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：^[1]

${\displaystyle u_{t}+\mu *u_{xxx}+2*\beta *u_{xxxxx}=0}$

解析解

${\displaystyle u(x,t)=(1/676)*(-338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*tanh(_{C}1-(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tanh(_{C}1-(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=(1/676)*(338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*coth(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*coth(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=(1/676)*(338*{\sqrt {(}}-13*\mu )*_{C}3-69*\mu ^{3})/\mu +(105/338)*\mu ^{2}*tanh(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tanh(_{C}1+(1/26)*{\sqrt {(}}-13*\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=-(1/676)*(-338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*cot(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*cot(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=-(1/676)*(-338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*tan(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tan(_{C}1-(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=-(1/676)*(338*{\sqrt {(}}13)*{\sqrt {(}}\mu )*_{C}3+69*\mu ^{3})/\mu -(105/338)*\mu ^{2}*tan(_{C}1+(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{2}-(105/676)*\mu ^{2}*tan(_{C}1+(1/26)*{\sqrt {(}}13)*{\sqrt {(}}\mu )*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=(1/209560)*(-(13/5)*(2015*\mu -(195*I)*\mu *{\sqrt {(}}31))^{(}3/2)*_{C}3-(4991/1300*(2015*\mu -(195*I)*\mu *{\sqrt {(}}31)))*\mu ^{3}+10478*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*_{C}3*\mu +961*\mu ^{4})/\mu ^{2}+(7/676)*\mu *((651/20)*\mu -(123/20*I)*\mu *{\sqrt {(}}31))*sech(_{C}1-(1/260)*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{2}-(651/1352)*\mu *((11/20)*\mu -(3/20*I)*\mu *{\sqrt {(}}31))*sech(_{C}1-(1/260)*{\sqrt {(}}2015*\mu -(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{4}}$

${\displaystyle u(x,t)=-(1/523900)*(-(13/2)*(2015*\mu +(195*I)*\mu *{\sqrt {(}}31))^{(}3/2)*_{C}3+(23529/650*(2015*\mu +(195*I)*\mu *{\sqrt {(}}31)))*\mu ^{3}-26195*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*_{C}3*\mu -175863*\mu ^{4})/(\mu *((21/10)*\mu +(3/10*I)*\mu *{\sqrt {(}}31)))+(-(217/338)*\mu ^{2}+(7/16900)*\mu *(2015*\mu +(195*I)*\mu *{\sqrt {(}}31)))*coth(_{C}1-(1/260)*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{2}-(651/1352)*\mu *((11/20)*\mu +(3/20*I)*\mu *{\sqrt {(}}31))*coth(_{C}1-(1/260)*{\sqrt {(}}2015*\mu +(195*I)*\mu *{\sqrt {(}}31))*x+_{C}3*t)^{4}}$

${\displaystyle p[46]:=8.0741721983973002048*10^{5}+11257.587038449976187*I+(1813.0402209066405653-19.865040422291120617*I)*JacobiNS(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{1}.5-1199.5620*JacobiNS(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{4}}$

${\displaystyle p[47]:=8.0741721983973002048*10^{5}+11257.587038449976187*I+(15.667532401561428902-15.998034611025966429*I)*JacobiSN(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{1}.5+(0.48758338627653809364e-2+0.84722715715396426655e-2*I)*JacobiSN(1.5250+1.7351587051052163701*x^{1}.25+1.9035752853902350521*t^{1}.25,0.21767841032926169436e-1-0.16688086862630055943e-1*I)^{4}}$

行波图

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

Karachawa equation traveling wave plot

参考文献

1. 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

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