库普-库珀施密特方程

利用Maple软件包TWSolution，随所选定展开函数不同，可得多种行波解[2]

tanh 展开

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

JacobiSN 展开

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

sech 展开

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

sec、coth 展开

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

csch 展开

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

1. Qinghua Feng New Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation,2012 International Conference on Computer Technology and Science (ICCTS 2012) IPCSIT vol. 47 (2012)
2. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27

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