布雷瑟顿方程

布雷瑟顿方程(Bretherton equation)是一个非线性偏微分方程： u t t + u x x + u x x x x − α ∗ u 3 = 0 {\displaystyle u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0} 利用Maple软件包TWSolution可得布雷瑟顿方程两个WeirstrassP函数行波解和六个Jacobi椭圆函数行波解。 u ( x , t ) = − ( 1 / 30 ) ∗ ( C 5 2 + C 4 2 ) ∗ ( 30 ) / ( ( α ) ∗ C 4 2 ) − 2 ∗ ( 30 ) ∗ C 4 2 ∗ W e i e r s t r a s s P ( C 3 + C 4 ∗ x + C 5 ∗ t , − ( 1 / 180 ) ∗ ( C 5 4 + 2 ∗ C 5 2 ∗ C 4 2 + C 4 4 ) / C 4 8 , ( 1 / 5400 ) ∗ ( C 5 2 + C 4 2 ) ∗ ( C 5 4 + 2 ∗ C 5 2 ∗ C 4 2 + C 4 4 ) / C 4 1 2 ) / ( α ) {\displaystyle {u(x,t)=-(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})-2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}} u ( x , t ) = ( 1 / 30 ) ∗ ( C 5 2 + C 4 2 ) ∗ ( 30 ) / ( ( α ) ∗ C 4 2 ) + 2 ∗ ( 30 ) ∗ C 4 2 ∗ W e i e r s t r a s s P ( C 3 + C 4 ∗ x + C 5 ∗ t , − ( 1 / 180 ) ∗ ( C 5 4 + 2 ∗ C 5 2 ∗ C 4 2 + C 4 4 ) / C 4 8 , ( 1 / 5400 ) ∗ ( C 5 2 + C 4 2 ) ∗ ( C 5 4 + 2 ∗ C 5 2 ∗ C 4 2 + C 4 4 ) / C 4 1 2 ) / ( α ) {\displaystyle {u(x,t)=(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})+2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}} p [ 3 ] := − 4.4229081351691113421 − 0.93307517430300270455 e − 1 ∗ I + ( 18.208764436791314548 + 0. ∗ I ) ∗ J a c o b i D N ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[3]:=-4.4229081351691113421-0.93307517430300270455e-1*I+(18.208764436791314548+0.*I)*JacobiDN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} p [ 4 ] := − 5.1919544739096094160 + 5.3436314406870407268 ∗ I + ( 18.208764436791314548 + 0. ∗ I ) ∗ J a c o b i N S ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[4]:=-5.1919544739096094160+5.3436314406870407268*I+(18.208764436791314548+0.*I)*JacobiNS(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} p [ 5 ] := − 2.5743416547838020433 − 3.2877514938561477464 ∗ I + ( 2.3878725879366620662 − 40.119130323671954306 ∗ I ) ∗ J a c o b i N D ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[5]:=-2.5743416547838020433-3.2877514938561477464*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiND(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} p [ 6 ] := − 5.5945307529851545322 + 1.6067031040792677714 ∗ I + ( 2.3878725879366620662 − 40.119130323671954306 ∗ I ) ∗ J a c o b i N C ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[6]:=-5.5945307529851545322+1.6067031040792677714*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiNC(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} p [ 7 ] := − 5.1805015655073574818 + 2.2007257608074972052 ∗ I + ( 21.217747598613725681 − 27.288660748456732356 ∗ I ) ∗ J a c o b i S N ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[7]:=-5.1805015655073574818+2.2007257608074972052*I+(21.217747598613725681-27.288660748456732356*I)*JacobiSN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} p [ 8 ] := − 5.7120768471542134149 − .77489365858670457610 ∗ I + ( 21.217747598613725681 − 27.288660748456732356 ∗ I ) ∗ J a c o b i C N ( 1.22 + 1.3 ∗ x + ( 15.095721921379631782 + 24.271454992000839308 ∗ I ) ∗ t , 1.1984050731412980386 − .42555954136146448708 ∗ I ) 1 .5 {\displaystyle p[8]:=-5.7120768471542134149-.77489365858670457610*I+(21.217747598613725681-27.288660748456732356*I)*JacobiCN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5} {\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》 152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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