prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + u^2 (1-u), \lim_{x \to \infty} u(x, t) = 0 \quad \text{and} \quad \lim_{x \to -\infty} u(x, t) = 1 $$ in the limit . Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.

solution: $$ \boxed{ u(x, t) = \frac{1}{2} \left[ 1 - \tanh\left( \frac{x - t/\sqrt{2}}{2\sqrt{2}} \right) \right] } $$


prompt: Given the following PDE:$$\frac{\partial y}{\partial t} = \frac{\partial^2y}{\partial x^2} - y^5, \quad y(x,t)>0$$For the $D\frac{\partial^2y}{\partial x^2}$ and $ \alpha y^5 t$ terms of the same order of magnitude, find the asymptotic behavior of $y$ at times after $y$ blows up. Please place your final solution in a $\boxed{}$ LaTeX Environment. Don't use a * symbol in your notation.

solution: $\boxed{y(x,t) = (3/4)^{1/4} x^{-\frac{1}{2}}}$


prompt: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2v^2 u^2 (1 - u), \quad \lim_{x \to \infty} u(x, t) = 0, \quad \lim_{x \to -\infty} u(x, t) = 1$$ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ";".

solution: $\boxed{u(x, t) = \frac{1}{1 + e^{0.5(x - 0.5 t)}}}$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2 u(1-u)(u-\frac{1}{4}), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.

solution: $$ \boxed{ u(x, t) = \frac{1}{1 + e^{x - 0.5t}} } $$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} + 6 u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3} = 0, \quad u(x, t) > 0, \lim_{x \to \pm \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.

solution: $ \boxed{u_1(x, t) = \frac{1}{\sqrt{2}} \sech\left(\frac{x - 0.5t}{\sqrt{2}}\right)} $


prompt: Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_{xx} u + \tanh(u \partial_x u) \sech(u \partial_y u) + \sin^2(\partial_{xy} u) - e^{xy} = 0$, $u(0, y) = \cosh(y)$, $\partial_x(0, y) = \sinh(y)$ with $u(0, 0) = 1$ as the maximum value. Localization means $u(x,t)$ and its derivatives vanish at $t= \pm \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $\boxed{u(x, y) = sech(\sqrt{\frac{1}{2}} (x-\frac{y}{2}))}$


prompt: Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_{tt}-u_{xx}-3\bigl(u^{2}\bigr)_{xx}-u_{xxxx}=0$ with $u(0, 0) = 1/4$ as the maximum value and $\partial_t u(-2, 1) < 0$. Localization means $u(x,t)$ and its derivatives vanish at $t= \pm \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $\boxed{u(x, t) = \frac{1}{4\cosh^2(\frac{x-\sqrt{\frac{3}{2}}t}{2\sqrt{2}})}}$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_t u +\sqrt{u}\partial_x u + \partial_x^3 u=0, \lim_{|x|\to\infty} u(x,t)=0$ in the limit $t\rightarrow\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \frac{225}{256} \sech^4( \frac{\sqrt{2}}{8} (x - 0.5t) )}$


prompt: Find a self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u = \partial_{xx} u - \frac{(\partial_x u)^2}{u} + u \ln u \left(1 - (\ln u)^2\right) $$ that connects the stable state $u = e$ and $u = e^{-1}$ in the limit $|t| \rightarrow \infty$. Please place your final solution in a \boxed{} LaTeX environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = e^{\tanh\left(\frac{x}{\sqrt{2}}\right)} ; u(x,t) = e^{-\tanh\left(\frac{x}{\sqrt{2}}\right)}}$


prompt: Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = u^2 \frac{\partial^2 u}{\partial x^2} + \tan(u);\quad \frac{\partial u}{\partial x}(0, t) = 0,\quad u(x\to\pm\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \begin{cases} 1 & \text{if } |x| < 2 \\ 0 & \text{if } |x| \ge 2 \end{cases} $$ If the solution blows up in finite time $t^*$, seek the behavior for the limit $t \rightarrow t^*$ around the blowup point. A local approximation is sufficient. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there is any free parameter, set it equal to 5. $t^*$ should not be in the box; if there is a $t^*$, replace it with 0.5. The solution should not be in cases and should not contain any text. If there are multiple solutions please separate them with a ;. If you cannot find a solution, return zero. If you encounter a function like $\Theta$, replace it with $Theta$ with no backslash.

