庫普-庫珀施密特方程

庫普-庫珀施密特方程(Kaup-Kupershmidt Equation)是一個非線性偏微分方程：

$$\frac{\partial^4 u(x,t)}{\partial x^4} +\frac{\partial u(x,t)}{\partial x} +45(\frac{\partial u(x,t)}{\partial x}*u(x,t)^2 -(75/2)*\frac{\partial^2 u(x,t)}{\partial x^2}*\frac{\partial u(x,t)}{\partial x} -15*u(x,t)*\frac{\partial^3 u(x,t)}{\partial x^3}$$

## 行波解

利用Maple軟體包TWSolution，隨所選定展開函數不同，可得多種行波解

tanh 展開

$$g[2] := {u(x, t) = -(2/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*tanh(_C1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[3] := {u(x, t) = -(2/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*tanh(_C1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[4] := {u(x, t) = -(2/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*tanh(_C1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[5] := {u(x, t) = -(2/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*tanh(_C1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[6] := {u(x, t) = -(4/3)*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2+2*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*tanh(_C1+(-(1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[7] := {u(x, t) = -(4/3)*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2+2*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*tanh(_C1+(-(1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[8] := {u(x, t) = -(4/3)*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2+2*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*tanh(_C1+((1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[9] := {u(x, t) = -(4/3)*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2+2*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*tanh(_C1+((1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$

JacobiSN 展開

$$g[2] := {u(x, t) = -(1/2)*_C3^2-(1/6)*sqrt(-3*_C3^4-4)+((1/2)*_C3^2+(1/2)*sqrt(-3*_C3^4-4))*JacobiSN(_C2+_C3*x+_C4*t, (1/2)*sqrt(2*_C3^2+2*sqrt(-3*_C3^4-4))/_C3)^2}$$ $$g[3] := {u(x, t) = -(1/2)*_C3^2+(1/6)*sqrt(-3*_C3^4-4)+((1/2)*_C3^2-(1/2)*sqrt(-3*_C3^4-4))*JacobiSN(_C2+_C3*x+_C4*t, (1/2)*sqrt(2*_C3^2-2*sqrt(-3*_C3^4-4))/_C3)^2}$$ $$g[4] := {u(x, t) = -4*_C3^2-(2/33)*sqrt(-1452*_C3^4-11)+(4*_C3^2+(2/11)*sqrt(-1452*_C3^4-11))*JacobiSN(_C2+_C3*x+_C4*t, (1/22)*sqrt(242*_C3^2+11*sqrt(-1452*_C3^4-11))/_C3)^2}$$ $$g[5] := {u(x, t) = -4*_C3^2+(2/33)*sqrt(-1452*_C3^4-11)+(4*_C3^2-(2/11)*sqrt(-1452*_C3^4-11))*JacobiSN(_C2+_C3*x+_C4*t, (1/22)*sqrt(242*_C3^2-11*sqrt(-1452*_C3^4-11))/_C3)^2}$$

sech 展開

$$g[2] := {u(x, t) = (1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2-(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*sech(_C1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[3] := {u(x, t) = (1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2-(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*sech(_C1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[4] := {u(x, t) = (1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2-((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*sech(_C1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[5] := {u(x, t) = (1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2-((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*sech(_C1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[6] := {u(x, t) = (2/3)*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2-2*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*sech(_C1+(-(1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[7] := {u(x, t) = (2/3)*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2-2*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*sech(_C1+(-(1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[8] := {u(x, t) = (2/3)*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2-2*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*sech(_C1+((1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[9] := {u(x, t) = (2/3)*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2-2*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*sech(_C1+((1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$

sec、coth 展開

$$g[2] := {u(x, t) = -(1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*sec(_C1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[3] := {u(x, t) = -(1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*sec(_C1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[4] := {u(x, t) = -(1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*sec(_C1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[5] := {u(x, t) = -(1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*sec(_C1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[6] := {u(x, t) = -(2/3)*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2+2*(-(1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*sec(_C1+(-(1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[7] := {u(x, t) = -(2/3)*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2+2*(-(1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*sec(_C1+(-(1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[8] := {u(x, t) = -(2/3)*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2+2*((1/22)*sqrt(2)*11^(3/4)-(1/22*I)*sqrt(2)*11^(3/4))^2*sec(_C1+((1/44)*sqrt(2)*11^(3/4)-(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[9] := {u(x, t) = -(2/3)*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2+2*((1/22)*sqrt(2)*11^(3/4)+(1/22*I)*sqrt(2)*11^(3/4))^2*sec(_C1+((1/44)*sqrt(2)*11^(3/4)+(1/44*I)*sqrt(2)*11^(3/4))*x+_C3*t)^2}$$ $$g[10] := {u(x, t) = -(2/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*coth(_C1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$

csch 展開

$${u(x, t) = _C4}$$ $$g[2] := {u(x, t) = (1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*csch(_C1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[3] := {u(x, t) = (1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*csch(_C1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[4] := {u(x, t) = (1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^2*csch(_C1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_C3*t)^2}$$ $$g[5] := {u(x, t) = (1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^2*csch(_C1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_C3*t)^2}$$

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