2.15.11 Degasperis Procesi $u_t-u_{xxt}+4u{u}_x=3u_x{u}_{x}x+u{u}_{xxx}$

problem number 120

Added December 27, 2018.

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

Degasperis Procesi. Solve for $u(x,t)$ $$u_t-u_{xxt}+4u{u}_x=3u_x{u}_{x}x+u{u}_{xxx}$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] - D[D[u[x, t], {x, 2}], t] + 4*u[x, t]*D[u[x, t], x] == 3*D[u[x, t], x]*D[u[x, t], {x, 2}] + u[x, t]*D[u[x, t], {x, 3}]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(u(x,t),t)-diff(u(x,t),x,x,t)+4*u(x,t)*diff(u(x,t),x)=3*diff(u(x,t),x)*diff(u(x,t),x$2)+u(x,t)*diff(u(x,t),x$3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
 

$$\mathit{P}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}(u(x,t)=\frac{\mathit{\_}\mathit{F}\mathit{1}\left(x\right)}{-{\mathit{\_}\mathit{c}}_{2}t+\mathit{\_}\mathit{C}\mathit{2}},[\left\{\{\mathit{\_}\mathit{F}\mathit{1}\left(x\right)=\mathit{O}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}(\mathit{\_}\mathit{a},[\{\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathit{\_}\mathit{a}}^2}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right){\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2\mathit{\_}\mathit{a}+{\left(\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{a}}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathit{\_}\mathit{a}+\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\left(\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{a}}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)({\mathit{\_}\mathit{c}}_2+3\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right))-\mathit{\_}\mathit{a}({\mathit{\_}\mathit{c}}_2+4\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right))}{\mathit{\_}\mathit{a}}=0\},\{\mathit{\_}\mathit{a}=\mathit{\_}\mathit{F}\mathit{1}\left(x\right),\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=\frac{\mathrm{d}}{\mathrm{d}x}\mathit{\_}\mathit{F}\mathit{1}\left(x\right)\},\{x=\int {\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1},\mathit{\_}\mathit{F}\mathit{1}\left(x\right)=\mathit{\_}\mathit{a}\}])\}\right\}])$$ But still has unresolved ODE’s in solution

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