KdV方程

科特韦赫-德弗里斯方程（英語：Korteweg-De Vries equation），一般简称KdV方程，是1895年由荷兰数学家科特韦赫和德弗里斯共同发现的一种偏微分方程。关于实自变量x 和t 的函数φ所满足的KdV方程形式如下：

\partial _{t}\phi -6\phi \partial _{x}\phi +\partial _{x}^{3}\phi =0

KdV方程的解为簇集的孤立子（又称孤子，孤波）。

KdV方程的行波解

KdV 方程有多种孤波解^[1]^[2]。

\phi (x,t)={\frac {1}{2}}\,c\,\mathrm {sech} ^{2}\left[{{\sqrt {c}} \over 2}(x-c\,t-a)\right]

\phi (x,t)=k\,\mathrm {tanh} [k(x+2tk^{2}+c)]

$\phi (x,t)=a+b\,\mathrm {tanh} (1+cx+dt)^{2}$

利用Maple tanh 法可得孤立子解：^[3]。

{u(x,t)=(1/6)*(4*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=(1/6)*(4*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}}

u(x,t)=-(1/6)*(4*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}

{u(x,t)=-(1/6)*(4*_{C}2^{3}+_{C}3)/_{C}2+2*_{C}2^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=(1/6)*(8*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=(1/6)*(8*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=-(1/6)*(8*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*cot(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=-(1/6)*(8*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2}}

{u(x,t)=(1/6)*(-8*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+2*_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

{u(x,t)=(1/6)*(-8*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+(2*_{C}3^{2}-2*_{C}3^{2}*_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

{u(x,t)=(1/6)*(4*_{C}3^{3}*_{C}1^{2}+4*_{C}3^{3}-_{C}4)/_{C}3-2*_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

{u(x,t)=(1/6)*(4*_{C}3^{3}*_{C}1^{2}+4*_{C}3^{3}-_{C}4)/_{C}3-2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

{u(x,t)=-(1/6)*(8*_{C}3^{3}*_{C}1^{2}-4*_{C}3^{3}+_{C}4)/_{C}3+(-2*_{C}3^{2}+2*_{C}3^{2}*_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

{u(x,t)=-(1/6)*(8*_{C}3^{3}*_{C}1^{2}-4*_{C}3^{3}+_{C}4)/_{C}3+2*_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}

9.81207-7.70406*I+5.44331*arctanh(10.4881/{\sqrt {(}}-110.*csc(1.40000+1.50000*x+1.60000*t)^{2}+110.))

9.81207-7.70406*I-5.44331*arctan(10.4881/{\sqrt {(}}-110.*csch(1.40000+1.50000*x+1.60000*t)^{2}-110.))

9.81207-7.70406*I+5.44331*arctan(10.4881/{\sqrt {(}}-110.*csch(1.40000+1.50000*x+1.60000*t)^{2}-110.))

KdV方程行波图

KdV方程在物理学的许多领域都有应用，例如等离子体磁流波、离子声波、非谐振晶格振动、低温非线性晶格声子波包的热激发、液体气体混合物的压力表等。

KdV方程也可以用逆散射技术求解。

Korteweg, D. J. and de Vries, F. "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves." Philosophical Magazine, 39, 422--443, 1895.
P. G. Drazin. Solitons. Cambridge University Press, 1983.

^ 阎振亚著《复杂非线性波动构造性理论及其应用》 29页科学出版社 2007
^ Graham W.Griffiths William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p422-430
^ Graham W.Griffiths William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p391-404

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