____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t+u u_x = \nu u_{xx} \]
Mathematica ✓
ClearAll[u, x, t, v]; pde = D[u[x, t], t] + u[x, t]*D[u[x, t], x] == v*D[u[x, t], {x, 2}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to -\frac {2 c_1^2 v \tanh \left (c_2 t+c_1 x+c_3\right )+c_2}{c_1}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)+u(x,t)*diff(u(x,t),x)=v*diff(u(x,t),x$2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =-2\,v{\it \_C2}\,\tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\frac {{\it \_C3}}{{\it \_C2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t+u_x + u u+x - u_{xxt} = 0 \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[u[x, t], t] + D[u[x, t], x] + u[x, t]*D[u[x, t], x] - D[D[u[x, t], {x, 2}], t] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to \frac {12 c_2 c_1^2 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_2 c_1^2-c_1-c_2}{c_1}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)+diff(u(x,t),x)+u(x,t)*diff(u(x,t),x)-diff(u(x,t),x,x,t)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =12\,{\it \_C2}\,{\it \_C3}\, \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}-{\frac {8\,{{\it \_C2}}^{2}{\it \_C3}+{\it \_C2}+{\it \_C3}}{{\it \_C2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t+H u_{xx} +u u_x = 0 \]
Important note. \(H\) above is meant to be Hilbert transform. https://en.wikipedia.org/wiki/Benjamin%E2%80%93Ono_equation However, here in the code below it is taken as just a scalar.
Mathematica ✓
ClearAll[u, x, t, h]; pde = D[u[x, t], t] + h*D[u[x, t], {x, 2}] + u[x, t]*D[u[x, t], x] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to \frac {2 c_1^2 h \tanh \left (c_2 t+c_1 x+c_3\right )-c_2}{c_1}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)+H*diff(u(x,t),x$2)+u(x,t)*diff(u(x,t),x)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =2\,H\,{\it \_C2}\,\tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\frac {{\it \_C3}}{{\it \_C2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
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[u, x, t]; 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]];
\[ \left \{\left \{u(x,t)\to c_1(t+x)+c_2(t-x)\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; 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'));
\[ u \left ( x,t \right ) ={\it \_C7}\, \left ( \tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) +{\it \_C4} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0 \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to -\frac {12 c_1^4 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_1^4+c_1^2-c_2^2}{6 c_1^2}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t$2)-diff(u(x,t),x$2)-diff(u(x,t),x$4)- 3 * diff( u(x,t)^2, x$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/6\,{\frac {8\,{{\it \_C2}}^{4}-{{\it \_C2}}^{2}+{{\it \_C3}}^{2}}{{{\it \_C2}}^{2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0 \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to -\frac {12 c_1^4 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_1^4+c_1^2-c_2^2}{6 c_1^2}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t';alpha:='alpha';beta:='beta'; pde:=diff(u(x,t),t$2)-diff(u(x,t),x$2)-2*alpha*diff( (u(x,t)*diff(u(x,t),x)) ,x) - beta*diff(u(x,t),x,x,t,t)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =-6\,{\frac {{{\it \_C3}}^{2}\beta \, \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}}{\alpha }}+1/2\,{\frac {8\,{{\it \_C2}}^{2}{{\it \_C3}}^{2}\beta -{{\it \_C2}}^{2}+{{\it \_C3}}^{2}}{\alpha \,{{\it \_C2}}^{2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t = (u^4)_{xx} + (u^3)_x \]
Mathematica ✗
ClearAll[u, x, t]; pde = D[u[x, t], t] == D[u[x, t]^4, {x, 2}] + D[u[x, t]^3, x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)= diff(u(x,t)^4,x$2)+diff(u(x,t)^3,x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =\RootOf \left ( {\it \_C1}\,x+{\it \_C2}\,t+{\it \_C3}+\int ^{{\it \_Z}}\!4\,{\frac {{{\it \_C1}}^{2}{{\it \_f}}^{3}}{{\it \_C1}\,{{\it \_f}}^{3}+4\,{\it \_C3}\,{{\it \_C1}}^{2}-{\it \_C2}\,{\it \_f}}}{d{\it \_f}}+{\it \_C4} \right ) \] Answer in terms of RootOf.
