33 Nonlinear PDE’s

 33.1 Bateman-Burgers equation
 33.2 Benjamin Bona Mahony
 33.3 Benjamin Ono
 33.4 Born Infeld
 33.5 Boussinesq
 33.6 Boussinesq type PDE
 33.7 Buckmaster
 33.8 Camassa Holm
 33.9 Chaffee Infante equation
 33.10 Clarke’s equation
 33.11 Degasperis Procesi
 33.12 Dym equation
 33.13 Estevez Mansfield Clarkson equation
 33.14 Fisher’s equation
 33.15 Hunter Saxton
 33.16 Kadomtsev Petviashvili
 33.17 Klein Gordon (nonlinear)
 33.18 special case Klein Gordon (nonlinear)
 33.19 Khokhlov Zabolotskaya
 33.20 Korteweg de Vries (KdV)
 33.21 Lin Tsien equation
 33.22 Liouville equation
 33.23 Plateau
 33.24 Rayleigh
 33.25 Sawada Kotera
 33.26 Sine Gordon
 33.27 Sinh Gordon
 33.28 Sinh Poisson
 33.29 Thomas equation
 33.30 phi equation

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33.1 Bateman-Burgers equation

problem number 179

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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33.2 Benjamin Bona Mahony

problem number 180

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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33.3 Benjamin Ono

problem number 181

Added December 27, 2018.

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

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

\[ u_t+H u_{xx} +u u_x = 0 \]

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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33.4 Born Infeld

problem number 182

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

\[ u \left ( x,t \right ) ={\it \_C7}\, \left ( \tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{3}+{\it \_C5}\,\tanh \left ( -{\it \_C2}\,t+{\it \_C2}\,x+{\it \_C1} \right ) +{\it \_C4} \] Mathematica’s solution is simpler. Verified both correct.

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33.5 Boussinesq

problem number 183

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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33.6 Boussinesq type PDE

problem number 184

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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33.7 Buckmaster

problem number 185

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

\[ \text{DSolve}\left [u^{(0,1)}(x,t)=4 u^{(2,0)}(x,t) u(x,t)^3+12 u^{(1,0)}(x,t)^2 u(x,t)^2+3 u^{(1,0)}(x,t) u(x,t)^2,u(x,t),\{x,t\}\right ] \]

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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33.8 Camassa Holm

problem number 186

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

\[ \text{DSolve}\left [2 k u^{(1,0)}(x,t)+u^{(0,1)}(x,t)+3 u(x,t) u^{(1,0)}(x,t)-u^{(2,1)}(x,t)=2 u^{(1,0)}(x,t) u^{(2,0)}(x,t)+u(x,t) u^{(3,0)}(x,t),u(x,t),\{x,t\}\right ] \]

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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33.9 Chaffee Infante equation

problem number 187

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

\[ \text{DSolve}\left [\lambda \left (u(x,t)^3-u(x,t)\right )+u^{(0,1)}(x,t)-u^{(2,0)}(x,t)=0,u(x,t),\{x,t\}\right ] \]

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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33.10 Clarke’s equation

problem number 188

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

\[ \text{DSolve}\left [\gamma \left (-e^{\theta (x,t)} \theta ^{(0,1)}(x,t)^2-e^{\theta (x,t)} \theta ^{(0,2)}(x,t)\right )+\theta ^{(0,3)}(x,t)=-e^{\theta (x,t)} \theta ^{(1,0)}(x,t)^2-e^{\theta (x,t)} \theta ^{(2,0)}(x,t)+\theta ^{(2,1)}(x,t),\theta (x,t),\{x,t\}\right ] \]

Maple

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

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33.11 Degasperis Procesi

problem number 189

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

\[ \text{DSolve}\left [u^{(0,1)}(x,t)+4 u(x,t) u^{(1,0)}(x,t)-u^{(2,1)}(x,t)=3 u^{(1,0)}(x,t) u^{(2,0)}(x,t)+u(x,t) u^{(3,0)}(x,t),u(x,t),\{x,t\}\right ] \]

Maple

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

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33.12 Dym equation

problem number 190

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

\[ \text{DSolve}\left [u^{(0,1)}(x,t)=u(x,t)^3 u^{(3,0)}(x,t),u(x,t),\{x,t\}\right ] \]

Maple

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

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33.13 Estevez Mansfield Clarkson equation

problem number 191

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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33.14 Fisher’s equation

problem number 192

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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33.15 Hunter Saxton

problem number 193

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

\[ \text{DSolve}\left [u^{(1,0)}(x,t)^2+u^{(1,1)}(x,t)+u(x,t) u^{(2,0)}(x,t)=\frac{1}{2} u^{(1,0)}(x,t)^2,u(x,t),\{x,t\}\right ] \]

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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33.16 Kadomtsev Petviashvili

problem number 194

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

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

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33.17 Klein Gordon (nonlinear)

problem number 195

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

\[ \text{DSolve}\left [\lambda u(x,y)^p+u^{(0,2)}(x,y)+u^{(2,0)}(x,y)=0,u(x,y),\{x,y\}\right ] \]

Maple

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

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33.18 special case Klein Gordon (nonlinear)

problem number 196

Added December 27, 2018.

