Karachawa方程

Karachawa方程是一个模拟有表面张力的水波运动的非线性偏微分方程： u t + μ ∗ u x x x + 2 ∗ β ∗ u x x x x x = 0 {\displaystyle u_{t}+\mu *u_{xxx}+2*\beta *u_{xxxxx}=0} u ( x , t ) = ( 1 / 676 ) ∗ ( − 338 ∗ ( − 13 ∗ μ ) ∗ C 3 − 69 ∗ μ 3 ) / μ + ( 105 / 338 ) ∗ μ 2 ∗ t a n h ( C 1 − ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ t a n h ( C 1 − ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=(1/676)*(-338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*tanh(_{C}1-(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tanh(_{C}1-(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}} u ( x , t ) = ( 1 / 676 ) ∗ ( 338 ∗ ( − 13 ∗ μ ) ∗ C 3 − 69 ∗ μ 3 ) / μ + ( 105 / 338 ) ∗ μ 2 ∗ c o t h ( C 1 + ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ c o t h ( C 1 + ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=(1/676)*(338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*coth(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*coth(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}} u ( x , t ) = ( 1 / 676 ) ∗ ( 338 ∗ ( − 13 ∗ μ ) ∗ C 3 − 69 ∗ μ 3 ) / μ + ( 105 / 338 ) ∗ μ 2 ∗ t a n h ( C 1 + ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ t a n h ( C 1 + ( 1 / 26 ) ∗ ( − 13 ∗ μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=(1/676)*(338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*tanh(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tanh(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}} u ( x , t ) = − ( 1 / 676 ) ∗ ( − 338 ∗ ( 13 ) ∗ ( μ ) ∗ C 3 + 69 ∗ μ 3 ) / μ − ( 105 / 338 ) ∗ μ 2 ∗ c o t ( C 1 − ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ c o t ( C 1 − ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=-(1/676)*(-338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*cot(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*cot(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}} u ( x , t ) = − ( 1 / 676 ) ∗ ( − 338 ∗ ( 13 ) ∗ ( μ ) ∗ C 3 + 69 ∗ μ 3 ) / μ − ( 105 / 338 ) ∗ μ 2 ∗ t a n ( C 1 − ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ t a n ( C 1 − ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=-(1/676)*(-338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*tan(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tan(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}} u ( x , t ) = − ( 1 / 676 ) ∗ ( 338 ∗ ( 13 ) ∗ ( μ ) ∗ C 3 + 69 ∗ μ 3 ) / μ − ( 105 / 338 ) ∗ μ 2 ∗ t a n ( C 1 + ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 2 − ( 105 / 676 ) ∗ μ 2 ∗ t a n ( C 1 + ( 1 / 26 ) ∗ ( 13 ) ∗ ( μ ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=-(1/676)*(338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*tan(_{C}1+(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tan(_{C}1+(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}} u ( x , t ) = ( 1 / 209560 ) ∗ ( − ( 13 / 5 ) ∗ ( 2015 ∗ μ − ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ( 3 / 2 ) ∗ C 3 − ( 4991 / 1300 ∗ ( 2015 ∗ μ − ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ) ∗ μ 3 + 10478 ∗ ( 2015 ∗ μ − ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ C 3 ∗ μ + 961 ∗ μ 4 ) / μ 2 + ( 7 / 676 ) ∗ μ ∗ ( ( 651 / 20 ) ∗ μ − ( 123 / 20 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ s e c h ( C 1 − ( 1 / 260 ) ∗ ( 2015 ∗ μ − ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ x + C 3 ∗ t ) 2 − ( 651 / 1352 ) ∗ μ ∗ ( ( 11 / 20 ) ∗ μ − ( 3 / 20 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ s e c h ( C 1 − ( 1 / 260 ) ∗ ( 2015 ∗ μ − ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=(1/209560)*(-(13/5)*(2015*\mu -(195*I)*\mu *{\sqrt {(}}31))^{(}3/2)*_{C}3-(4991/1300*(2015*\mu -(195*I)*\mu *{\sqrt {(}}31)))*\mu ^{3}+10478*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*_{C}3*\mu +961*\mu ^{4})/\mu ^{2}+(7/676)*\mu *((651/20)*\mu -(123/20*I)*\mu *{\sqrt {(}}31))*sech(_{C}1-(1/260)*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{2}-(651/1352)*\mu *((11/20)*\mu -(3/20*I)*\mu *{\sqrt {(}}31))*sech(_{C}1-(1/260)*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{4}} u ( x , t ) = − ( 1 / 523900 ) ∗ ( − ( 13 / 2 ) ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ( 3 / 2 ) ∗ C 3 + ( 23529 / 650 ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ) ∗ μ 3 − 26195 ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ C 3 ∗ μ − 175863 ∗ μ 4 ) / ( μ ∗ ( ( 21 / 10 ) ∗ μ + ( 3 / 10 ∗ I ) ∗ μ ∗ ( 31 ) ) ) + ( − ( 217 / 338 ) ∗ μ 2 + ( 7 / 16900 ) ∗ μ ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ) ∗ c o t h ( C 1 − ( 1 / 260 ) ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ x + C 3 ∗ t ) 2 − ( 651 / 1352 ) ∗ μ ∗ ( ( 11 / 20 ) ∗ μ + ( 3 / 20 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ c o t h ( C 1 − ( 1 / 260 ) ∗ ( 2015 ∗ μ + ( 195 ∗ I ) ∗ μ ∗ ( 31 ) ) ∗ x + C 3 ∗ t ) 4 {\displaystyle u(x,t)=-(1/523900)*(-(13/2)*(2015*\mu +(195*I)*\mu *{\sqrt {(}}31))^{(}3/2)*_{C}3+(23529/650*(2015*\mu +(195*I)*\mu *{\sqrt {(}}31)))*\mu ^{3}-26195*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*_{C}3*\mu -175863*\mu ^{4})/(\mu *((21/10)*\mu +(3/10*I)*\mu *{\sqrt {(}}31)))+(-(217/338)*\mu ^{2}+(7/16900)*\mu *(2015*\mu +(195*I)*\mu *{\sqrt {(}}31)))*coth(_{C}1-(1/260)*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{2}-(651/1352)*\mu *((11/20)*\mu +(3/20*I)*\mu *{\sqrt {(}}31))*coth(_{C}1-(1/260)*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{4}} p [ 46 ] := 8.0741721983973002048 ∗ 10 5 + 11257.587038449976187 ∗ I + ( 1813.0402209066405653 − 19.865040422291120617 ∗ I ) ∗ J a c o b i N S ( 1.5250 + 1.7351587051052163701 ∗ x 1 .25 + 1.9035752853902350521 ∗ t 1 .25 , 0.21767841032926169436 e − 1 − 0.16688086862630055943 e − 1 ∗ I ) 1 .5 − 1199.5620 ∗ J a c o b i N S ( 1.5250 + 1.7351587051052163701 ∗ x 1 .25 + 1.9035752853902350521 ∗ t 1 .25 , 0.21767841032926169436 e − 1 − 0.16688086862630055943 e − 1 ∗ I ) 4 {\displaystyle p[46]:=8.0741721983973002048*10^{5}+11257.587038449976187*I+(1813.0402209066405653-19.865040422291120617*I)*JacobiNS(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{1}.5-1199.5620*JacobiNS(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{4}} p [ 47 ] := 8.0741721983973002048 ∗ 10 5 + 11257.587038449976187 ∗ I + ( 15.667532401561428902 − 15.998034611025966429 ∗ I ) ∗ J a c o b i S N ( 1.5250 + 1.7351587051052163701 ∗ x 1 .25 + 1.9035752853902350521 ∗ t 1 .25 , 0.21767841032926169436 e − 1 − 0.16688086862630055943 e − 1 ∗ I ) 1 .5 + ( 0.48758338627653809364 e − 2 + 0.84722715715396426655 e − 2 ∗ I ) ∗ J a c o b i S N ( 1.5250 + 1.7351587051052163701 ∗ x 1 .25 + 1.9035752853902350521 ∗ t 1 .25 , 0.21767841032926169436 e − 1 − 0.16688086862630055943 e − 1 ∗ I ) 4 {\displaystyle p[47]:=8.0741721983973002048*10^{5}+11257.587038449976187*I+(15.667532401561428902-15.998034611025966429*I)*JacobiSN(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{1}.5+(0.48758338627653809364e-2+0.84722715715396426655e-2*I)*JacobiSN(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{4}} 李志斌编著 《非线性数学物理方程的行波解》 127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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