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx} \]
Mathematica ✗
ClearAll[u, x, t, k]; pde = D[u[x, t], t] + 2*k*D[u[x, t], x] - D[D[u[x, t], {x, 2}], t] + 3*u[x, t]*D[u[x, t], x] == 2*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]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)+2*k*diff(u(x,t),x)- diff(u(x,t),x,x,t)+3*u(x,t)*diff(u(x,t),x)=2*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))),output='realtime'));
\[ u \left ( x,t \right ) ={\frac {1}{{\it \_C1}} \left ( \left ( \RootOf \left ( -i{\it \_C1}\,x-i{\it \_C2}\,t-i{\it \_C3}+\int ^{-{\frac {-{{\it \_Z}}^{2}+{\it \_C2}}{{\it \_C1}}}}\!{\frac {\sqrt {{\it \_C1}\,{\it \_f}+{\it \_C2}}}{\sqrt {{{\it \_C1}}^{3}{\it \_C3}\,{\it \_f}+{{\it \_C1}}^{2}{\it \_C2}\,{\it \_C3}-{\it \_C1}\,{{\it \_f}}^{3}-2\,{\it \_C1}\,{{\it \_f}}^{2}k+{\it \_C4}\,{{\it \_C1}}^{2}-{\it \_C2}\,{{\it \_f}}^{2}}}}{d{\it \_f}}{\it \_C1}+{\it \_C5}\,{\it \_C1} \right ) \right ) ^{2}-{\it \_C2} \right ) } \] Answer in terms of RootOf.
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t = u_{xx} + \lambda (u^3 - u) = 0 \]
Mathematica ✗
ClearAll[u, x, t, lambda]; pde = D[u[x, t], t] - D[u[x, t], {x, 2}] + lambda*(u[x, t]^3 - u[x, t]) == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)-diff(u(x,t),x$2)+lambda*(u(x,t)^3-u(x,t))=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =1/2\,\tanh \left ( -3/4\,\lambda \,t+1/4\,\sqrt {2}\sqrt {\lambda }x+{\it \_C1} \right ) -1/2 \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(\theta (x,t)\)
\[ \left ( \theta _t - \gamma e^\theta \right )_{tt} = \left ( \theta _t - e^\theta \right )_{xx} \]
Mathematica ✗
ClearAll[theta, x, t, gamma]; pde = D[D[theta[x, t], t] - gamma*Exp[theta[x, t]], {t, 2}] == D[D[theta[x, t], t] - Exp[theta[x, t]], {x, 2}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, theta[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✗
theta:='theta';x:='x';t:='t';g:='g'; pde := diff(diff(theta(x,t),t)-g*exp(theta(x,t)),t$2) = diff( diff(theta(x,t),t)-exp(theta(x,t)),x$2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,theta(x,t))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx} \]
Mathematica ✗
ClearAll[u, x, t]; 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]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; 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'));
\[ {\it PDESolStruc} \left ( u \left ( x,t \right ) ={\frac {{\it \_F1} \left ( x \right ) }{-{\it \_c}_{{2}}t+{\it \_C2}}},[ \left \{ \left \{ {\it \_F1} \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ \left ( {\frac {{\rm d}^{2}}{{\rm d}{{\it \_a}}^{2}}}{\it \_b} \left ( {\it \_a} \right ) \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}+{\frac { \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{\it \_b} \left ( {\it \_a} \right ) {\it \_a}+3\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) +{\it \_b} \left ( {\it \_a} \right ) \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_c}_{{2}}-4\,{\it \_b} \left ( {\it \_a} \right ) {\it \_a}-{\it \_a}\,{\it \_c}_{{2}}}{{\it \_a}}}=0 \right \} , \left \{ {\it \_a}={\it \_F1} \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}x}}{\it \_F1} \left ( x \right ) \right \} , \left \{ x=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1},{\it \_F1} \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \right \} ] \right ) \] But still has unresolved ODE’s in solution
