### 33 Nonlinear PDE’s

33.3 Benjamin Ono
33.4 Born Infeld
33.5 Boussinesq

33.7 Buckmaster
33.8 Camassa Holm

33.10 Clarke’s equation
33.11 Degasperis Procesi
33.12 Dym equation

33.14 Fisher’s equation
33.15 Hunter Saxton

33.21 Lin Tsien equation
33.22 Liouville equation
33.23 Plateau
33.24 Rayleigh
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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problem number 194

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

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

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

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

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

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

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

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

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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problem number 203

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

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

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

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

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

Solve for $$u(x,t)$$
$\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0$
$\left \{\left \{\phi (x,t)\to -\tanh \left (c_2 t-\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t-\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to -\tanh \left (c_2 t+\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \},\left \{\phi (x,t)\to \tanh \left (c_2 t+\frac{\sqrt{2 c_2^2+1} x}{\sqrt{2}}+c_3\right )\right \}\right \}$
$\left \{ \left \{ \phi \left ( x,t \right ) =-1 \right \} , \left \{ \phi \left ( x,t \right ) =1 \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( -1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =-\tanh \left ( 1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( -1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} , \left \{ \phi \left ( x,t \right ) =\tanh \left ( 1/2\,\sqrt{4\,{{\it \_C2}}^{2}-2}t+{\it \_C2}\,x+{\it \_C1} \right ) \right \} \right \}$