卡马萨-霍尔姆方程

卡馬薩-霍爾姆方程(Camassa Holm equation)是流體力學中的一個非線性偏微分方程

${\displaystyle u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}.\,}$

1993年卡馬薩和霍爾姆以此偏微分方程模擬淺水波^[1]，

其中κ是大於0的參數。

行波解

卡馬薩-霍爾姆方程3D動畫

卡馬薩-霍爾姆方程有行波解^[2]：

${\displaystyle u2:=(3/2)*{\frac {c*(c-2*\kappa )}{(cosh((1/2)*{\sqrt {((c-2*\kappa )/c)}}*(-x-x0+c*t))^{2}*(\kappa +c))}}}$

參數：c = 1, x0 = 1, kappa = .3 代人得：

${\displaystyle u(x,t)={\frac {.463}{cosh(-.316*x-.316+.316*t)^{2}}}}$

Maple TWSolution

Maple軟體包TWSolution可提供多種行波解^[3]。

sech 展開

${\displaystyle g[2]:={u(x,t)=-(1/2)*kappa+_{C}5*sech(_{C}1+(1/2)*sqrt(3)*x-(1/4)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[3]:={u(x,t)=-(1/2)*kappa+_{C}5*sech(_{C}1-(1/2)*sqrt(3)*x+(1/4)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[4]:={u(x,t)=_{C}4+(-(3/2)*_{C}4-(3/4)*kappa)*sech(_{C}1+(1/2*I)*sqrt(3)*x-(1/4*I)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[5]:={u(x,t)=_{C}4+(-(3/2)*_{C}4-(3/4)*kappa)*sech(_{C}1-(1/2*I)*sqrt(3)*x+(1/4*I)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[6]:={u(x,t)=-(_{C}3-4*_{C}3*_{C}2^{2}+2*kappa*_{C}2)/(_{C}2*(3+4*_{C}2^{2}))+24*_{C}2*(2*_{C}3+kappa*_{C}2)*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/(16*_{C}2^{4}-9)}}$

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  Camassa Holm equation traveling wave sech plot5
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  Camassa Holm equation traveling wave sech plot4
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  Camassa Holm equation traveling wave sech plot6

exp 展開

${\displaystyle g[2]:={u(x,t)=-(1/9)*sqrt(3)*_{C}3-(1/3)*kappa+_{C}5*exp(_{C}1-sqrt(3)*x+_{C}3*t)}}$ ${\displaystyle g[3]:={u(x,t)=(1/9)*sqrt(3)*_{C}3-(1/3)*kappa+_{C}5*exp(_{C}1+sqrt(3)*x+_{C}3*t)}}$ ${\displaystyle g[5]:={u(x,t)=-(1/3)*sqrt(3)*_{C}3-(1/3)*kappa+_{C}7*(exp(_{C}1-(1/3)*sqrt(3)*x+_{C}3*t))^{3}}}$

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  Camassa Holm equation traveling wave exp plot2
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  Camassa Holm equation traveling wave exp plot3
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  Camassa Holm equation traveling wave exp plot5

csch 展開

${\displaystyle g[2]:={u(x,t)=-(1/2)*kappa+_{C}5*csch(_{C}1+(1/2)*sqrt(3)*x-(1/4)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[4]:={u(x,t)=_{C}4+((3/2)*_{C}4+(3/4)*kappa)*csch(_{C}1+(1/2*I)*sqrt(3)*x-(1/4*I)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[6]:={u(x,t)=-(_{C}3-4*_{C}3*_{C}2^{2}+2*kappa*_{C}2)/(_{C}2*(4*_{C}2^{2}+3))-24*_{C}2*(2*_{C}3+kappa*_{C}2)*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/(16*_{C}2^{4}-9)}}$

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  Camassa Holm equation traveling wave csch plot2
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  Camassa Holm equation traveling wave csch plot4
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  Camassa Holm equation traveling wave csch plot6

sec 展開

${\displaystyle g[3]:={u(x,t)=-(1/2)*kappa+_{C}5*sec(_{C}1-(1/2*I)*sqrt(3)*x+(1/4*I)*kappa*sqrt(3)*t)^{2}}}$ ${\displaystyle g[5]:={u(x,t)=_{C}4+(-(3/2)*_{C}4-(3/4)*kappa)*sec(_{C}1-(1/2)*sqrt(3)*x+(1/4)*kappa*sqrt(3)*t)^{2}}}$

${\displaystyle g[6]:={u(x,t)=(_{C}3+4*_{C}3*_{C}2^{2}+2*kappa*_{C}2)/(_{C}2*(4*_{C}2^{2}-3))-24*_{C}2*(2*_{C}3+kappa*_{C}2)*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/(16*_{C}2^{4}-9)}}$

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  Camassa Holm equation traveling wave sec plot3
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  Camassa Holm equation traveling wave sec plot5

JacobiSN 展開

${\displaystyle g[3]:={u(x,t)=(1/9*I)*sqrt(3)*_{C}4-(1/3)*kappa+_{C}6*sin(_{C}2-I*sqrt(3)*x+_{C}4*t)}}$ ${\displaystyle g[4]:={u(x,t)=-(2/9*I)*sqrt(3)*_{C}4-(1/2)*_{C}7-(1/3)*kappa+_{C}7*sin(_{C}2+(1/2*I)*sqrt(3)*x+_{C}4*t)^{2}}}$

Camassa Holm equation traveling wave Jacobish plot4

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