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：

$$\begin{cases} u_t-\frac{1}{2}u_{xxx}+3uu_x-3(vw)_x=0,\\ v_t+v_{xxx}-3uv_x=0,\\ w_t+w_{xxx}-3uw_x=0.\end{cases}$$

## 解析解

$$u(x, t) = -(1/3)*(-_C3+_C2^3)/_C2, v(x, t) = 0, w(x, t) = _C7+_C8*cos(_C1+_C2*x+_C3*t)$$

$$u(x, t) = -(1/3)*(-_C3+_C2^3)/_C2, v(x, t) = 0, w(x, t) = _C7+_C8*sin(_C1+_C2*x+_C3*t)$$

$$u(x, t) = (1/3)*(_C3+_C2^3)/_C2, v(x, t) = 0, w(x, t) = _C7+_C8*cosh(_C1+_C2*x+_C3*t)$$

$$u(x, t) = (1/3)*(_C3+_C2^3)/_C2, v(x, t) = 0, w(x, t) = _C7+_C8*sinh(_C1+_C2*x+_C3*t)$$

$$u(x, t) = -(1/3)*(9*_C3^3-_C4)/_C3, v(x, t) = _C6-(3/4)*_C9*cos(_C2+_C3*x+_C4*t)+_C9*cos(_C2+_C3*x+_C4*t)^3, w(x, t) = 0$$

$$u(x, t) = -_C2^2+2*_C2^2*coth(_C1+_C2*x-_C2^3*t)^2, v(x, t) = _C5+_C6*coth(_C1+_C2*x-_C2^3*t), w(x, t) = 0$$

$${u(x, t) = -(1/2)*_C2^2+2*_C2^2*csc(_C1+_C2*x-(1/2)*_C2^3*t)^2, v(x, t) = _C5+_C6*csc(_C1+_C2*x-(1/2)*_C2^3*t), w(x, t) = 0}$$

$${u(x, t) = (1/2)*_C2^2+2*_C2^2*csch(_C1+_C2*x+(1/2)*_C2^3*t)^2, v(x, t) = _C5+_C6*csch(_C1+_C2*x+(1/2)*_C2^3*t), w(x, t) = 0}$$

$$u(x, t) = -(1/3)*(-_C4-2*_C3^3+_C3^3*_C1^2)/_C3-2*_C3^2*JacobiDN(_C2+_C3*x+_C4*t, _C1)^2, v(x, t) = (2/3)*_C3*(2*_C4-2*_C3^3+_C3^3*_C1^2)*_C8/_C9^2-(2/3)*_C3*(2*_C4-2*_C3^3+_C3^3*_C1^2)*JacobiDN(_C2+_C3*x+_C4*t, _C1)/_C9, w(x, t) = _C8+_C9*JacobiDN(_C2+_C3*x+_C4*t, _C1)$$

$$u(x, t) = -(1/3)*(-_C4+_C3^3*_C1^2+_C3^3)/_C3+2*_C3^2*_C1^2*JacobiSN(_C2+_C3*x+_C4*t, _C1)^2, v(x, t) = -(2/3)*_C1^2*_C3*(_C3^3*_C1^2+_C3^3+2*_C4)*_C10/_C11^2+(2/3)*_C3*_C1^2*(_C3^3*_C1^2+_C3^3+2*_C4)*JacobiSN(_C2+_C3*x+_C4*t, _C1)/_C11, w(x, t) = _C10+_C11*JacobiSN(_C2+_C3*x+_C4*t, _C1)$$

$$u(x, t) = (1/3)*(-2*_C3^3+4*_C3^3*_C1^2-_C4)/_C3+(2*_C3^2-2*_C3^2*_C1^2)*JacobiNC(_C2+_C3*x+_C4*t, _C1)^2, v(x, t) = _C6, w(x, t) = _C10$$

$$u(x, t) = -(1/2)*_C3^2+_C3^2*_C1^2-2*_C3^2*_C1^2*JacobiCN(_C2+_C3*x+(-(1/2)*_C3^3+_C3^3*_C1^2)*t, _C1)^2, v(x, t) = _C6+_C7*JacobiCN(_C2+_C3*x+(-(1/2)*_C3^3+_C3^3*_C1^2)*t, _C1), w(x, t) = 0$$

## 行波圖

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## 參考文獻

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- G Haghighatdoost, M Bazghandi, F Pashaie, Differential Invariants of Coupled Hirota-Satsuma KdV Equations . Kragujevac Journal of Mathematics 49 (5), 793-805, 2025 10.46793/KgJMat2505.793H