广田-萨摩方程组

广田-萨摩方程(Hirota Satsuma equation)是一个三元非线性偏微分方程组： u t − 0.5 ∗ u x x x + 3 u u x − 3 ( v w ) x = 0 {\displaystyle u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0} v t + v x x x − 3 u v x = 0 {\displaystyle v_{t}+v_{xxx}-3uv_{x}=0} w t + w x x x − 3 u w x = 0 {\displaystyle w_{t}+w_{xxx}-3uw_{x}=0} u ( x , t ) = − ( 1 / 3 ) ∗ ( − C 3 + C 2 3 ) / C 2 , v ( x , t ) = 0 , w ( x , t ) = C 7 + C 8 ∗ c o s ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cos(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = − ( 1 / 3 ) ∗ ( − C 3 + C 2 3 ) / C 2 , v ( x , t ) = 0 , w ( x , t ) = C 7 + C 8 ∗ s i n ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sin(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = ( 1 / 3 ) ∗ ( C 3 + C 2 3 ) / C 2 , v ( x , t ) = 0 , w ( x , t ) = C 7 + C 8 ∗ c o s h ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cosh(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = ( 1 / 3 ) ∗ ( C 3 + C 2 3 ) / C 2 , v ( x , t ) = 0 , w ( x , t ) = C 7 + C 8 ∗ s i n h ( C 1 + C 2 ∗ x + C 3 ∗ t ) {\displaystyle u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sinh(_{C}1+_{C}2*x+_{C}3*t)} u ( x , t ) = − ( 1 / 3 ) ∗ ( 9 ∗ C 3 3 − C 4 ) / C 3 , v ( x , t ) = C 6 − ( 3 / 4 ) ∗ C 9 ∗ c o s ( C 2 + C 3 ∗ x + C 4 ∗ t ) + C 9 ∗ c o s ( C 2 + C 3 ∗ x + C 4 ∗ t ) 3 , w ( x , t ) = 0 {\displaystyle u(x,t)=-(1/3)*(9*_{C}3^{3}-_{C}4)/_{C}3,v(x,t)=_{C}6-(3/4)*_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)+_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)^{3},w(x,t)=0} u ( x , t ) = − C 2 2 + 2 ∗ C 2 2 ∗ c o t h ( C 1 + C 2 ∗ x − C 2 3 ∗ t ) 2 , v ( x , t ) = C 5 + C 6 ∗ c o t h ( C 1 + C 2 ∗ x − C 2 3 ∗ t ) , w ( x , t ) = 0 {\displaystyle u(x,t)=-_{C}2^{2}+2*_{C}2^{2}*coth(_{C}1+_{C}2*x-_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*coth(_{C}1+_{C}2*x-_{C}2^{3}*t),w(x,t)=0} u ( x , t ) = − ( 1 / 2 ) ∗ C 2 2 + 2 ∗ C 2 2 ∗ c s c ( C 1 + C 2 ∗ x − ( 1 / 2 ) ∗ C 2 3 ∗ t ) 2 , v ( x , t ) = C 5 + C 6 ∗ c s c ( C 1 + C 2 ∗ x − ( 1 / 2 ) ∗ C 2 3 ∗ t ) , w ( x , t ) = 0 {\displaystyle {u(x,t)=-(1/2)*_{C}2^{2}+2*_{C}2^{2}*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t),w(x,t)=0}} u ( x , t ) = ( 1 / 2 ) ∗ C 2 2 + 2 ∗ C 2 2 ∗ c s c h ( C 1 + C 2 ∗ x + ( 1 / 2 ) ∗ C 2 3 ∗ t ) 2 , v ( x , t ) = C 5 + C 6 ∗ c s c h ( C 1 + C 2 ∗ x + ( 1 / 2 ) ∗ C 2 3 ∗ t ) , w ( x , t ) = 0 {\displaystyle {u(x,t)=(1/2)*_{C}2^{2}+2*_{C}2^{2}*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t),w(x,t)=0}} u ( x , t ) = − ( 1 / 3 ) ∗ ( − C 4 − 2 ∗ C 3 3 + C 3 3 ∗ C 1 2 ) / C 3 − 2 ∗ C 3 2 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 , v ( x , t ) = ( 2 / 3 ) ∗ C 3 ∗ ( 2 ∗ C 4 − 2 ∗ C 3 3 + C 3 3 ∗ C 1 2 ) ∗ C 8 / C 9 2 − ( 2 / 3 ) ∗ C 3 ∗ ( 2 ∗ C 4 − 2 ∗ C 3 3 + C 3 3 ∗ C 1 2 ) ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) / C 9 , w ( x , t ) = C 8 + C 9 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) {\displaystyle u(x,t)=-(1/3)*(-_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})/_{C}3-2*_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*_{C}8/_{C}9^{2}-(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}9,w(x,t)=_{C}8+_{C}9*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)} u ( x , t ) = − ( 1 / 3 ) ∗ ( − C 4 + C 3 3 ∗ C 1 2 + C 3 3 ) / 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 ) = − ( 2 / 3 ) ∗ C 1 2 ∗ C 3 ∗ ( C 3 3 ∗ C 1 2 + C 3 3 + 2 ∗ C 4 ) ∗ C 10 / C 11 2 + ( 2 / 3 ) ∗ C 3 ∗ C 1 2 ∗ ( C 3 3 ∗ C 1 2 + C 3 3 + 2 ∗ C 4 ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) / C 11 , w ( x , t ) = C 10 + C 11 ∗ 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+_{C}3^{3}*_{C}1^{2}+_{C}3^{3})/_{C}3+2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(2/3)*_{C}1^{2}*_{C}3*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*_{C}10/_{C}11^{2}+(2/3)*_{C}3*_{C}1^{2}*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}11,w(x,t)=_{C}10+_{C}11*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)} u ( x , t ) = ( 1 / 3 ) ∗ ( − 2 ∗ C 3 3 + 4 ∗ C 3 3 ∗ C 1 2 − C 4 ) / C 3 + ( 2 ∗ C 3 2 − 2 ∗ C 3 2 ∗ C 1 2 ) ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 , v ( x , t ) = C 6 , w ( x , t ) = C 10 {\displaystyle u(x,t)=(1/3)*(-2*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+(2*_{C}3^{2}-2*_{C}3^{2}*_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=_{C}6,w(x,t)=_{C}10} u ( x , t ) = − ( 1 / 2 ) ∗ C 3 2 + C 3 2 ∗ C 1 2 − 2 ∗ C 3 2 ∗ C 1 2 ∗ J a c o b i C N ( C 2 + C 3 ∗ x + ( − ( 1 / 2 ) ∗ C 3 3 + C 3 3 ∗ C 1 2 ) ∗ t , C 1 ) 2 , v ( x , t ) = C 6 + C 7 ∗ J a c o b i C N ( C 2 + C 3 ∗ x + ( − ( 1 / 2 ) ∗ C 3 3 + C 3 3 ∗ C 1 2 ) ∗ t , C 1 ) , w ( x , t ) = 0 {\displaystyle u(x,t)=-(1/2)*_{C}3^{2}+_{C}3^{2}*_{C}1^{2}-2*_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1)^{2},v(x,t)=_{C}6+_{C}7*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1),w(x,t)=0} {\displaystyle } {\displaystyle } {\displaystyle } 李志斌编著 《非线性数学物理方程的行波解》 140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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