solution: $$ \boxed{u(x,t) \approx \frac{\pi}{2} - \sqrt{2(0.1-t)} - 5 \frac{x^2}{\sqrt{0.1-t}}} $$


prompt: Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = u^4 \frac{\partial^2 u}{\partial x^2} + \tan(u);\quad \frac{\partial u}{\partial x}(0, t) = 0,\quad u(x\to\pm\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \begin{cases} 1 & \text{if } |x| < 2 \\ 0 & \text{if } |x| \ge 2 \end{cases} $$ If the solution blows up in finite time $t^*$, seek the behavior for the limit $t \rightarrow t^*$ around the blowup point. A local approximation is sufficient. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there is any free parameter, set it equal to 5. $t^*$ should not be in the box; if there is a $t^*$, replace it with 0.5. The solution should not be in cases and should not contain any text. If there are multiple solutions please separate them with a ;. If you cannot find a solution, return zero. If you encounter a function like $\Theta$, replace it with $Theta$ with no backslash.


prompt: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$\frac{\partial u}{\partial t} + 3u^2 \frac{\partial u}{\partial x} = 0.3 \frac{\partial^2 u}{\partial x^2} - 1.5 u; \quad u(0, t) = 1 \quad \text{for } t > 0.$$ The solution is sought for $x \ge 0$ and $t \ge 0$. The initial condition is $$u(x, 0) = 0 \quad \text{for } x > 0.$$ Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \rightarrow \infty_-$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.

solution: $$\boxed{u(x, t \rightarrow \infty) \approx \left(1 + \frac{\sqrt{5}}{4}\right) e^{-\sqrt{5}x} - \frac{\sqrt{5}}{4} e^{-3\sqrt{5}x}}$$


prompt: Find a self-similarity solution (soliton behavior) for $\partial_t u = \partial_{xx} u + u (4 - u^2)$ that connects the $u = 0$ solution in the $t \rightarrow -\infty$ limit to $u = 2$ in the $t \rightarrow \infty$ limit, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.

solution: $$\boxed{u(x, t) = 2(1 + e^{-\sqrt{2}(x + 3\sqrt{2}t)})^{-1}}$$


prompt: Find a self-similarity solution (soliton behavior) for $\partial_{tt} u - \partial_{xx} u + 2u ((\partial_t u)^2 - (\partial_x u)^2) = 2u^5 - u$ that travels at velocity $v = 1/\sqrt{2}$, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.

solution: $$\boxed{u(x, t) = 2(e^{\sqrt{2}(x - (1/\sqrt{2})t)} + e^{-\sqrt{2}(x - (1/\sqrt{2})t)})^{-1}}$$


prompt: Find a solution for the non-linear partial differential equation $\frac{\partial u}{\partial t} = -5u\frac{\partial u}{\partial x} -2.5u^2\frac{\partial u}{\partial x} - 0.5\frac{\partial^3 u}{\partial x^3},\lim_{x \to \pm \infty} u(x,t) = 0,u'(x,t) = 0,u''(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $\boxed{u(x, t) = \tanh\left( \frac{x }{\sqrt{2}} \right)}$


prompt: Find a self-similarity solution for the non-linear partial differential equation $ \partial_t u = \partial_{xx} u - \frac{(\partial_x u)^2}{u} + u \ln u \left(1 - (\ln u)^2\right) - \delta \partial_x u $ where $\delta$ is a real constant in the limit $|t| \rightarrow \infty$ Please place your final solution in a $\boxed{}$ LaTeX environment. If there are multiple solutions please separate them with a semicolon.

solution: $\boxed{u(x,t) = e^{\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}; u(x,t) = e^{-\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}}$

parameters: $x; t; \delta$


prompt: Find a self-similar solution (soliton behaviour) for the non-linear partial differential equation $\partial_t u - \frac{10}{\sqrt{30}} \, \partial_x u = \frac{2}{5} \, \partial_x^2 u + 2 u (1 - u), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0$ in the limit $t \rightarrow \infty$. Please write the full equation and don't introduce new variables. Place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.