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t =u^3 u_{xxx} \]
Mathematica ✗
ClearAll[u, x, t]; pde = D[u[x, t], t] == u[x, t]^3*D[u[x, t], {x, 3}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:= diff(u(x,t),t)=u(x,t)^3 * diff(u(x,t),x$3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[ u \left ( x,t \right ) ={\frac {1}{\sqrt [3]{-3\,{\it \_c}_{{1}}t+{\it \_C4}}}\RootOf \left ( -\int ^{{\it \_Z}}\! \left ( \RootOf \left ( -\ln \left ( {\it \_f} \right ) +2\,\int ^{{\it \_Z}}\!{\frac {{\it \_h}}{2\,\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}\RootOf \left ( \AiryBi \left ( {\it \_Z} \right ) \sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_C1}\,{\it \_h}+\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_h}\,\AiryAi \left ( {\it \_Z} \right ) +2\,\AiryBi \left ( 1,{\it \_Z} \right ) {\it \_C1}\,{\it \_c}_{{1}}+2\,\AiryAi \left ( 1,{\it \_Z} \right ) {\it \_c}_{{1}} \right ) +{{\it \_h}}^{2}}}{d{\it \_h}}+{\it \_C2} \right ) \right ) ^{-1}{d{\it \_f}}+x+{\it \_C3} \right ) } \] has RootOf
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y,t)\)
\[ u_{tyyy} + \beta u_y u_{yt} + \beta u_{yy} u_t + u_{tt} = 0 \]
Mathematica ✓
ClearAll[u, x, t, y, beta]; pde = D[D[u[x, y, t], t], {y, 3}] + beta*D[u[x, y, t], y]*D[D[u[x, y, t], y], t] + beta*D[u[x, y, t], {y, 2}]*D[u[x, y, t], t] + D[u[x, y, t], {t, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];
\[ \left \{\left \{u(x,y,t)\to \frac {\beta c_4(x)+6 c_1(x) \tanh \left (-4 t c_1(x){}^3+y c_1(x)+c_3(x)\right )}{\beta }\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t';y:='y';beta='beta'; pde:= diff(u(x,y,t),t,y,y,y)+ beta*diff(u(x,y,t),y)*diff(u(x,y,t),y,t) + beta*diff(u(x,y,t),y$2)*diff(u(x,y,t),t) + diff(u(x,y,t),t$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));
\[ u \left ( x,y,t \right ) =6\,{\frac {{\it \_C3}\,\tanh \left ( -4\,{{\it \_C3}}^{3}t+{\it \_C2}\,x+{\it \_C3}\,y+{\it \_C1} \right ) }{\beta }}+{\it \_C5} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t = u(1-u)+u_{xx} \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[u[x, t], t] == u[x, t]*(1 - u[x, t]) + D[u[x, t], {x, 2}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to \frac {1}{4} \left (\tanh \left (-c_3+\frac {5 t}{12}-\frac {x}{2 \sqrt {6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (-c_3+\frac {5 t}{12}-\frac {i x}{2 \sqrt {6}}\right )\right ) \left (1+\tanh \left (-c_3+\frac {5 t}{12}-\frac {i x}{2 \sqrt {6}}\right )\right )\right \},\left \{u(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (-c_3+\frac {5 t}{12}+\frac {i x}{2 \sqrt {6}}\right )\right ) \left (1+\tanh \left (-c_3+\frac {5 t}{12}+\frac {i x}{2 \sqrt {6}}\right )\right )\right \},\left \{u(x,t)\to \frac {1}{4} \left (\tanh \left (-c_3+\frac {5 t}{12}+\frac {x}{2 \sqrt {6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to \frac {1}{4} \left (\tanh \left (c_3+\frac {5 t}{12}-\frac {x}{2 \sqrt {6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (c_3+\frac {5 t}{12}-\frac {i x}{2 \sqrt {6}}\right )\right ) \left (1+\tanh \left (c_3+\frac {5 t}{12}-\frac {i x}{2 \sqrt {6}}\right )\right )\right \},\left \{u(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (c_3+\frac {5 t}{12}+\frac {i x}{2 \sqrt {6}}\right )\right ) \left (1+\tanh \left (c_3+\frac {5 t}{12}+\frac {i x}{2 \sqrt {6}}\right )\right )\right \},\left \{u(x,t)\to \frac {1}{4} \left (\tanh \left (c_3+\frac {5 t}{12}+\frac {x}{2 \sqrt {6}}\right )+1\right ){}^2\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:= diff(u(x,t),t)= u(x,t)*(1-u(x,t))+ diff(u(x,t),x$2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',{PDEtools:-TWSolutions(pde,u(x,t))}),output='realtime'));
