36 Nonlinear PDE’s

 36.1 Bateman-Burgers equation
 36.2 Benjamin Bona Mahony
 36.3 Benjamin Ono
 36.4 Born Infeld
 36.5 Boussinesq
 36.6 Boussinesq type PDE
 36.7 Buckmaster
 36.8 Camassa Holm
 36.9 Chaffee Infante equation
 36.10 Clarke’s equation
 36.11 Degasperis Procesi
 36.12 Dym equation
 36.13 Estevez Mansfield Clarkson equation
 36.14 Fisher’s equation
 36.15 Hunter Saxton
 36.16 Kadomtsev Petviashvili
 36.17 Klein Gordon (nonlinear)
 36.18 special case Klein Gordon (nonlinear)
 36.19 Khokhlov Zabolotskaya
 36.20 Korteweg de Vries (KdV)
 36.21 Lin Tsien equation
 36.22 Liouville equation
 36.23 Plateau
 36.24 Rayleigh
 36.25 Sawada Kotera
 36.26 Sine Gordon
 36.27 Sinh Gordon
 36.28 Sinh Poisson
 36.29 Thomas equation
 36.30 phi equation

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36.1 Bateman-Burgers equation

problem number 195

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

\[ \left \{\left \{u(x,t)\to -\frac{2 c_1^2 v \tanh \left (c_2 t+c_1 x+c_3\right )+c_2}{c_1}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-2\,v{\it \_C2}\,\tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\frac{{\it \_C3}}{{\it \_C2}}} \]

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36.2 Benjamin Bona Mahony

problem number 196

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

\[ \left \{\left \{u(x,t)\to \frac{12 c_2 c_1^2 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_2 c_1^2-c_1-c_2}{c_1}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =12\,{\it \_C2}\,{\it \_C3}\, \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}-{\frac{8\,{{\it \_C2}}^{2}{\it \_C3}+{\it \_C2}+{\it \_C3}}{{\it \_C2}}} \]

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36.3 Benjamin Ono

problem number 197

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

\[ \left \{\left \{u(x,t)\to \frac{2 c_1^2 h \tanh \left (c_2 t+c_1 x+c_3\right )-c_2}{c_1}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =2\,H\,{\it \_C2}\,\tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\frac{{\it \_C3}}{{\it \_C2}}} \]

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36.4 Born Infeld

problem number 198

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

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

Maple

\[ u \left ( x,t \right ) ={\it \_C7}\, \left ( \tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) +{\it \_C4} \]

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36.5 Boussinesq

problem number 199

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

\[ \left \{\left \{u(x,t)\to -\frac{12 c_1^4 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_1^4+c_1^2-c_2^2}{6 c_1^2}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/6\,{\frac{8\,{{\it \_C2}}^{4}-{{\it \_C2}}^{2}+{{\it \_C3}}^{2}}{{{\it \_C2}}^{2}}} \]

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36.6 Boussinesq type PDE

problem number 200

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

\[ \left \{\left \{u(x,t)\to -\frac{12 c_1^4 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_1^4+c_1^2-c_2^2}{6 c_1^2}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-6\,{\frac{{{\it \_C3}}^{2}\beta \, \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}}{\alpha }}+1/2\,{\frac{8\,{{\it \_C2}}^{2}{{\it \_C3}}^{2}\beta -{{\it \_C2}}^{2}+{{\it \_C3}}^{2}}{\alpha \,{{\it \_C2}}^{2}}} \]

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36.7 Buckmaster

problem number 201

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) =\RootOf \left ({\it \_C1}\,x+{\it \_C2}\,t+{\it \_C3}+\int ^{{\it \_Z}}\!4\,{\frac{{{\it \_C1}}^{2}{{\it \_f}}^{3}}{{\it \_C1}\,{{\it \_f}}^{3}+4\,{\it \_C3}\,{{\it \_C1}}^{2}-{\it \_C2}\,{\it \_f}}}{d{\it \_f}}+{\it \_C4} \right ) \] Answer in terms of RootOf.