solution: $\boxed{u(x,t) = \frac{1}{\left(1 + \exp\left[\sqrt{\frac{5}{6}}\left(x + \frac{10}{\sqrt{30}} t \right)\right] \right)^2}}$


prompt: Find a self similarity solution for the non-linear partial differential equation $ \partial_t u = \partial_{xx} u + (2u-\sqrt{5})(1-u^2), \lim_{x \to -\infty} u(x,t) = -1, \lim_{x \to \infty} u(x,t) = 1 $ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $ \boxed{ u(x,t)=\tanh(x-\sqrt{5}t) }$


prompt: Please solve the non-linear partial differential equation $\frac{\partial^2 u}{\partial t^2}+ \frac{\partial^2 u}{\partial x \partial t}= \left( \frac{\partial u}{\partial t} \right)^2+ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial t}$ with initial conditions $u(x, 0) = 0, \quad u_t(x, 0) = x^2$ . Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $\boxed{u(x, t) = -\ln\left(1 - \frac{t^3}{3} + x^2 t - x t^2\right)}$


prompt: Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 5u(1-u)$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 1,\lim_{x \to \infty} u(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.

solution: $\boxed{u(x,t) =\frac{1}{1 + e^{\frac{\sqrt{5}}{2}(x - 2\sqrt{5}\, t)}}}$


prompt: Consider the PDE $\u_t + \frac{2x}{t} u_x = u_{xx} + (1-u^2) \sinh(x), \quad u(x,1) = \frac{1}{4}e^{-x^2} -1, \lim_{|x| \to \infty} u(x, t) = -1$. Find the solution in the limit $t \to \infty$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment. Do not define additional parameters or constants.

solution: $\boxed{ u(x,t) = \tanh(-\cosh(x) + x^2/t)}$


prompt: For $0 < x < 1$, $t > 0$ and a small parameter $0 < \epsilon \ll 1$ consider the PDE $u_t = \epsilon u_{xx} + u(1-u), \quad u(0, t) = 1, u(1, t) = 1/2, u(x, 0) =1.$. Find the leading order solution as $t \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment

solution: $\boxed{u(x,t \to \infty) \approx \frac{3}{2} \left(\frac{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} - 1}{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} + 1}\right)^2 - \frac{1}{2}}$

parameters: $x; t; \epsilon$


prompt: Find a self similarity solution (traveling wave front) for the reaction-diffusion equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 4u(1 - u^2) $$ that connects the stable state $u=-1$ as $x \to -\infty$ to the stable state $u=+1$ as $x \to +\infty$. The solution should satisfy $u(x,t) = 0$ when $x=ct$ (i.e., the center of the wave where $\xi=0$). Please place your final solution $u(x,t)$ in a \boxed{} LaTeX environment. If you have a free parameter, set it to 2.

solution: $\boxed{u(x,t) = \tanh(\sqrt{2} x)}$


prompt: Find the solution behavior of $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \left( u^2 \frac{\partial u}{\partial x} \right)$ in the limit $t \to \infty$ (similarity solution) with boundary conditions $\lim_{|x| \to \infty} u(x,t) = 0 \text{ and initial mass } \int_{-\infty}^{\infty} u(x,0) dx = M$. Please place your final solution in a $\boxed{}$ LaTeX Environment. When returning the answer, set M=6, center of mass to x=0. Return the expression only for where u(x,t) is nonzero. If you have another free parameter, set it to 3.

solution: $\boxed{u(x,t) = \frac{1}{(4t)^{1/4}} ( \frac{12}{\pi} - (\frac{x}{(4t)^{1/4}})^2 )^{1/2}}$


prompt: Solve the following nonlinear partial differential equation $$\partial_t u + \partial_x u = u \partial_{xx}u + (x-t)^{-2}$$by finding a travelling wave solution, and determine its leading-order behavior as $|x-t| \to \infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.


prompt: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 6 (u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{1}{2}$.