\[ \left \{ \left \{ u \left ( x,t \right ) =1 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( -{\frac {5\,t}{12}}-1/12\,\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac {5\,t}{12}}-1/12\,\sqrt {6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( -{\frac {5\,t}{12}}+1/12\,\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac {5\,t}{12}}+1/12\,\sqrt {6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac {5\,t}{12}}-i/12\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac {5\,t}{12}}-i/12\sqrt {6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac {5\,t}{12}}+i/12\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac {5\,t}{12}}+i/12\sqrt {6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( {\frac {5\,t}{12}}-1/12\,\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ( {\frac {5\,t}{12}}-1/12\,\sqrt {6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( {\frac {5\,t}{12}}+1/12\,\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ( {\frac {5\,t}{12}}+1/12\,\sqrt {6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( {\frac {5\,t}{12}}-i/12\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ( {\frac {5\,t}{12}}-i/12\sqrt {6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( {\frac {5\,t}{12}}+i/12\sqrt {6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ( {\frac {5\,t}{12}}+i/12\sqrt {6}x+{\it \_C1} \right ) +3/4 \right \} \right \} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ \left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2 \]
Mathematica ✗
ClearAll[u, x, t]; pde = D[D[u[x, t], t] + u[x, t]*D[u[x, t], x], x] == (1*D[u[x, t], x]^2)/2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:= diff( (diff(u(x,t),t)+ u(x,t)* diff(u(x,t),x)) , x) = 1/2* (diff(u(x,t),x))^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[ u \left ( x,t \right ) =2\,{\frac {\RootOf \left ( -{\it \_C2}\,{{\it \_c}_{{1}}}^{3}-x{{\it \_c}_{{1}}}^{3}-2\,{\it \_C1}\,\sqrt {{\it \_Z}}{\it \_c}_{{1}}+2\,{{\it \_C1}}^{2}\ln \left ( \sqrt {{\it \_Z}}{\it \_c}_{{1}}+{\it \_C1} \right ) +{\it \_Z}\,{{\it \_c}_{{1}}}^{2} \right ) }{{\it \_c}_{{1}}t+2\,{\it \_C3}} \left ( {\frac {{\it \_c}_{{1}}t}{{\it \_c}_{{1}}t+2\,{\it \_C3}}}+2\,{\frac {{\it \_C3}}{{\it \_c}_{{1}}t+2\,{\it \_C3}}} \right ) ^{-1}} \] with RootOf
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y,t)\)
\[ \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[D[u[x, y, t], t] + u[x, y, t]*D[u[x, y, t], x] + eps^2*D[u[x, y, t], {x, 3}], t] + lambda*D[u[x, y, t], {y, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];
\[ \left \{\left \{u(x,y,t)\to -\frac {12 c_3 c_1^3 \text {eps}^2 \tanh ^2\left (c_3 t+c_1 x+c_2 y+c_4\right )-8 c_3 c_1^3 \text {eps}^2+c_2^2 \lambda +c_3^2}{c_1 c_3}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t';y:='y';lambda:='lambda';epsilon:='epsilon'; pde:= diff( diff(u(x,y,t),t)+u(x,y,t)*diff(u(x,y,t),x)+epsilon^2* diff(u(x,y,t),x$3),x)+ lambda*diff(u(x,y,t),y$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));
