2.15.4 Born Infeld \((1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0\)

problem number 113

Added December 27, 2018.

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

Born Infeld. Solve for \(u(x,t)\)

\[ (1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (1 - D[u[x, t], t]^2)*D[u[x, t], {x, 2}] + 2*D[u[x, t], x]*D[u[x, t], t]*D[D[u[x, t], x], t] - (1 + D[u[x, t], x]^2)*D[u[x, t], {t, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\[\{\{u(x,t)\to c_1(t+x)+c_2(t-x)\}\}\]

Maple ✓

restart; 
pde :=(1-diff(u(x,t),t)^2)*diff(u(x,t),x$2)+2*diff(u(x,t),x)*diff(u(x,t),t)*diff(u(x,t),x,t)-(1+diff(u(x,t),x)^2)*diff(u(x,t),t$2)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = c_7 \tanh \left (\left (-t +x \right ) c_2 +c_1 \right )^{3}+c_5 \tanh \left (\left (-t +x \right ) c_2 +c_1 \right )+c_4\]

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