2.15.25 Sawada Kotera \(u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \)

problem number 134

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Sawada Kotera. Solve for \(u(x,t)\) \[ u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \]

Mathematica


\begin {align*} & \left \{u(x,t)\to -\frac {4}{3} c_1{}^2 \left (-2+3 \tanh ^2\left (-16 c_1{}^5 t+c_1 x+c_3\right )\right )\right \}\\& \left \{u(x,t)\to \frac {-30 c_1{}^{5/2} \tanh ^2(c_2 t+c_1 x+c_3)+20 c_1{}^{5/2}+\sqrt {20 c_1{}^5-5 c_2}}{15 \sqrt {c_1}}\right \}\\& \left \{u(x,t)\to \frac {20 c_1{}^{5/2}-\sqrt {20 c_1{}^5-5 c_2}}{15 \sqrt {c_1}}-2 c_1{}^2 \tanh ^2(c_2 t+c_1 x+c_3)\right \}\\ \end {align*}

Maple


\begin {align*} & \{u \left (x , t\right ) = c_{4}\}\\& \left \{u \left (x , t\right ) = -4 c_{2}^{2} \left (\tanh ^{2}\left (-16 c_{2}^{5} t +c_{2} x +c_{1}\right )\right )+\frac {8 c_{2}^{2}}{3}\right \}\\& \left \{u \left (x , t\right ) = -2 c_{2}^{2} \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )-\frac {-20 c_{2}^{3}+\sqrt {20 c_{2}^{6}-5 c_{3} c_{2}}}{15 c_{2}}\right \}\\& \left \{u \left (x , t\right ) = -2 c_{2}^{2} \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )+\frac {20 c_{2}^{3}+\sqrt {20 c_{2}^{6}-5 c_{3} c_{2}}}{15 c_{2}}\right \}\\ \end {align*}

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