\[ u \left ( x,y,t \right ) =-12\,{\epsilon }^{2}{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,y+{\it \_C4}\,t+{\it \_C1} \right ) \right ) ^{2}+{\frac {8\,{{\it \_C2}}^{4}{\epsilon }^{2}-{{\it \_C3}}^{2}\lambda -{\it \_C4}\,{\it \_C2}}{{{\it \_C2}}^{2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y)\)
\[ u_{xx}+u_{yy}+ \lambda u^p=0 \]
Mathematica ✗
ClearAll[u, x, y, lambda]; pde = Laplacian[u[x, y], {x, y}] + lambda*u[x, y]^p == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
u:='u';x:='x';y:='y';lambda:='lambda'; pde:=diff(u(x,y),x$2)+diff(u(x,y),y$2)+lambda*u(x,y)^p=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),'build')),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Solve for \(u(x,y)\)
\[ u_{xx}+u_{yy}+ u^2=0 \]
Mathematica ✗
ClearAll[u, x, y, lambda]; pde = Laplacian[u[x, y], {x, y}] + u[x, y]^2 == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';y:='y';lambda:='lambda'; pde:=diff(u(x,y),x$2)+diff(u(x,y),y$2)+u(x,y)^2=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[ u \left ( x,y \right ) =-6\,{\it WeierstrassP} \left ( {\it \_C1}\,x+{\it \_C2}\,y+2\,{\it \_C3},0,{\it \_C4} \right ) \left ( {{\it \_C1}}^{2}+{{\it \_C2}}^{2} \right ) \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y,t)\)
\[ u_{x t} - (u u_x)_x = u_{yy} \]
Mathematica ✗
ClearAll[u, x, y, t]; 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]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';y:='y';t:='t'; 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'));
\[ u \left ( x,y,t \right ) ={\frac {{\it \_C3}\,{\it \_C1}-{{\it \_C2}}^{2}+\sqrt {2\, \left ( {\it \_C1}\,x+{\it \_C2}\,y+{\it \_C3}\,t+{\it \_C4} \right ) {{\it \_C1}}^{2}{\it \_C4}+{{\it \_C1}}^{2}{{\it \_C3}}^{2}-2\,{\it \_C1}\,{{\it \_C2}}^{2}{\it \_C3}+{{\it \_C2}}^{4}+2\,{{\it \_C1}}^{2}{\it \_C5}}}{{{\it \_C1}}^{2}}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_t + (u_x)^3+ 6 u u_x = 0 \]
Mathematica ✓
ClearAll[u, x, t]; pde = D[u[x, t], t] + D[u[x, t], x]^3 + 6*u[x, t]*D[u[x, t], x] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \left \{\left \{u(x,t)\to \frac {-18 c_1 t x-18 c_2 t-9 c_1^2 x^2-18 c_1 c_2 x-9 c_2^2-c_1-9 t^2}{6 c_1^2}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),t)+ diff( u(x,t),x )^3 + 6 * u(x,t)* diff(u(x,t),x) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ u \left ( x,t \right ) =-3/2\,{{\it \_C1}}^{2}+3\,{\it \_C1}\, \left ( {\it \_c}_{{2}}t+x \right ) -3/2\, \left ( {\it \_c}_{{2}}t+x \right ) ^{2}-1/6\,{\it \_c}_{{2}} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y,t)\)
\[ 2 u_{tx} + u_x u_{xx} - u_{yy} = 0 \]
Mathematica ✗
ClearAll[u, x, t]; pde = 2*D[u[x, y, t], t, x] + D[u[x, y, t], x]*D[u[x, y, t], {x, 2}] - D[u[x, y, t], {y, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t';y:='y'; pde:=2*diff(u(x,y,t),t,x)+ diff(u(x,y,t),x)* diff(u(x,y,t),x$2) - diff(u(x,y,t),y$2) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));
\[ u \left ( x,y,t \right ) ={\it \_C4}+{\it \_C5}\, \left ( {\it \_C1}\,x+{\it \_C2}\,y+{\it \_C3}\,t+{\it \_C4} \right ) \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y)\)
\[ u_{xx} + u_{yy} +e^{\lambda u} = 0 \]
Mathematica ✗
ClearAll[u, x, lam, y]; pde = Laplacian[u[x, y], {x, y}] + Exp[lam*u[x, y]] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
u:='u';x:='x';y:='y';lambda:='lambda'; pde:=diff(u(x,y),x$2)+ diff(u(x,y),y$2)+exp(lambda*u(x,y))=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,y)\)
