2.15.19 Khokhlov Zabolotskaya \(u_{x t} - (u u_x)_x = u_{yy}\)

problem number 128

Added December 27, 2018.

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

Khokhlov Zabolotskaya. Solve for \(u(x,y,t)\)

\[ u_{x t} - (u u_x)_x = u_{yy} \]

Mathematica ✗

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

Failed

Maple ✓

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

\[u \left (x , y , t\right ) = \frac {c_3 c_1 -c_2^{2}+\sqrt {2 c_1^{3} c_{4} x +\left (2 c_2 c_{4} y +2 c_3 c_{4} t +c_3^{2}+2 c_{4}^{2}+2 c_{5} \right ) c_1^{2}-2 c_1 \,c_2^{2} c_3 +c_2^{4}}}{c_1^{2}}\]

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