Karachawa方程是一个模拟有表面张力的水波运动的非线性偏微分方程：

u_{t}+\muu_{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^2tanh(_C1-(1/26)\sqrt(-13\mu)x+_C3t)^2-(105/676)\mu^2tanh(_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^2coth(_C1+(1/26)\sqrt(-13\mu)x+_C3t)^2-(105/676)\mu^2coth(_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^2tanh(_C1+(1/26)\sqrt(-13\mu)x+_C3t)^2-(105/676)\mu^2tanh(_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^2cot(_C1-(1/26)\sqrt(13)\sqrt(\mu)x+_C3t)^2-(105/676)\mu^2cot(_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^2tan(_C1-(1/26)\sqrt(13)\sqrt(\mu)x+_C3t)^2-(105/676)\mu^2tan(_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^2tan(_C1+(1/26)\sqrt(13)\sqrt(\mu)x+_C3t)^2-(105/676)\mu^2tan(_C1+(1/26)\sqrt(13)\sqrt(\mu)x+_C3*t)^4
:u(x, t) = (1/209560)(-(13/5)(2015\mu-(195I)\mu\sqrt(31))^(3/2)_C3-(4991/1300(2015\mu-(195I)\mu\sqrt(31)))\mu^3+10478\sqrt(2015\mu-(195I)\mu\sqrt(31))_C3\mu+961\mu^4)/\mu^2+(7/676)\mu((651/20)\mu-(123/20I)\mu\sqrt(31))sech(_C1-(1/260)\sqrt(2015\mu-(195I)\mu\sqrt(31))x+_C3t)^2-(651/1352)\mu((11/20)\mu-(3/20I)\mu\sqrt(31))sech(_C1-(1/260)\sqrt(2015\mu-(195I)\mu\sqrt(31))x+_C3*t)^4
:u(x, t) = -(1/523900)(-(13/2)(2015\mu+(195I)\mu\sqrt(31))^(3/2)_C3+(23529/650(2015\mu+(195I)\mu\sqrt(31)))\mu^3-26195\sqrt(2015\mu+(195I)\mu\sqrt(31))_C3\mu-175863\mu^4)/(\mu((21/10)\mu+(3/10I)\mu\sqrt(31)))+(-(217/338)\mu^2+(7/16900)\mu(2015\mu+(195I)\mu\sqrt(31)))coth(_C1-(1/260)\sqrt(2015\mu+(195I)\mu\sqrt(31))x+_C3t)^2-(651/1352)\mu((11/20)\mu+(3/20I)\mu\sqrt(31))coth(_C1-(1/260)\sqrt(2015\mu+(195I)\mu\sqrt(31))x+_C3*t)^4
:p[46] := 8.074172198397300204810^5+11257.587038449976187I+(1813.0402209066405653-19.865040422291120617I)JacobiNS(1.5250+1.7351587051052163701x^1.25+1.9035752853902350521t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1I)^1.5-1199.5620JacobiNS(1.5250+1.7351587051052163701x^1.25+1.9035752853902350521t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1*I)^4
:p[47] := 8.074172198397300204810^5+11257.587038449976187I+(15.667532401561428902-15.998034611025966429I)JacobiSN(1.5250+1.7351587051052163701x^1.25+1.9035752853902350521t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1I)^1.5+(0.48758338627653809364e-2+0.84722715715396426655e-2I)JacobiSN(1.5250+1.7351587051052163701x^1.25+1.9035752853902350521t^1.25, 0.21767841032926169436e-1-0.16688086862630055943e-1I)^4

行波图
参考文献

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