這張在法國大西洋 岸雷島 (RHE)鯨魚燈塔拍攝的照片,顯示淺海上田字形的橢圓餘弦波列。這種淺水中的孤波 可以由卡東穆夫-彼得韋亞斯維利方程模擬。
卡東穆塞夫-彼得韋亞斯維利方程 (Kadomtsev-Petviashvili equation),簡稱KP方程,是1970年蘇聯物理學家波里斯·卡東穆塞夫 和弗拉基米爾-彼得韋亞斯維利創立以模擬非線性波動的非線性偏微分方程[ 1] :
∂
x
(
∂
t
u
+
u
∂
x
u
+
ϵ
2
∂
x
x
x
u
)
+
λ
∂
y
y
u
=
0
{\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0}
其中
λ
=
±
1
{\displaystyle \lambda =\pm 1}
.
解析解
卡東穆塞夫-彼得韋亞斯維利方程有解析解[ 2]
行波解
u
(
x
,
y
,
t
)
=
C
5
+
12.
∗
C
2
∗
tanh
(
C
1
+
C
2
∗
x
+
C
3
∗
y
−
(
.50000000000000000000
∗
(
8.
∗
C
2
4
+
C
3
2
)
)
∗
t
/
C
2
)
{\displaystyle u(x,y,t)=C5+12.*_{C}2*\tanh(_{C}1+_{C}2*x+_{C}3*y-(.50000000000000000000*(8.*_{C}2^{4}+_{C}3^{2}))*t/_{C}2)}
代人參數: C5 = 1, _C1 = 0, _C2 = 1, _C3 = 3
得:
u
=
1
+
12.
∗
t
a
n
h
(
x
+
3
∗
y
−
8.5000000000000000000
∗
t
)
{\displaystyle u=1+12.*tanh(x+3*y-8.5000000000000000000*t)}
Sech 函數亮孤立子解
利用sech函數展開法可得卡東穆塞夫-彼得韋亞斯維利方程的sech函數解和tanh函數解[ 3] 。
u
:=
a
∗
s
e
c
h
(
a
∗
x
+
b
∗
y
+
c
∗
z
−
(
a
4
+
3
∗
b
2
+
3
∗
c
2
)
/
a
)
∗
t
{\displaystyle u:=a*sech(a*x+b*y+c*z-(a^{4}+3*b^{2}+3*c^{2})/a)*t}
參數:a = -2 .. 2, b = -2 .. 2, c = 0
tanh 函數解
u
:=
2
∗
a
2
∗
t
a
n
h
(
a
∗
x
+
b
∗
y
+
(
8
∗
a
4
−
3
∗
b
2
)
/
a
)
2
∗
t
{\displaystyle u:=2*a^{2}*tanh(a*x+b*y+(8*a^{4}-3*b^{2})/a)^{2}*t}
[ 4] 。
參數:a = 2, b = -2;
雅可比橢圓函數解
通過朗斯基行列式 展開法可得卡東塞穆夫-彼得韋亞斯維利方程多個雅可比橢圓函數解[ 5] 。
u
4
:=
(
−
4
∗
m
2
∗
k
[
1
]
2
∗
g
)
(
1
−
m
2
∗
s
n
(
ξ
[
1
]
,
k
)
∗
s
i
n
(
ξ
[
2
]
)
+
d
n
(
ξ
[
1
]
,
k
)
∗
c
o
s
(
ξ
[
2
]
)
∗
c
n
(
ξ
[
1
]
,
k
)
)
2
)
{\displaystyle u4:={\frac {(-4*m^{2}*k[1]^{2}*g)}{({\sqrt {1-m^{2}}}*sn(\xi [1],k)*sin(\xi [2])+dn(\xi [1],k)*cos(\xi [2])*cn(\xi [1],k))^{2})}}}
其中:
g
=
(
m
2
−
1
)
∗
s
n
(
ξ
[
1
]
,
k
)
2
+
(
2
−
2
∗
m
2
)
∗
s
n
(
ξ
[
1
]
,
k
)
4
+
c
o
s
(
ξ
[
2
]
)
2
;
−
2
∗
s
n
(
ξ
[
1
]
,
k
)
2
∗
c
o
s
(
ξ
[
2
]
)
2
+
m
2
∗
s
n
(
ξ
[
1
]
,
k
)
4
∗
c
o
s
(
ξ
[
2
]
)
2
{\displaystyle g=(m^{2}-1)*sn(\xi [1],k)^{2}+(2-2*m^{2})*sn(\xi [1],k)^{4}+cos(\xi [2])^{2};-2*sn(\xi [1],k)^{2}*cos(\xi [2])^{2}+m^{2}*sn(\xi [1],k)^{4}*cos(\xi [2])^{2}}
ξ
[
1
]
=
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
{\displaystyle \xi [1]=k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1]}
ξ
[
2
]
=
1
−
m
2
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
{\displaystyle \xi [2]={\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2]}
代入後得:
f
4
:=
−
4
∗
m
2
∗
k
[
1
]
2
∗
(
(
m
2
−
1
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
{\displaystyle f4:=-4*m^{2}*k[1]^{2}*((m^{2}-1)*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}}
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
2
+
(
2
−
2
∗
m
2
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
{\displaystyle -3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{2}+(2-2*m^{2})*JacobiSN(k[1]*x+\lambda [1]*y}
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
4
+
{\displaystyle +(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{4}+}
c
o
s
(
(
1
−
m
2
)
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
{\displaystyle cos({\sqrt {(1-m^{2})}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)}
−
γ
[
2
]
)
2
)
/
(
s
q
r
t
(
1
−
m
2
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
{\displaystyle -\gamma [2])^{2})/(sqrt(1-m^{2})*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-}
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
∗
s
i
n
(
(
1
−
m
2
)
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle 3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*sin({\sqrt {(}}1-m^{2})*(k[1]*x+\lambda [1]*y+}
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
)
+
J
a
c
o
b
i
D
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])+JacobiDN(k[1]*x+\lambda [1]*y+}
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
∗
c
o
s
(
1
−
m
2
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle (4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*cos({\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+}
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
)
∗
J
a
c
o
b
i
C
N
(
k
[
1
]
∗
x
+
{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])*JacobiCN(k[1]*x+}
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
)
2
{\displaystyle \lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k))^{2}}
參考文獻
^ Kodomtsev,B.B and Petviashivili V.I. On the stability of solitary waves in weakly dispersive media Dokl. Akad Nauk SSSR 192 753-6(1970) Soviet Phys. Dok 15,539-41(1970)
^ Erk Infeld & George Rowlands, Nonlinear Waves,Solitons and Chaos p224-233 Cambridge University Press,2000
^ AHMET BEKIR and ÖZKAN GÜNER Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation,journal of Physics, August 2013 Vol. 81, No. 2, pp. 203–214
^ AHMET BEKIR and ÖZKAN GÜNER
^ 呂大昭等 Novel Interaction Solutions to Kadomtsev–Petviashvili Equation,Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 484–488
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