广田-萨摩方程组

广田-萨摩方程(Hirota Satsuma equation)是一个三元非线性偏微分方程组：^[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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行波图

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