卡东穆塞夫-彼得韦亚斯维利方程

卡东穆塞夫-彼得韦亚斯维利方程(Kadomtsev-Petviashvili equation)，简称KP方程，是1970年苏联物理学家波里斯·卡东穆塞夫 和弗拉基米尔-彼得韦亚斯维利创立以模拟非线性波动的非线性偏微分方程^[1]：

这张在法国大西洋岸雷岛（RHE）鲸鱼灯塔拍摄的照片，显示浅海上田字形的椭圆余弦波列。这种浅水中的孤波可以由卡东穆夫-彼得韦亚斯维利方程模拟。

${\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0}$

其中 ${\displaystyle \lambda =\pm 1}$.

解析解

卡东穆塞夫-彼得韦亚斯维利方程有解析解^[2]

行波解

${\displaystyle u(x,y,t)=C5+12.*_{C}2*\tanh(_{C}1+_{C}2*x+_{C}3*y-(.50000000000000000000*(8.*_{C}2^{4}+_{C}3^{2}))*t/_{C}2)}$ 

代人参数： C5 = 1, _C1 = 0, _C2 = 1, _C3 = 3

得：${\displaystyle u=1+12.*tanh(x+3*y-8.5000000000000000000*t)}$ 

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  Kadomtsev-Petviahivili equation 3D plot

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  Kadomtsev-Petviashivili pde animation

Sech 函数亮孤立子解

利用sech函数展开法可得卡东穆塞夫-彼得韦亚斯维利方程的sech函数解和tanh函数解^[3]。

${\displaystyle u:=a*sech(a*x+b*y+c*z-(a^{4}+3*b^{2}+3*c^{2})/a)*t}$ 

参数：a = -2 .. 2, b = -2 .. 2, c = 0

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  Kadomtsev Petviashivili pde sech solution 3d plot

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  卡东穆塞夫-彼得韦亚斯维利方程sech函数亮孤立子

tanh 函数解

${\displaystyle u:=2*a^{2}*tanh(a*x+b*y+(8*a^{4}-3*b^{2})/a)^{2}*t}$ ^[4]。

参数：a = 2, b = -2；

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  KP方程暗tanh函数暗孤立子

雅可比橢圓函數解

通过朗斯基行列式展开法可得卡东塞穆夫-彼得韦亚斯维利方程多个雅可比橢圓函數解^[5]。

${\displaystyle u4:={\frac {(-4*m^{2}*k[1]^{2}*g)}{({\sqrt {1-m^{2}}}*sn(\xi [1],k)*sin(\xi [2])+dn(\xi [1],k)*cos(\xi [2])*cn(\xi [1],k))^{2})}}}$ 

其中：

${\displaystyle g=(m^{2}-1)*sn(\xi [1],k)^{2}+(2-2*m^{2})*sn(\xi [1],k)^{4}+cos(\xi [2])^{2};-2*sn(\xi [1],k)^{2}*cos(\xi [2])^{2}+m^{2}*sn(\xi [1],k)^{4}*cos(\xi [2])^{2}}$ 

${\displaystyle \xi [1]=k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1]}$ 

${\displaystyle \xi [2]={\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2]}$ 

代入后得：

${\displaystyle f4:=-4*m^{2}*k[1]^{2}*((m^{2}-1)*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}}$ 

${\displaystyle -3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{2}+(2-2*m^{2})*JacobiSN(k[1]*x+\lambda [1]*y}$ 

${\displaystyle +(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{4}+}$ 

${\displaystyle cos({\sqrt {(1-m^{2})}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)}$ 

${\displaystyle -\gamma [2])^{2})/(sqrt(1-m^{2})*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-}$ 

${\displaystyle 3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*sin({\sqrt {(}}1-m^{2})*(k[1]*x+\lambda [1]*y+}$ 

${\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])+JacobiDN(k[1]*x+\lambda [1]*y+}$ 

${\displaystyle (4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*cos({\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+}$ 

${\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])*JacobiCN(k[1]*x+}$ 

${\displaystyle \lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k))^{2}}$ 

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  Kadomtsev Petviashivili pde elliptic function solution 3d plot

参考文献

1. 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)
2. Erk Infeld & George Rowlands, Nonlinear Waves,Solitons and Chaos p224-233 Cambridge University Press,2000
3. 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
4. AHMET BEKIR and ÖZKAN GÜNER
5. 吕大昭等 Novel Interaction Solutions to Kadomtsev–Petviashvili Equation，Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 484–488

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