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36.8 Camassa Holm

problem number 202

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) ={\frac{1}{{\it \_C1}} \left ( \left ( \RootOf \left ( -i{\it \_C1}\,x-i{\it \_C2}\,t-i{\it \_C3}+\int ^{-{\frac{-{{\it \_Z}}^{2}+{\it \_C2}}{{\it \_C1}}}}\!{\frac{\sqrt{{\it \_C1}\,{\it \_f}+{\it \_C2}}}{\sqrt{{{\it \_C1}}^{3}{\it \_C3}\,{\it \_f}+{{\it \_C1}}^{2}{\it \_C2}\,{\it \_C3}-{\it \_C1}\,{{\it \_f}}^{3}-2\,{\it \_C1}\,{{\it \_f}}^{2}k+{\it \_C4}\,{{\it \_C1}}^{2}-{\it \_C2}\,{{\it \_f}}^{2}}}}{d{\it \_f}}{\it \_C1}+{\it \_C5}\,{\it \_C1} \right ) \right ) ^{2}-{\it \_C2} \right ) } \] Answer in terms of RootOf.

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36.9 Chaffee Infante equation

problem number 203

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) =1/2\,\tanh \left ( -3/4\,\lambda \,t+1/4\,\sqrt{2}\sqrt{\lambda }x+{\it \_C1} \right ) -1/2 \]

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36.10 Clarke’s equation

problem number 204

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.11 Degasperis Procesi

problem number 205

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

\[ \text{Failed} \]

Maple

\[{\it PDESolStruc} \left ( u \left ( x,t \right ) ={\frac{{\it \_F1} \left ( x \right ) }{-{\it \_c}_{{2}}t+{\it \_C2}}},[ \left \{ \left \{{\it \_F1} \left ( x \right ) ={\it ODESolStruc} \left ({\it \_a},[ \left \{ \left ({\frac{{\rm d}^{2}}{{\rm d}{{\it \_a}}^{2}}}{\it \_b} \left ({\it \_a} \right ) \right ) \left ({\it \_b} \left ({\it \_a} \right ) \right ) ^{2}+{\frac{ \left ({\frac{\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ({\it \_a} \right ) \right ) ^{2}{\it \_b} \left ({\it \_a} \right ){\it \_a}+3\, \left ({\it \_b} \left ({\it \_a} \right ) \right ) ^{2}{\frac{\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ({\it \_a} \right ) +{\it \_b} \left ({\it \_a} \right ) \left ({\frac{\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ({\it \_a} \right ) \right ){\it \_c}_{{2}}-4\,{\it \_b} \left ({\it \_a} \right ){\it \_a}-{\it \_a}\,{\it \_c}_{{2}}}{{\it \_a}}}=0 \right \} , \left \{{\it \_a}={\it \_F1} \left ( x \right ) ,{\it \_b} \left ({\it \_a} \right ) ={\frac{\rm d}{{\rm d}x}}{\it \_F1} \left ( x \right ) \right \} , \left \{ x=\int \! \left ({\it \_b} \left ({\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1},{\it \_F1} \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \right \} ] \right ) \] But still has unresolved ODE’s in solution

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36.12 Dym equation

problem number 206

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) ={\frac{1}{\sqrt [3]{-3\,{\it \_c}_{{1}}t+{\it \_C4}}}\RootOf \left ( -\int ^{{\it \_Z}}\! \left ( \RootOf \left ( -\ln \left ({\it \_f} \right ) +2\,\int ^{{\it \_Z}}\!{\frac{{\it \_h}}{2\,\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}\RootOf \left ( \AiryBi \left ({\it \_Z} \right ) \sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_C1}\,{\it \_h}+\sqrt [3]{2}\sqrt [3]{-{{\it \_c}_{{1}}}^{2}}{\it \_h}\,\AiryAi \left ({\it \_Z} \right ) +2\,\AiryBi \left ( 1,{\it \_Z} \right ){\it \_C1}\,{\it \_c}_{{1}}+2\,\AiryAi \left ( 1,{\it \_Z} \right ){\it \_c}_{{1}} \right ) +{{\it \_h}}^{2}}}{d{\it \_h}}+{\it \_C2} \right ) \right ) ^{-1}{d{\it \_f}}+x+{\it \_C3} \right ) } \] has RootOf

