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 ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] + 45*u[x, t]^2*D[u[x, t], x] + 15*D[u[x, t], x]*D[u[x, t], {x, 2}] + 15*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\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 ✓

restart; 
pde := diff(u(x,t),t)+45* u(x,t)^2* diff(u(x,t),x)+ 15* diff(u(x,t),x)*diff(u(x,t),x$2)+15*u(x,t)*diff(u(x,t),x$3)+diff(u(x,t),x$5); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',PDEtools:-TWSolutions(pde,u(x,t))),output='realtime'));
 

\begin{align*}& \{u \left (x , t\right ) = c_4\}\\& \left \{u \left (x , t\right ) = -4 \tanh \left (-16 c_2^{5} t +c_2 x +c_1 \right )^{2} c_2^{2}+\frac {8 c_2^{2}}{3}\right \}\\& \left \{u \left (x , t\right ) = -2 \tanh \left (c_2 x +c_3 t +c_1 \right )^{2} c_2^{2}-\frac {-20 c_2^{3}+\sqrt {20 c_2^{6}-5 c_2 c_3}}{15 c_2}\right \}\\& \left \{u \left (x , t\right ) = -2 \tanh \left (c_2 x +c_3 t +c_1 \right )^{2} c_2^{2}+\frac {20 c_2^{3}+\sqrt {20 c_2^{6}-5 c_2 c_3}}{15 c_2}\right \}\\ \end{align*}

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