广义伯格斯-KdV方程

广义伯格斯-KdV方程 (Generalized Burgers-KdV equation)是一个非线性偏微分方程： U [ t ] − α ∗ ∂ n u ( x , t ) ∂ x n − β ∗ u ( x , t ) ∗ ∂ u ( x , t ) ∂ x = 0 {\displaystyle U[t]-\alpha *{\frac {\partial ^{n}u(x,t)}{\partial x^{n}}}-\beta *u(x,t)*{\frac {\partial u(x,t)}{\partial x}}=0} 当 n=7， 有下列特解： u ( x , t ) = ( 71280 ∗ α ∗ C 4 7 ∗ C 1 + C 5 ) / ( β ∗ C 4 ) − 665280 ∗ α ∗ C 4 6 ∗ W e i e r s t r a s s P ( C 3 + C 4 ∗ x + C 5 ∗ t , 0 , C 1 ) 3 / β {\displaystyle u(x,t)=(71280*\alpha *_{C}4^{7}*_{C}1+_{C}5)/(\beta *_{C}4)-665280*\alpha *_{C}4^{6}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,0,_{C}1)^{3}/\beta } u ( x , t ) = ( C 4 − 42240 ∗ α ∗ C 3 7 + 84480 ∗ α ∗ C 3 7 ∗ ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) / ( β ∗ C 3 ) − 665280 ∗ α ∗ C 3 6 ∗ ( − 1 + ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 / β + 665280 ∗ α ∗ C 3 6 ∗ ( ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 − 2 ) ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 4 / β + 665280 ∗ α ∗ C 3 6 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 6 / β {\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta } u ( x , t ) = ( C 4 − 42240 ∗ α ∗ C 3 7 + 84480 ∗ α ∗ C 3 7 ∗ ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) / ( β ∗ C 3 ) − 665280 ∗ α ∗ C 3 6 ∗ ( − 1 + ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 / β + 665280 ∗ α ∗ C 3 6 ∗ ( ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 − 2 ) ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 4 / β + 665280 ∗ α ∗ C 3 6 ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 6 / β {\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta } u ( x , t ) = ( C 4 − 42240 ∗ α ∗ C 3 7 + 84480 ∗ α ∗ C 3 7 ∗ ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) / ( β ∗ C 3 ) − 665280 ∗ C 3 6 ∗ α ∗ ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 / β + 665280 ∗ α ∗ C 3 6 ∗ ( 1 + ( − ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 ) ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 4 / β − 665280 ∗ α ∗ C 3 6 ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 6 / β {\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta } u ( x , t ) = ( C 4 − 42240 ∗ α ∗ C 3 7 + 84480 ∗ α ∗ C 3 7 ∗ ( − ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 ) / ( β ∗ C 3 ) − 665280 ∗ C 3 6 ∗ α ∗ ( − ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 2 / β + 665280 ∗ α ∗ C 3 6 ∗ ( 1 + ( − ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 ) ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 4 / β − 665280 ∗ α ∗ C 3 6 ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) − 1 / 2 ∗ I ) 6 / β {\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{6}/\beta } u ( x , t ) = ( C 4 − 42240 ∗ α ∗ C 3 7 + 84480 ∗ α ∗ C 3 7 ∗ ( ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 ) / ( β ∗ C 3 ) − 665280 ∗ α ∗ C 3 6 ∗ ( − 1 + ( ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 / β + 665280 ∗ α ∗ C 3 6 ∗ ( ( ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 2 − 2 ) ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 4 / β + 665280 ∗ α ∗ C 3 6 ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , ( 1 / 2 ) ∗ ( 3 ) + 1 / 2 ∗ I ) 6 / β {\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*((1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+((1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(((1/2)*{\sqrt {(}}3)+1/2*I)^{2}-2)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta } 广义伯格斯-KdV方程之部分通解为： u ( x , t ) = C 1 ( n − 1 ) ∗ ( x + C 1 ) ∗ ( C 1 ∗ x + b ∗ C 1 ∗ C 2 ∗ t + C 3 ) / ( b ∗ ( C 2 + t ) ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2} u ( x , t ) = C 1 ( n − 1 ) ∗ ( x + C 1 ) ∗ ( C 1 ∗ x + b ∗ C 1 ∗ C 2 ∗ t + C 3 ) / ( b ∗ ( C 2 + t ) ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2} u ( x , t ) = C 1 ( n − 1 ) ∗ ( ( − 1 ) n ∗ a ∗ ( 2 ∗ n − 1 ) ! / ( b ∗ ( n − 1 ) ! ∗ ( x + b ∗ C 1 ∗ t + C 2 ) ( n − 1 ) ) + C 1 ) ∗ ( C 1 ∗ x + b ∗ C 1 ∗ C 2 ∗ t + C 3 ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2} u ( x , t ) = C 1 ( n − 1 ) ∗ ( ( − 1 ) n ∗ a ∗ ( 2 ∗ n − 1 ) ! / ( b ∗ ( n − 1 ) ! ∗ ( x + b ∗ C 1 ∗ t + C 2 ) ( n − 1 ) ) + C 1 ) ∗ ( C 1 ∗ x + b ∗ C 1 ∗ C 2 ∗ t + C 3 ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2} u ( x , t ) = C 1 ( n − 1 ) ∗ ( x + C 1 ) ∗ ( C 1 n ∗ t + C 4 ) / ( b ∗ ( C 2 + t ) ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2} u ( x , t ) = C 1 ( n − 1 ) ∗ ( x + C 1 ) ∗ ( C 1 n ∗ t + C 4 ) / ( b ∗ ( C 2 + t ) ) + C 2 {\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2} {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } {\displaystyle } Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS,（《非线性偏微分方程手册》） SECOND EDITION p1045 CRC PRESS *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. 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