库普-库珀施密特方程

库普-库珀施密特方程(Kaup-Kupershmidt Equation)是一个非线性偏微分方程： ∂ 4 u ( x , t ) ∂ x 4 + ∂ u ( x , t ) ∂ x + 45 ( ∂ u ( x , t ) ∂ x ∗ u ( x , t ) 2 − ( 75 / 2 ) ∗ ∂ 2 u ( x , t ) ∂ x 2 ∗ ∂ u ( x , t ) ∂ x − 15 ∗ u ( x , t ) ∗ ∂ 3 u ( x , t ) ∂ x 3 {\displaystyle {\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}}}} 利用Maple软件包TWSolution，随所选定展开函数不同，可得多种行波解 tanh 展开 g [ 2 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ t a n h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) ∗ x + C 3 ∗ t ) 2 {\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}}} g [ 3 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ t a n h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 4 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ t a n h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 5 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ t a n h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 6 ] := u ( x , t ) = − ( 4 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ t a n h ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) ∗ 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}}} g [ 7 ] := u ( x , t ) = − ( 4 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ t a n h ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 8 ] := u ( x , t ) = − ( 4 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ t a n h ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 9 ] := u ( x , t ) = − ( 4 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ t a n h ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} Kaup Kupershmidt eq tanh method animation2 Kaup Kupershmidt eq tanh method animation7 Kaup Kupershmidt eq tanh method animation8 JacobiSN 展开 g [ 2 ] := u ( x , t ) = − ( 1 / 2 ) ∗ C 3 2 − ( 1 / 6 ) ∗ s q r t ( − 3 ∗ C 3 4 − 4 ) + ( ( 1 / 2 ) ∗ C 3 2 + ( 1 / 2 ) ∗ s q r t ( − 3 ∗ C 3 4 − 4 ) ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ s q r t ( 2 ∗ C 3 2 + 2 ∗ s q r t ( − 3 ∗ C 3 4 − 4 ) ) / C 3 ) 2 {\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}}} g [ 3 ] := u ( x , t ) = − ( 1 / 2 ) ∗ C 3 2 + ( 1 / 6 ) ∗ s q r t ( − 3 ∗ C 3 4 − 4 ) + ( ( 1 / 2 ) ∗ C 3 2 − ( 1 / 2 ) ∗ s q r t ( − 3 ∗ C 3 4 − 4 ) ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ s q r t ( 2 ∗ C 3 2 − 2 ∗ s q r t ( − 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}}} g [ 4 ] := u ( x , t ) = − 4 ∗ C 3 2 − ( 2 / 33 ) ∗ s q r t ( − 1452 ∗ C 3 4 − 11 ) + ( 4 ∗ C 3 2 + ( 2 / 11 ) ∗ s q r t ( − 1452 ∗ C 3 4 − 11 ) ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 22 ) ∗ s q r t ( 242 ∗ C 3 2 + 11 ∗ s q r t ( − 1452 ∗ C 3 4 − 11 ) ) / 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}}} g [ 5 ] := u ( x , t ) = − 4 ∗ C 3 2 + ( 2 / 33 ) ∗ s q r t ( − 1452 ∗ C 3 4 − 11 ) + ( 4 ∗ C 3 2 − ( 2 / 11 ) ∗ s q r t ( − 1452 ∗ C 3 4 − 11 ) ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 22 ) ∗ s q r t ( 242 ∗ C 3 2 − 11 ∗ s q r t ( − 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}}} Kaup Kupershmidt JacobiSN method animation2 Kaup Kupershmidt JacobiSN method animation3 Kaup Kupershmidt JacobiSN method animation4 sech 展开 g [ 2 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 − ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) ∗ x + C 3 ∗ t ) 2 {\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}}} g [ 3 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 − ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 4 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 − ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 5 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 − ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 6 ] := u ( x , t ) = ( 2 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 − 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c h ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) ∗ 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}}} g [ 7 ] := u ( x , t ) = ( 2 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 − 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c h ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 8 ] := u ( x , t ) = ( 2 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 − 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c h ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 9 ] := u ( x , t ) = ( 2 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 − 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c h ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} Kaup Kupershmidt sech method animation2 Kaup Kupershmidt sech method animation4 Kaup Kupershmidt sech method animation8 sec、coth 展开 g [ 2 ] := u ( x , t ) = − ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) ∗ x + C 3 ∗ t ) 2 {\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}}} g [ 3 ] := u ( x , t ) = − ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 4 ] := u ( x , t ) = − ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 5 ] := u ( x , t ) = − ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ s e c ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 6 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) ∗ 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}}} g [ 7 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( − ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c ( C 1 + ( − ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 8 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) − ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 9 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 + 2 ∗ ( ( 1 / 22 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 22 ∗ I ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) ) 2 ∗ s e c ( C 1 + ( ( 1 / 44 ) ∗ s q r t ( 2 ) ∗ 11 ( 3 / 4 ) + ( 1 / 44 ∗ I ) ∗ s q r t ( 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}}} g [ 10 ] := u ( x , t ) = − ( 2 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ c o t h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) ∗ 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}}} Kaup Kupershmidt sec method animation3 Kaup Kupershmidt sec method animation5 Kaup Kupershmidt sec method animation10 csch 展开 u ( x , t ) = C 4 {\displaystyle {u(x,t)=_{C}4}} g [ 2 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ c s c h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) ∗ x + C 3 ∗ t ) 2 {\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}}} g [ 3 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ c s c h ( C 1 + ( − ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 4 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ c s c h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) − ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} g [ 5 ] := u ( x , t ) = ( 1 / 3 ) ∗ ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 2 ) ) 2 ∗ c s c h ( C 1 + ( ( 1 / 2 ) ∗ s q r t ( 2 ) + ( 1 / 2 ∗ I ) ∗ s q r t ( 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}}} Kaup Kupershmidt csch method animation2 Kaup Kupershmidt csch method animation3 Kaup Kupershmidt csch method animation5 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) Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27 *谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社 *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年 李志斌编著 《非线性数学物理方程的行波解》 科学出版社 王东明著 《消去法及其应用》 科学出版社 2002 *何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445 Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000 Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 Dongming Wang, Elimination Practice,Imperial College Press 2004 David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004 George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759