广田-萨摩方程组

**廣田 - 薩摩方程**(Hirota Satsuma equation) 是一個三元非線性偏微分方程組：

　
: $u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0$

　
: $v_{t}+v_{xxx}-3uv_{x}=0$

　
: $w_{t}+w_{xxx}-3uw_{x}=0$

 

 

## 解析解

　
: $u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cos(_{C}1+_{C}2*x+_{C}3*t)$

　
: $u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sin(_{C}1+_{C}2*x+_{C}3*t)$

　
: $u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cosh(_{C}1+_{C}2*x+_{C}3*t)$

　
: $u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sinh(_{C}1+_{C}2*x+_{C}3*t)$

　
: $u(x,t)=-(1/3)*(9*_{C}3^{3}-_{C}4)/_{C}3,v(x,t)=_{C}6-(3/4)*_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)+_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)^{3},w(x,t)=0$

　
: $u(x,t)=-_{C}2^{2}+2*_{C}2^{2}*coth(_{C}1+_{C}2*x-_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*coth(_{C}1+_{C}2*x-_{C}2^{3}*t),w(x,t)=0$

　
: ${u(x,t)=-(1/2)*_{C}2^{2}+2*_{C}2^{2}*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t),w(x,t)=0}$

　
: ${u(x,t)=(1/2)*_{C}2^{2}+2*_{C}2^{2}*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t),w(x,t)=0}$

　
: $u(x,t)=-(1/3)*(-_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})/_{C}3-2*_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*_{C}8/_{C}9^{2}-(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}9,w(x,t)=_{C}8+_{C}9*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)$

　
: $u(x,t)=-(1/3)*(-_{C}4+_{C}3^{3}*_{C}1^{2}+_{C}3^{3})/_{C}3+2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(2/3)*_{C}1^{2}*_{C}3*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*_{C}10/_{C}11^{2}+(2/3)*_{C}3*_{C}1^{2}*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}11,w(x,t)=_{C}10+_{C}11*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)$

　
: $u(x,t)=(1/3)*(-2*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+(2*_{C}3^{2}-2*_{C}3^{2}*_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=_{C}6,w(x,t)=_{C}10$

　
: $u(x,t)=-(1/2)*_{C}3^{2}+_{C}3^{2}*_{C}1^{2}-2*_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1)^{2},v(x,t)=_{C}6+_{C}7*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1),w(x,t)=0$

　
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## 行波圖

## 參考文獻

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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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