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36.13 Estevez Mansfield Clarkson equation

problem number 207

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

\[ \left \{\left \{u(x,y,t)\to \frac{\beta c_4(x)+6 c_1(x) \tanh \left (-4 t c_1(x){}^3+y c_1(x)+c_3(x)\right )}{\beta }\right \}\right \} \]

Maple

\[ u \left ( x,y,t \right ) =6\,{\frac{{\it \_C3}\,\tanh \left ( -4\,{{\it \_C3}}^{3}t+{\it \_C2}\,x+{\it \_C3}\,y+{\it \_C1} \right ) }{\beta }}+{\it \_C5} \]

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36.14 Fisher’s equation

problem number 208

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

\[ \left \{\left \{u(x,t)\to \frac{1}{4} \left (\tanh \left (-c_3+\frac{5 t}{12}-\frac{x}{2 \sqrt{6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to -\frac{1}{4} \left (-3+\tanh \left (-c_3+\frac{5 t}{12}-\frac{i x}{2 \sqrt{6}}\right )\right ) \left (1+\tanh \left (-c_3+\frac{5 t}{12}-\frac{i x}{2 \sqrt{6}}\right )\right )\right \},\left \{u(x,t)\to -\frac{1}{4} \left (-3+\tanh \left (-c_3+\frac{5 t}{12}+\frac{i x}{2 \sqrt{6}}\right )\right ) \left (1+\tanh \left (-c_3+\frac{5 t}{12}+\frac{i x}{2 \sqrt{6}}\right )\right )\right \},\left \{u(x,t)\to \frac{1}{4} \left (\tanh \left (-c_3+\frac{5 t}{12}+\frac{x}{2 \sqrt{6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to \frac{1}{4} \left (\tanh \left (c_3+\frac{5 t}{12}-\frac{x}{2 \sqrt{6}}\right )+1\right ){}^2\right \},\left \{u(x,t)\to -\frac{1}{4} \left (-3+\tanh \left (c_3+\frac{5 t}{12}-\frac{i x}{2 \sqrt{6}}\right )\right ) \left (1+\tanh \left (c_3+\frac{5 t}{12}-\frac{i x}{2 \sqrt{6}}\right )\right )\right \},\left \{u(x,t)\to -\frac{1}{4} \left (-3+\tanh \left (c_3+\frac{5 t}{12}+\frac{i x}{2 \sqrt{6}}\right )\right ) \left (1+\tanh \left (c_3+\frac{5 t}{12}+\frac{i x}{2 \sqrt{6}}\right )\right )\right \},\left \{u(x,t)\to \frac{1}{4} \left (\tanh \left (c_3+\frac{5 t}{12}+\frac{x}{2 \sqrt{6}}\right )+1\right ){}^2\right \}\right \} \]

Maple

\[ \left \{ \left \{ u \left ( x,t \right ) =1 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( -{\frac{5\,t}{12}}-1/12\,\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac{5\,t}{12}}-1/12\,\sqrt{6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ( -{\frac{5\,t}{12}}+1/12\,\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac{5\,t}{12}}+1/12\,\sqrt{6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac{5\,t}{12}}-i/12\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac{5\,t}{12}}-i/12\sqrt{6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac{5\,t}{12}}+i/12\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac{5\,t}{12}}+i/12\sqrt{6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ({\frac{5\,t}{12}}-1/12\,\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ({\frac{5\,t}{12}}-1/12\,\sqrt{6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =1/4\, \left ( \tanh \left ({\frac{5\,t}{12}}+1/12\,\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ({\frac{5\,t}{12}}+1/12\,\sqrt{6}x+{\it \_C1} \right ) +1/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ({\frac{5\,t}{12}}-i/12\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ({\frac{5\,t}{12}}-i/12\sqrt{6}x+{\it \_C1} \right ) +3/4 \right \} , \left \{ u \left ( x,t \right ) =-1/4\, \left ( \tanh \left ({\frac{5\,t}{12}}+i/12\sqrt{6}x+{\it \_C1} \right ) \right ) ^{2}+1/2\,\tanh \left ({\frac{5\,t}{12}}+i/12\sqrt{6}x+{\it \_C1} \right ) +3/4 \right \} \right \} \]

