Karachawa方程

Karachawa方程是一個模擬有表面張力的水波運動的非線性偏微分方程：

$$u_{t}+\mu*u_{xxx}+2*\beta*u_{xxxxx}=0$$

## 解析解

$$u(x, t) = (1/676)*(-338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*tanh(_C1-(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*tanh(_C1-(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/676)*(338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*coth(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*coth(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/676)*(338*\sqrt(-13*\mu)*_C3-69*\mu^3)/\mu+(105/338)*\mu^2*tanh(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^2-(105/676)*\mu^2*tanh(_C1+(1/26)*\sqrt(-13*\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(-338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*cot(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*cot(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(-338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*tan(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*tan(_C1-(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = -(1/676)*(338*\sqrt(13)*\sqrt(\mu)*_C3+69*\mu^3)/\mu-(105/338)*\mu^2*tan(_C1+(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^2-(105/676)*\mu^2*tan(_C1+(1/26)*\sqrt(13)*\sqrt(\mu)*x+_C3*t)^4$$

$$u(x, t) = (1/209560)*(-(13/5)*(2015*\mu-(195*I)*\mu*\sqrt(31))^(3/2)*_C3-(4991/1300*(2015*\mu-(195*I)*\mu*\sqrt(31)))*\mu^3+10478*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*_C3*\mu+961*\mu^4)/\mu^2+(7/676)*\mu*((651/20)*\mu-(123/20*I)*\mu*\sqrt(31))*sech(_C1-(1/260)*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*x+_C3*t)^2-(651/1352)*\mu*((11/20)*\mu-(3/20*I)*\mu*\sqrt(31))*sech(_C1-(1/260)*\sqrt(2015*\mu-(195*I)*\mu*\sqrt(31))*x+_C3*t)^4$$

$$u(x, t) = -(1/523900)*(-(13/2)*(2015*\mu+(195*I)*\mu*\sqrt(31))^(3/2)*_C3+(23529/650*(2015*\mu+(195*I)*\mu*\sqrt(31)))*\mu^3-26195*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*_C3*\mu-175863*\mu^4)/(\mu*((21/10)*\mu+(3/10*I)*\mu*\sqrt(31)))+(-(217/338)*\mu^2+(7/16900)*\mu*(2015*\mu+(195*I)*\mu*\sqrt(31)))*coth(_C1-(1/260)*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*x+_C3*t)^2-(651/1352)*\mu*((11/20)*\mu+(3/20*I)*\mu*\sqrt(31))*coth(_C1-(1/260)*\sqrt(2015*\mu+(195*I)*\mu*\sqrt(31))*x+_C3*t)^4$$

$$p[46] := 8.0741721983973002048*10^5+11257.587038449976187*I+(1813.0402209066405653-19.865040422291120617*I)*JacobiNS(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^1.5-1199.5620*JacobiNS(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^4$$

$$p[47] := 8.0741721983973002048*10^5+11257.587038449976187*I+(15.667532401561428902-15.998034611025966429*I)*JacobiSN(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^1.5+(0.48758338627653809364e-2+0.84722715715396426655e-2*I)*JacobiSN(1.5250+1.7351587051052163701*x^1.25+1.9035752853902350521*t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^4$$

## 參考文獻

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