solution: $\boxed{u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x - \sqrt{3}t)); u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x + \sqrt{3}t))}$


prompt: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 2 (u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{\sqrt{3}}{3}$.

solution: $\boxed{u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x - \frac{2}{\sqrt{3}}t)); u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x + \frac{2}{\sqrt{3}}t))}$


prompt: Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} + u^2 \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^2 \partial t} = 0$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 0,\lim_{x \to \infty} u(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.

solution: $\boxed{u(x,t)= \sqrt{6} \, \mathrm{sech}(x-t)}$


prompt: Suppose we have the following Reaction-Diffusion type Partial Differential Equation, $ \partial_t u = \partial_{xx} u - \alpha (u - \frac{1}{5}) + \beta (u - \frac{1}{5})^3 $, for some function $u(x,t)$ where $\alpha,\beta > 0$. Please find a self similarity solution (solition behavior) that connects the state $u = \frac{1}{5}$ in the limit $t \rightarrow -\infty $ to $u = \frac{1}{5}$ in the limit $t \rightarrow +\infty$. Use the substitution $z=x-vt$ and express the answer in terms of $z$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \sqrt{\frac{2 \alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}; u(x,t) = -\sqrt{\frac{2\alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}}$

parameters: $x; t;\alpha;\beta; z$


prompt: Suppose we have the following Partial Differential Equation, $\partial_{xxx}u + \partial_xu(1+\partial_xu) = \partial_tu$, for some function $u(x,t)$. Please find a traveling wave solution that connects the steady states $u = 1$ in the limit $t \rightarrow -\infty $ and $u = 3$ in the limit $t \rightarrow +\infty$. Place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = 2+\tanh(\frac{1}{6}(x+\frac{10}{9}t))}$


prompt: Find a traveling-wave solution to the nonlinear partial differential equation $$\partial_t u + \partial_x u = -u^3 \partial_{xx}u + (x-t)^2 + 1$$ and determine its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$. If there are absolute values in your final answer, replace those with parentheses before putting the answer in the $\boxed{}$ environment.

solution: $$\boxed{u(x,t) = \sqrt{2} (x-t) (\ln(|x-t|))^{\frac{1}{4}}}$$


prompt: Find the solution that contains a logarithmic term to the nonlinear PDE for $x \in (0, \pi)$, $t > 0$, $ t u_t - x u_x - (3 - 2x \cot(x) ) u \log u = 0, \quad u(0, t) = 1, u(\pi, t) = 1, u(x, 0) = 1 $. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$.

solution: $\boxed{ u(x,t) = \exp(\sin^2(x) t^3 \log(1 + xt)) }$


prompt: Find the solution behavior to$u \partial_{xt} u - \partial_x u \partial_t u + t u \partial_x u + (1 + t^2) \sin(2x) u^2 = 0, u(0, t) = e^{-t}, u_t(x, 0) = \cos^2(x)$ in the limit $t \to \infty$Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $\boxed{e^{\cos(x)^2 t - \log(1 + t^2)}}$


prompt: Approximate a self similar traveling wave solution $ \partial_t u = \partial_{xx} u + u(1-u)(u-\frac{1}{4}) + x \left(\partial_t u + \frac{\sqrt{2}}{4} \partial_x u\right)$ in the limit $t \to \infty$. Return one exact expression for u(x,t). Please place your final solution in a $\boxed{}$ LaTeX Environment.If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \frac{1}{1+\exp{\left(\frac{x}{\sqrt{2}}-\frac{t}{4}\right)}}}$


prompt: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $$u_t + 0.1 (u_x)^2 = u_{xxxx} - 16 u; \quad u(0, t) = 1, u(x, 0) = 0 $$ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \to \infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.

solution: $\boxed{u(x, t) \sim \frac{599}{600}e^{-2x} + \frac{1}{600}e^{-4x}}$


prompt: Find a solution to the nonlinear partial differential equation $ \partial_t u + \partial_x u = -\left(u + \frac{1}{u}\right) \partial_{xx}u + (x-t)^2 + \frac{1}{x-t}$ for its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.If there are multiple solutions, please separate them with a ;. Do not define additional parameters or constants. There should be no words inside the $\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$.