\[ (1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0 \]
Mathematica ✗
ClearAll[u, x, y]; pde = (1 + D[u[x, y], y]^2)*D[u[x, y], {x, 2}] - 2*D[u[x, y], x]*D[u[x, y], y]*D[u[x, y], x, y] + (1 + D[u[x, y], x]^2)*D[u[x, y], {y, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';y:='y'; pde:=(1+diff(u(x,y),y)^2)* diff(u(x,y),x$2)-2* diff(u(x,y),x)* diff(u(x,y),y)*diff(u(x,y),x,y)+ (1+diff(u(x,y),x)^2)*diff(u(x,y),y$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[ u \left ( x,y \right ) ={\it \_C7}\, \left ( \tanh \left ( {\it \_C2}\,x-i{\it \_C2}\,y+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ( {\it \_C2}\,x-i{\it \_C2}\,y+{\it \_C1} \right ) +{\it \_C4} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{tt} - u_{xx} = \epsilon (u_t - u_t^3) \]
Mathematica ✗
ClearAll[u, x, t, epsilon]; pde = D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] == epsilon*(D[u[x, t], t] - D[u[x, t], t]^3); sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';t:='t';epsilon:='epsilon'; pde:=diff(u(x,t),t$2)-diff(u(x,t),x$2)=epsilon*(diff(u(x,t),t)-diff(u(x,t),t)^3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[ u \left ( x,t \right ) =1/2\,{\it \_c}_{{1}}{x}^{2}+{\it \_C1}\,x+{\it \_C2}+\int \!\RootOf \left ( t+\int ^{{\it \_Z}}\! \left ( {{\it \_f}}^{3}\epsilon -{\it \_f}\,\epsilon -{\it \_c}_{{1}} \right ) ^{-1}{d{\it \_f}}+{\it \_C3} \right ) \,{\rm d}t+{\it \_C4} \] Has RootOf
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
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[phi, x, t]; 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]];
\[ \left \{\left \{u(x,t)\to -\frac {4}{3} c_1^2 \left (3 \tanh ^2\left (-16 c_1^5 t+c_1 x+c_3\right )-2\right )\right \},\left \{u(x,t)\to \frac {-30 c_1^{5/2} \tanh ^2\left (c_2 t+c_1 x+c_3\right )+20 c_1^{5/2}+\sqrt {5} \sqrt {4 c_1^5-c_2}}{15 \sqrt {c_1}}\right \},\left \{u(x,t)\to -\frac {30 c_1^{5/2} \tanh ^2\left (c_2 t+c_1 x+c_3\right )-20 c_1^{5/2}+\sqrt {5} \sqrt {4 c_1^5-c_2}}{15 \sqrt {c_1}}\right \}\right \} \]
Maple ✓
u:='u';x:='x';t:='t'; 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'));
\[ \left \{ \left \{ u \left ( x,t \right ) ={\it \_C4} \right \} , \left \{ u \left ( x,t \right ) =-4\,{{\it \_C2}}^{2} \left ( \tanh \left ( -16\,{{\it \_C2}}^{5}t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{2}+8/3\,{{\it \_C2}}^{2} \right \} , \left \{ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}-1/15\,{\frac {-20\,{{\it \_C2}}^{3}+\sqrt {20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}}}{{\it \_C2}}} \right \} , \left \{ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/15\,{\frac {20\,{{\it \_C2}}^{3}+\sqrt {20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}}}{{\it \_C2}}} \right \} \right \} \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ \phi _{tt} - \phi _{xx} + \sin \phi = 0 \]
Mathematica ✗
ClearAll[phi, x, t]; pde = D[phi[x, t], {t, 2}] - D[phi[x, t], {x, 2}] + Sin[phi[x, t]] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, phi[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✗
phi:='phi';x:='x';t:='t'; pde:=diff(phi(x,t),t$2)-diff(phi(x,t),x$2)+sin(phi(x,t))=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,phi(x,t))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{xt} = \sinh u \]
Mathematica ✗