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36.15 Hunter Saxton

problem number 209

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) =2\,{\frac{\RootOf \left ( -{\it \_C2}\,{{\it \_c}_{{1}}}^{3}-x{{\it \_c}_{{1}}}^{3}-2\,{\it \_C1}\,\sqrt{{\it \_Z}}{\it \_c}_{{1}}+2\,{{\it \_C1}}^{2}\ln \left ( \sqrt{{\it \_Z}}{\it \_c}_{{1}}+{\it \_C1} \right ) +{\it \_Z}\,{{\it \_c}_{{1}}}^{2} \right ) }{{\it \_c}_{{1}}t+2\,{\it \_C3}} \left ({\frac{{\it \_c}_{{1}}t}{{\it \_c}_{{1}}t+2\,{\it \_C3}}}+2\,{\frac{{\it \_C3}}{{\it \_c}_{{1}}t+2\,{\it \_C3}}} \right ) ^{-1}} \] with RootOf

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36.16 Kadomtsev Petviashvili

problem number 210

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

\[ \left \{\left \{u(x,y,t)\to -\frac{12 c_3 c_1^3 \text{eps}^2 \tanh ^2\left (c_3 t+c_1 x+c_2 y+c_4\right )-8 c_3 c_1^3 \text{eps}^2+c_2^2 \lambda +c_3^2}{c_1 c_3}\right \}\right \} \]

Maple

\[ u \left ( x,y,t \right ) =-12\,{\epsilon }^{2}{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,y+{\it \_C4}\,t+{\it \_C1} \right ) \right ) ^{2}+{\frac{8\,{{\it \_C2}}^{4}{\epsilon }^{2}-{{\it \_C3}}^{2}\lambda -{\it \_C4}\,{\it \_C2}}{{{\it \_C2}}^{2}}} \]

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36.17 Klein Gordon (nonlinear)

problem number 211

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.18 special case Klein Gordon (nonlinear)

problem number 212

Added December 27, 2018.

Solve for \(u(x,y)\)

\[ u_{xx}+u_{yy}+ u^2=0 \]

Mathematica

\[ \text{Failed} \]

Maple

\[ u \left ( x,y \right ) =-6\,{\it WeierstrassP} \left ({\it \_C1}\,x+{\it \_C2}\,y+2\,{\it \_C3},0,{\it \_C4} \right ) \left ({{\it \_C1}}^{2}+{{\it \_C2}}^{2} \right ) \]

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36.19 Khokhlov Zabolotskaya

problem number 213

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,y,t \right ) ={\frac{{\it \_C3}\,{\it \_C1}-{{\it \_C2}}^{2}+\sqrt{2\, \left ({\it \_C1}\,x+{\it \_C2}\,y+{\it \_C3}\,t+{\it \_C4} \right ){{\it \_C1}}^{2}{\it \_C4}+{{\it \_C1}}^{2}{{\it \_C3}}^{2}-2\,{\it \_C1}\,{{\it \_C2}}^{2}{\it \_C3}+{{\it \_C2}}^{4}+2\,{{\it \_C1}}^{2}{\it \_C5}}}{{{\it \_C1}}^{2}}} \]

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36.20 Korteweg de Vries (KdV)

problem number 214

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

\[ \left \{\left \{u(x,t)\to \frac{-18 c_1 t x-18 c_2 t-9 c_1^2 x^2-18 c_1 c_2 x-9 c_2^2-c_1-9 t^2}{6 c_1^2}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-3/2\,{{\it \_C1}}^{2}+3\,{\it \_C1}\, \left ({\it \_c}_{{2}}t+x \right ) -3/2\, \left ({\it \_c}_{{2}}t+x \right ) ^{2}-1/6\,{\it \_c}_{{2}} \]

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36.21 Lin Tsien equation

problem number 215

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,y,t \right ) ={\it \_C4}+{\it \_C5}\, \left ({\it \_C1}\,x+{\it \_C2}\,y+{\it \_C3}\,t+{\it \_C4} \right ) \]

