耦合KdV方程组

耦合KdV方程组(Coupled KdV equation)是一个二元非线性偏微分方程组： u t + 6 ∗ α ∗ u ∗ u x − 6 ∗ v ∗ v x + α ∗ u x x x = 0 {\displaystyle u_{t}+6*\alpha *u*u_{x}-6*v*v_{x}+\alpha *u_{xxx}=0} v t + 3 ∗ α ∗ u ∗ v x + α ∗ v x x x = 0 {\displaystyle v_{t}+3*\alpha *u*v_{x}+\alpha *v_{xxx}=0} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 3 + α ∗ C 2 3 ) / ( α ∗ C 2 ) − 2 ∗ C 2 2 ∗ c s c ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 12 ∗ α ∗ C 2 4 + 6 ∗ C 2 ∗ C 3 ) ∗ c s c ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(-_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}2^{4}+6*_{C}2*_{C}3)*csc(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 3 + α ∗ C 2 3 ) / ( α ∗ C 2 ) − 2 ∗ C 2 2 ∗ s e c ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 12 ∗ α ∗ C 2 4 + 6 ∗ C 2 ∗ C 3 ) ∗ s e c ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(-_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}2^{4}+6*_{C}2*_{C}3)*sec(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 3 + 2 ∗ α ∗ C 2 3 ) / ( α ∗ C 2 ) − 2 ∗ C 2 2 ∗ c o t h ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 6 ∗ C 2 ∗ C 3 + 24 ∗ α ∗ C 2 4 ) ∗ c o t h ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(-_{C}3+2*\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3+24*\alpha *_{C}2^{4})*coth(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 3 + 2 ∗ α ∗ C 2 3 ) / ( a l p h a ∗ C 2 ) − 2 ∗ C 2 2 ∗ t a n h ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 6 ∗ C 2 ∗ C 3 + 24 ∗ a l p h a ∗ C 2 4 ) ∗ t a n h ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(-_{C}3+2*\alpha *_{C}2^{3})/(alpha*_{C}2)-2*_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3+24*alpha*_{C}2^{4})*tanh(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = − ( 1 / 3 ) ∗ ( C 3 + α ∗ C 2 3 ) / ( α ∗ C 2 ) − 2 ∗ C 2 2 ∗ c s c h ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 6 ∗ C 2 ∗ C 3 − 12 ∗ a l p h a ∗ C 2 4 ) ∗ c s c h ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=-(1/3)*(_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3-12*alpha*_{C}2^{4})*csch(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = − ( 1 / 3 ) ∗ ( C 3 + 8 ∗ α ∗ C 2 3 ) / ( α ∗ C 2 ) − 4 ∗ C 2 2 ∗ t a n ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 , v ( x , t ) = − ( 1 / 6 ) ∗ ( C 3 + 8 ∗ α ∗ C 2 3 ) ∗ ( 2 ) / ( C 2 ∗ ( α ) ) − 2 ∗ ( 2 ) ∗ ( α ) ∗ C 2 2 ∗ t a n ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 {\displaystyle u(x,t)=-(1/3)*(_{C}3+8*\alpha *_{C}2^{3})/(\alpha *_{C}2)-4*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/6)*(_{C}3+8*\alpha *_{C}2^{3})*{\sqrt {(}}2)/(_{C}2*{\sqrt {(}}\alpha ))-2*{\sqrt {(}}2)*{\sqrt {(}}\alpha )*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2}} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 4 + α ∗ C 3 3 + α ∗ C 3 3 ∗ C 1 2 ) / ( α ∗ C 3 ) − 2 ∗ C 3 2 ∗ J a c o b i N S ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 12 ∗ α ∗ C 3 4 ∗ C 1 2 + 12 ∗ α ∗ C 3 4 + 6 ∗ C 3 ∗ C 4 ) ∗ J a c o b i N S ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) {\displaystyle u(x,t)=(1/3)*(-_{C}4+\alpha *_{C}3^{3}+\alpha *_{C}3^{3}*_{C}1^{2})/(\alpha *_{C}3)-2*_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}3^{4}*_{C}1^{2}+12*\alpha *_{C}3^{4}+6*_{C}3*_{C}4)*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)} u ( x , t ) = ( 1 / 3 ) ∗ ( − C 4 + α ∗ C 3 3 + α ∗ C 3 3 ∗ C 1 2 ) / ( α ∗ C 3 ) − 2 ∗ C 3 2 ∗ C 1 2 ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 , v ( x , t ) = − ( 1 / 3 ) ∗ ( 12 ∗ α ∗ C 3 4 ∗ C 1 2 + 12 ∗ α ∗ C 3 4 + 6 ∗ C 3 ∗ C 4 ) ∗ C 1 ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) {\displaystyle u(x,t)=(1/3)*(-_{C}4+\alpha *_{C}3^{3}+\alpha *_{C}3^{3}*_{C}1^{2})/(\alpha *_{C}3)-2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}3^{4}*_{C}1^{2}+12*\alpha *_{C}3^{4}+6*_{C}3*_{C}4)*_{C}1*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)} {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } 李志斌编著 《非线性数学物理方程的行波解》 38页 科学出版社 2008 *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. 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