ClearAll[u, x, t]; pde = D[u[x, t], x, t] == Sinh[u[x, t]]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[ \text {Failed} \]
Maple ✗
u:='u';x:='x';t:='t'; pde:=diff(u(x,t),x,t)=sinh(u(x,t)); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{xx}+u_{yy} + \sinh u=0 \]
Mathematica ✗
ClearAll[u, x, y]; pde = Laplacian[u[x, y], {x, y}] + Sinh[u[x, y]] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
u:='u';x:='x';y:='y'; pde:=diff(u(x,y),x$2)+diff(u(x,y),y$2)+ sinh(u(x,y))=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0 \]
Mathematica ✗
ClearAll[u, x, y, alpha, beta, nu]; pde = D[u[x, y], x, y] + alpha*D[u[x, y], x] + beta*D[u[x, y], y] + nu*D[u[x, y], x]*D[u[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
u:='u';x:='x';y:='y';beta:='beta';alpha:='alpha';nu:='nu'; pde:=diff(u(x,y),x,y)+alpha*diff(u(x,y),x)+beta*diff(u(x,y),y) +nu* diff(u(x,y),x)*diff(u(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),'build')),output='realtime'));
\[ u \left ( x,y \right ) =-1/2\,{\frac {\sqrt {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }x}{\nu }}+1/2\,{\frac {\sqrt {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }y}{\nu }}-1/2\,{\frac {\sqrt {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }x}{\nu }}-1/2\,{\frac {\sqrt {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }y}{\nu }}-{\frac {\alpha \,y}{\nu }}-{\frac {\beta \,x}{\nu }}-2\,{\frac {\ln \left ( 2 \right ) }{\nu }}-1/2\,{\frac {1}{\nu }\ln \left ( {\frac {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ( {\it \_C3}\,{{\rm e}^{2\, \left ( x/2+y/2 \right ) \sqrt {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C4} \right ) ^{2}}} \right ) }-1/2\,{\frac {1}{\nu }\ln \left ( {\frac {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ( {\it \_C1}\,{{\rm e}^{2\, \left ( x/2-y/2 \right ) \sqrt {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C2} \right ) ^{2}}} \right ) } \]
____________________________________________________________________________________
Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Solve for \(u(x,t)\)
\[ \phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0 \]
Mathematica ✓
ClearAll[u, x, y, alpha, beta, nu]; pde = D[phi[x, t], t, t] - D[phi[x, t], x, x] - phi[x, t] + phi[x, t]^3 == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, phi[x, t], {x, t}], 60*10]];
\[ \left \{\left \{\phi (x,t)\to -\tanh \left (c_2 t-\frac {\sqrt {2 c_2^2+1} x}{\sqrt {2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t-\frac {\sqrt {2 c_2^2+1} x}{\sqrt {2}}+c_3\right )\right \},\left \{\phi (x,t)\to -\tanh \left (c_2 t+\frac {\sqrt {2 c_2^2+1} x}{\sqrt {2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t+\frac {\sqrt {2 c_2^2+1} x}{\sqrt {2}}+c_3\right )\right \}\right \} \]
Maple ✓
phi:='phi';x:='x';t:='t'; pde:=diff(phi(x,t),t$2)-diff(phi(x,t),x$2) - phi(x,t) + phi(x,t)^3=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',{PDEtools:-TWSolutions(pde,phi(x,t))}),output='realtime'));
\[ \left \{ \left \{ \phi \left ( x,t \right ) =-1 \right \} , \left \{ \phi \left ( x,t \right ) =1 \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( -1/2\,\sqrt {4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( 1/2\,\sqrt {4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( -1/2\,\sqrt {4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( 1/2\,\sqrt {4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} \right \} \]