Karachawa方程

**Karachawa 方程**是一個模擬有表面張力的水波運動的非線性偏微分方程：

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

## 解析解

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

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

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

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

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

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

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

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

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

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

## 行波圖

## 參考文獻

1. \* 谷超豪 《孤立子理論中的達布變換及其幾何應用》 上海科學技術出版社
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3. 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
4. 王東明著 《消去法及其應用》 科學出版社 2002
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6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
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10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
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