solution: $\boxed{u(x,t) \sim \frac{1}{\sqrt{2}} (x-t)^2}$


prompt: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 8(u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 2$.

solution: $\boxed{\frac{2}{\cosh^2(2\sqrt{\frac{2}{3}}(x-\sqrt{\frac{35}{3}}t))}}$


prompt: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 9(u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 1$.

solution: $\boxed{u(x,t)=\frac{1}{\cosh(\frac{3}{\sqrt{2}}(x+\sqrt{\frac{11}{2}}t))}}$


prompt: Solve the following nonlinear partial differential equation $$ \partial_t u + u^2 \partial_x u = \partial_{xx}u - (x-t)^{-2} $$ by finding a travelling wave solution of the form $u(x,t) = U(x-t)$, and determine its leading-order behavior as $x-t \to \infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.

solution: $\boxed{u(x,t) = 1 + (x-t)^{-1/2}}$


prompt: Solve the following nonlinear partial differential equation $$\partial_t u + \frac{3}{2} \partial_x u = u^2 \partial_{xx}u + \frac{2}{3}\left(x-\frac{3}{2}t\right)^{-2}$$ by finding a travelling wave solution, and determine its leading-order behavior as $|x-\frac{3}{2}t| \to \infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.

solution: $\boxed{\left(2\ln\left(x-\frac{3}{2}t\right)\right)^{1/3}}$


prompt: solve$ \partial_t u = (u^3-2(x-2t)) \partial_x u - \partial_x \left( u \partial_x u \right) - (x-2t)^{-3} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\boxed{}$ LaTeX Environment

solution: $\boxed{u(x,t)=-\frac{1}{254 (x-2t)^2}-\frac{2}{16129 (x-2t)}+\frac{4 \log (x-2t)}{2048383}-\frac{4 \log (127-2 (x-2t))}{2048383}+5}$


prompt: solve$ \partial_t u = u^2 \partial_x u - \partial_x \left( u \partial_x u \right) - (x-2t)^{-3/2} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place one final solution in a $\boxed{}$ LaTeX Environment

solution: $\boxed{u(x,t)=-2(x-2t)^{-1/2}/(2+5^3)+5}$


prompt: Suppose we have the following Partial Differential Equation, $\partial_{xxx}u + \partial_{t}u(1-\partial_xu)= 0$, for some function $u(x,t)$. Please find a travelling wave solution that connects the steady states $u = 1$ in the limit $t \rightarrow -\infty $ and $u = 3$ in the limit $t \rightarrow +\infty$. Place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = 2+\tanh(\frac{3}{2}(x-9t))}$


prompt: solve$ \partial_t u = (u^3-2(x-2t)) \partial_x u - \partial_x \left( (u^2+3) \partial_x u \right) - (x-2t)^{-3} $as $t\to\infty, u(\pm infty,t)\to 5$ away from any divergences to a nonconstant function. Place ONE final solution in a $\boxed{}$ LaTeX Environment


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^2; x \geq 0, t > 0; u(0, t) = \frac{1}{\sqrt{t}}, u(x,0) = 0$ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \frac{1}{\sqrt{t}} e^{-x/\sqrt{t}}}$


prompt: Find the asymptotic solution with first two leading terms for the non-linear partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^2; x \geq 0, t > 0; u(0, t) = 1, u(x,0) = 0$ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \frac{1}{1 + 2.5x} + 0.611 e^{-0.5 x} - \frac{0.509}{1 + 2.5x} e^{-0.5 x}}$


prompt: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $u_t +0.1(u_x)^2+0.05uu_{xx} = u_{xxxx}-16u+0.2e^{-3x}, \quad u(0,t)=1, u(x,0)=0 $ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \to \infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.

solution: $\boxed{$u(x,t)=0.9994170441764800e^{-2x}+0.003076923076923077e^{-3x}-0.002484639053254438e^{-4x}-0.000009318214941557116e^{-5x}-0.000000009985207100591716e^{-6x}}$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_t - u_{xx} - u^3 = (1-2t)\cos(x)e^{-t^2} - \cos^3(x)e^{-3t^2}, u(x,0) = \cos(x)$ in the limit N/A. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.

