廣田-薩摩方程組

廣田-薩摩方程(Hirota Satsuma equation)是一個三元非線性偏微分方程組：^[1]

${\displaystyle {\begin{cases}u_{t}-{\frac {1}{2}}u_{xxx}+3uu_{x}-3(vw)_{x}=0,\\v_{t}+v_{xxx}-3uv_{x}=0,\\w_{t}+w_{xxx}-3uw_{x}=0.\end{cases}}}$

解析解

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

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

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

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

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

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

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

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

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

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

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

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

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行波圖

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot

Hirota Satsuma equations traveling wave plot