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36.22 Liouville equation

problem number 216

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.23 Plateau

problem number 217

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,y \right ) ={\it \_C7}\, \left ( \tanh \left ({\it \_C2}\,x-i{\it \_C2}\,y+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ({\it \_C2}\,x-i{\it \_C2}\,y+{\it \_C1} \right ) +{\it \_C4} \]

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36.24 Rayleigh

problem number 218

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,t \right ) =1/2\,{\it \_c}_{{1}}{x}^{2}+{\it \_C1}\,x+{\it \_C2}+\int \!\RootOf \left ( t+\int ^{{\it \_Z}}\! \left ({{\it \_f}}^{3}\epsilon -{\it \_f}\,\epsilon -{\it \_c}_{{1}} \right ) ^{-1}{d{\it \_f}}+{\it \_C3} \right ) \,{\rm d}t+{\it \_C4} \] Has RootOf

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36.25 Sawada Kotera

problem number 219

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

\[ \left \{\left \{u(x,t)\to -\frac{4}{3} c_1^2 \left (3 \tanh ^2\left (-16 c_1^5 t+c_1 x+c_3\right )-2\right )\right \},\left \{u(x,t)\to \frac{-30 c_1^{5/2} \tanh ^2\left (c_2 t+c_1 x+c_3\right )+20 c_1^{5/2}+\sqrt{5} \sqrt{4 c_1^5-c_2}}{15 \sqrt{c_1}}\right \},\left \{u(x,t)\to -\frac{30 c_1^{5/2} \tanh ^2\left (c_2 t+c_1 x+c_3\right )-20 c_1^{5/2}+\sqrt{5} \sqrt{4 c_1^5-c_2}}{15 \sqrt{c_1}}\right \}\right \} \]

Maple

\[ \left \{ \left \{ u \left ( x,t \right ) ={\it \_C4} \right \} , \left \{ u \left ( x,t \right ) =-4\,{{\it \_C2}}^{2} \left ( \tanh \left ( -16\,{{\it \_C2}}^{5}t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{2}+8/3\,{{\it \_C2}}^{2} \right \} , \left \{ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}-1/15\,{\frac{-20\,{{\it \_C2}}^{3}+\sqrt{20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}}}{{\it \_C2}}} \right \} , \left \{ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/15\,{\frac{20\,{{\it \_C2}}^{3}+\sqrt{20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}}}{{\it \_C2}}} \right \} \right \} \]

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36.26 Sine Gordon

problem number 220

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.27 Sinh Gordon

problem number 221

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.28 Sinh Poisson

problem number 222

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

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

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36.29 Thomas equation

problem number 223

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

\[ \text{Failed} \]

Maple

\[ u \left ( x,y \right ) =-1/2\,{\frac{\sqrt{{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }x}{\nu }}+1/2\,{\frac{\sqrt{{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }y}{\nu }}-1/2\,{\frac{\sqrt{{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }x}{\nu }}-1/2\,{\frac{\sqrt{{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }y}{\nu }}-{\frac{\alpha \,y}{\nu }}-{\frac{\beta \,x}{\nu }}-2\,{\frac{\ln \left ( 2 \right ) }{\nu }}-1/2\,{\frac{1}{\nu }\ln \left ({\frac{{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ({\it \_C3}\,{{\rm e}^{2\, \left ( x/2+y/2 \right ) \sqrt{{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C4} \right ) ^{2}}} \right ) }-1/2\,{\frac{1}{\nu }\ln \left ({\frac{{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ({\it \_C1}\,{{\rm e}^{2\, \left ( x/2-y/2 \right ) \sqrt{{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C2} \right ) ^{2}}} \right ) } \]

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36.30 phi equation

problem number 224

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

\[ \left \{\left \{\phi (x,t)\to -\tanh \left (c_2 t-\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t-\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to -\tanh \left (c_2 t+\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t+\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \}\right \} \]

Maple

\[ \left \{ \left \{ \phi \left ( x,t \right ) =-1 \right \} , \left \{ \phi \left ( x,t \right ) =1 \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( -1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( 1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( -1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( 1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} \right \} \]