Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
171
a(x, y, t) =
−
4α
φ(η, ζ, t)
(13)
Substituting equations (12) and (13), in Eq. (11), we get
(1 + ρ 1 +
2
ρ 2 +
3
ρ 3 + · · · )
−
3
v
2
αh 2 φ ηη +
5
v
48αh 2 φ ητ − · · ·
=
K
˜
m
−
3
4α
φ ηη −
5
α
φ ζ ζ − · · ·
(14)
Equating equal powers of on either sides of Eq. (14)
3
:
K
˜
m
=
v
2
h 2
(15)
4
: ρ 1 φ ηη = 2 ˙
φ ηζ
(16)
5
: −ρ 2 φ ηη +
1
12v
φ ητ = −φ τ ζ +
1
2
φ η φ ηη −
1
12
φ ηηηη
(17)
Using another change of variable
X = η + 12
ρ 2 (τ )dτ, T = τ, Y = y
(18)
and
U (X, Y, T ) = φ η (η, ζ, τ )
(19)
Then Eq. (17) reduces to
∂
∂ X
(U T − 6U X X + U X X X ) = −12U Y Y
(20)
which can be written as
U T X − 6U
2
x − 6UU X X + U X X X X + 12U Y Y = 0
(21)
Thus, for quadratic nonlinearity, the equation of motion for wave propagation through
2D lattice reduces to two-dimensional form of KdV equation which is now known
as KP equation.
Case(a): Quadratic nonlinearity α > 0, β = 0.
In this case, the equation of motion becomes
m i ¨
a =K (a i+1 − a i + α(a i+1 − a i )
2
+ β(a i+1 − a i )
3
+ · · · − (a i − a i−1 )−
α(a i − a i−1 )
2
− β(a i − a i−1 )
3
− · · ·
(22)