正则长波方程

正则长波方程(Regularized long wave equation)是一个非线性偏微分方程： u t + u x + α ∗ u ∗ u x − μ ∗ u t x x = 0 ; {\displaystyle u_{t}+u_{x}+\alpha *u*u_{x}-\mu *u_{txx}=0;} 当 α=1，μ=1，正则长波方程即本杰明-博纳-马奥尼方程。 u ( x , t ) = − ( C 5 + C 4 ) / ( α ∗ C 4 ) + 12 ∗ μ ∗ C 5 ∗ C 4 ∗ W e i e r s t r a s s P ( C 3 + C 4 ∗ x + C 5 ∗ t , C 2 , C 1 ) / α {\displaystyle {u(x,t)=-(_{C}5+_{C}4)/(\alpha *_{C}4)+12*\mu *_{C}5*_{C}4*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,_{C}2,_{C}1)/\alpha }} u ( x , t ) = ( 4 ∗ μ ∗ C 3 ∗ C 2 2 − C 3 − C 2 ) / ( α ∗ C 2 ) + 12 ∗ μ ∗ C 3 ∗ C 2 ∗ c s c h ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 / α {\displaystyle {u(x,t)=(4*\mu *_{C}3*_{C}2^{2}-_{C}3-_{C}2)/(\alpha *_{C}2)+12*\mu *_{C}3*_{C}2*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }} u ( x , t ) = ( 4 ∗ μ ∗ C 3 ∗ C 2 2 − C 3 − C 2 ) / ( α ∗ C 2 ) − 12 ∗ μ ∗ C 3 ∗ C 2 ∗ s e c h ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 / α {\displaystyle {u(x,t)=(4*\mu *_{C}3*_{C}2^{2}-_{C}3-_{C}2)/(\alpha *_{C}2)-12*\mu *_{C}3*_{C}2*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }} u ( x , t ) = − ( 4 ∗ μ ∗ C 3 ∗ C 2 2 + C 3 + C 2 ) / ( α ∗ C 2 ) + 12 ∗ μ ∗ C 3 ∗ C 2 ∗ c s c ( C 1 + C 2 ∗ x + C 3 ∗ t ) 2 / α {\displaystyle {u(x,t)=-(4*\mu *_{C}3*_{C}2^{2}+_{C}3+_{C}2)/(\alpha *_{C}2)+12*\mu *_{C}3*_{C}2*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }} u ( x , t ) = ( − 4 ∗ μ ∗ C 4 ∗ C 3 2 − C 4 − C 3 + 8 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 ) / ( α ∗ C 3 ) − 12 ∗ μ ∗ C 4 ∗ C 3 ∗ C 1 2 ∗ J a c o b i C N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=(-4*\mu *_{C}4*_{C}3^{2}-_{C}4-_{C}3+8*\mu *_{C}4*_{C}3^{2}*_{C}1^{2})/(\alpha *_{C}3)-12*\mu *_{C}4*_{C}3*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} u ( x , t ) = ( − 4 ∗ μ ∗ C 4 ∗ C 3 2 − C 4 − C 3 + 8 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 ) / ( α ∗ C 3 ) − 12 ∗ μ ∗ C 4 ∗ C 3 ∗ ( − 1 + C 1 2 ) ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=(-4*\mu *_{C}4*_{C}3^{2}-_{C}4-_{C}3+8*\mu *_{C}4*_{C}3^{2}*_{C}1^{2})/(\alpha *_{C}3)-12*\mu *_{C}4*_{C}3*(-1+_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} u ( x , t ) = − ( 4 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 + C 4 + C 3 − 8 ∗ μ ∗ C 4 ∗ C 3 2 ) / ( α ∗ C 3 ) − 12 ∗ μ ∗ C 4 ∗ C 3 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=-(4*\mu *_{C}4*_{C}3^{2}*_{C}1^{2}+_{C}4+_{C}3-8*\mu *_{C}4*_{C}3^{2})/(\alpha *_{C}3)-12*\mu *_{C}4*_{C}3*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} u ( x , t ) = − ( 4 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 + C 4 + C 3 − 8 ∗ μ ∗ C 4 ∗ C 3 2 ) / ( α ∗ C 3 ) + 12 ∗ μ ∗ C 4 ∗ C 3 ∗ ( − 1 + C 1 2 ) ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=-(4*\mu *_{C}4*_{C}3^{2}*_{C}1^{2}+_{C}4+_{C}3-8*\mu *_{C}4*_{C}3^{2})/(\alpha *_{C}3)+12*\mu *_{C}4*_{C}3*(-1+_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} u ( x , t ) = − ( 4 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 + 4 ∗ μ ∗ C 4 ∗ C 3 2 + C 4 + C 3 ) / ( α ∗ C 3 ) + 12 ∗ μ ∗ C 4 ∗ C 3 ∗ J a c o b i N S ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=-(4*\mu *_{C}4*_{C}3^{2}*_{C}1^{2}+4*\mu *_{C}4*_{C}3^{2}+_{C}4+_{C}3)/(\alpha *_{C}3)+12*\mu *_{C}4*_{C}3*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} u ( x , t ) = − ( 4 ∗ μ ∗ C 4 ∗ C 3 2 ∗ C 1 2 + 4 ∗ μ ∗ C 4 ∗ C 3 2 + C 4 + C 3 ) / ( α ∗ C 3 ) + 12 ∗ μ ∗ C 4 ∗ C 3 ∗ C 1 2 ∗ J a c o b i S N ( C 2 + C 3 ∗ x + C 4 ∗ t , C 1 ) 2 / α {\displaystyle {u(x,t)=-(4*\mu *_{C}4*_{C}3^{2}*_{C}1^{2}+4*\mu *_{C}4*_{C}3^{2}+_{C}4+_{C}3)/(\alpha *_{C}3)+12*\mu *_{C}4*_{C}3*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }} Griffith, chapter 13, p239-260 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