equation to a system of first order partial differential equations. It is found
that if α17 ≠ 0 and the data are non characteristic, then N.G KdV class (3+1)
equation reduced to a semi-linear system of first order partial differential equations
on its characteristics. The proof of this fact was carried out for the case where all
the characteristics are real and this proof can be done if some of these
characteristics are complex by reducing the system to two systems of real
characteristics and the reduction to systems of ordinary differential equations is
clearly obtained again.
Acknowledgements. Special thank for Prof. Gamal Samy Mokaddis and Prof.
Hamid moustafa El-sherbiny for their essential helps, encouragement, helpful
discussions, their kind advice and their careful review of the article.
K.A. Moustafa and H.A.F. Mahmoud
137
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