卡马萨-霍尔姆方程

卡马萨-霍尔姆方程(Camassa Holm equation)是流体力学中的一个非线性偏微分方程

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

1993年卡马萨和霍尔姆以此偏微分方程模拟浅水波^[1]，

其中κ是大于0的参数。

行波解

 

卡马萨-霍尔姆方程3D动画

卡马萨-霍尔姆方程有行波解^[2]：

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

参数：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软件包TWSolution可提供多种行波解^[3]。

sech 展开

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

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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 展开

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

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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 展开

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

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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 展开

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

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

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  Camassa Holm equation traveling wave sec plot3

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  Camassa Holm equation traveling wave sec plot5

JacobiSN 展开

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

 

Camassa Holm equation traveling wave Jacobish plot4

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参考文献

1. Camassa & Holm 1993
2. Beals, Sattinger & Szmigielski 1999
3. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27-35

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• Constantin, Adrian, On the scattering problem for the Camassa–Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2008), 2001, 457 (2008): 953–970, Bibcode:2001RSPSA.457..953C, doi:10.1098/rspa.2000.0701 
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• Eckhardt, Jonathan; Teschl, Gerald, On the isospectral problem of the dispersionless Camassa-Holm equation, Adv. Math. 235 (1), 2013, 235 (1): 469–495, arXiv:1205.5831  , doi:10.1016/j.aim.2012.12.006 
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