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

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

Mathematica

\[ \text{DSolve}\left [u^{(0,2)}(x,y)+u^{(2,0)}(x,y)+u(x,y)^2=0,u(x,y),\{x,y\}\right ] \]

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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33.19 Khokhlov Zabolotskaya

problem number 197

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

\[ \text{DSolve}\left [-u^{(1,0,0)}(x,y,t)^2+u^{(1,0,1)}(x,y,t)-u(x,y,t) u^{(2,0,0)}(x,y,t)=u^{(0,2,0)}(x,y,t),u(x,y,t),\{x,y,t\}\right ] \]

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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33.20 Korteweg de Vries (KdV)

problem number 198

Added December 27, 2018.

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

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

\[ u_t + (u_x)^3+ 6 u u_x = 0 \]

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

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

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33.21 Lin Tsien equation

problem number 199

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

\[ \text{DSolve}\left [-u^{(0,2,0)}(x,y,t)+2 u^{(1,0,1)}(x,y,t)+u^{(1,0,0)}(x,y,t) u^{(2,0,0)}(x,y,t)=0,u(x,y,t),\{x,y,t\}\right ] \]

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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33.22 Liouville equation

problem number 200

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

\[ \text{DSolve}\left [e^{\text{lam} u(x,y)}+u^{(0,2)}(x,y)+u^{(2,0)}(x,y)=0,u(x,y),\{x,y\}\right ] \]

Maple

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

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33.23 Plateau

problem number 201

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

\[ \text{DSolve}\left [u^{(0,2)}(x,y) \left (u^{(1,0)}(x,y)^2+1\right )-2 u^{(0,1)}(x,y) u^{(1,0)}(x,y) u^{(1,1)}(x,y)+\left (u^{(0,1)}(x,y)^2+1\right ) u^{(2,0)}(x,y)=0,u(x,y),\{x,y\}\right ] \]

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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33.24 Rayleigh

problem number 202

Added December 27, 2018.

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

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

\[ u_{tt} - u_{xx} = \epsilon (u_t - u_t^3) \]

Mathematica

\[ \text{DSolve}\left [u^{(0,2)}(x,t)-u^{(2,0)}(x,t)=\epsilon \left (u^{(0,1)}(x,t)-u^{(0,1)}(x,t)^3\right ),u(x,t),\{x,t\}\right ] \]

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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33.25 Sawada Kotera

problem number 203

Added December 27, 2018.

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

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

\[ u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \]

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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33.26 Sine Gordon

problem number 204

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

\[ \text{DSolve}\left [\phi ^{(0,2)}(x,t)-\phi ^{(2,0)}(x,t)+\sin (\phi (x,t))=0,\phi (x,t),\{x,t\}\right ] \]

Maple

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

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33.27 Sinh Gordon

problem number 205

Added December 27, 2018.

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

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

\[ u_{xt} = \sinh u \]

Mathematica

\[ \text{DSolve}\left [u^{(1,1)}(x,t)=\sinh (u(x,t)),u(x,t),\{x,t\}\right ] \]

Maple

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

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33.28 Sinh Poisson

problem number 206

Added December 27, 2018.

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

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

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

Mathematica

\[ \text{DSolve}\left [u^{(0,2)}(x,y)+u^{(2,0)}(x,y)+\sinh (u(x,y))=0,u(x,y),\{x,y\}\right ] \]

Maple

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

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33.29 Thomas equation

problem number 207

Added December 27, 2018.

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

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

\[ u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0 \]

Mathematica

\[ \text{DSolve}\left [\alpha u^{(1,0)}(x,y)+\beta u^{(0,1)}(x,y)+\nu u^{(1,0)}(x,y) u^{(0,1)}(x,y)+u^{(1,1)}(x,y)=0,u(x,y),\{x,y\}\right ] \]

Maple

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

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33.30 phi equation

problem number 208

Added December 27, 2018.

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

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

\[ \phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0 \]

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

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