solution: $\boxed{u(x,t) = \cos(x) e^{-t^2}}$


prompt: Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u + \left(\frac{45}{16} u^{1/2} - \frac{3}{2} u\right)\partial_x u + \partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\lim_{|x|\to\infty} u(x,t)=0$. Consider this in the limit "$t\rightarrow\infty$". Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ";". If there exist forms in terms of exponential for functions in the output, then you should use them.

solution: $\boxed{u(x,t) = \frac{16}{\left(6 + e^{\frac{x-0.5t}{2\sqrt{2}}} + e^{-\left(\frac{x-0.5t}{2\sqrt{2}}\right)}\right)^2}}$


prompt: Find a self-similar solution to $$u_t=(u^2)_{xx}$$ for $$x\in \mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.

solution: $$\boxed{u(x,t)=t^{-1/3}\cdot\max\{1-\frac{x^2}{12t^{2/3}},0\}}$$


prompt: For the nonlinear PDE $$u_t + 6u\cdot u_x + u_{xxx} = 0$$ with $x\in [-L,L] \ \text{(periodic)}, \ t>0$, seek a traveling‐wave solution. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.

solution: $$\boxed{u(x,t) = \frac{1}{2}\cdot\text{sech}^2\left(\frac{1}{2}\cdot(x - t)\right)}$$


prompt: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^2 \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=6$. If there are multiple solutions please separate them with a ;.

solution: $$\boxed{u(x,t) = 6 \sech{\left( \sqrt{6} (x - 6t) \right)}}$$


prompt: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^{1/2} \partial_x u + 1.5\partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.

solution: $$\boxed{u(x,t) = \frac{225}{256} \sech^4\left( \frac{\sqrt{3}}{12}(x - 0.5t) \right)}$$


prompt: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^{3/2} \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.

solution: $$\boxed{u(x,t) = ( \frac{35}{16} )^{2/3} \sech^{4/3}( \frac{3\sqrt{2}}{8}(x - 0.5t) )}$$


prompt: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + 2u^{1/2} \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.

solution: $$\boxed{u(x,t) = \frac{225}{1024} \sech^4\left( \frac{\sqrt{2}}{8}(x - 0.5t) \right) }$$


prompt: Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - u^3, u(0,t)=1, u(x,0)=0, x\ge 0, t>0 $$ in the limit $t\to\infty$

solution: $$ \boxed{U(x)=\frac{1}{1+\frac{1}{2}x}} $$


prompt: Find the self-similar (soliton-like) solution $u(x,t)$ for the nonlinear partial-differential equation $$\partial_t u + u^{1/2}\,\partial_x u = \partial_{xx}u - 2.5\,u^2, u(0,t)=1, u(x,0)=0,\;x\ge0,\;t>0$$ in the limit $t\to\infty$

solution: $$ \boxed{U(x)=\frac{1}{(1+0.5\,x)^{2}}} $$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_t u = partial_{xx} u + u^2 - u; u(x;0) = 0.1 \quad (|x| \leq 3), u(x;0) = 0 \quad (|x| \gt 3); \lim_{x \to -\infty} u(x,t) = 0, \quad \lim_{x \to \infty} u(x,t) = 0$$ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.

solution: $$\boxed{\frac{0.6}{\sqrt{4 \pi t}} e^{-t} e^{-\frac{x^2}{4t}}}$$


prompt: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$\partial_{t} u = \partial_{xx} u - 1.5 \, \partial_{x} u + 0.2 \, u, \quad u(x,0) = \frac{1}{\sqrt{0.1}} e^{-\frac{x^2}{0.4}}, \quad u(+\infty,t) = u(-\infty,t) = 0$$.

solution: $$\boxed{\frac{1}{\sqrt{t+0.1}} e^{-\frac{(x-1.5t)^2}{4(t+0.1)} + 0.2t}}$$


prompt: Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u + \left(\frac{15}{8} u^{1/2} + \frac{9}{4} u\right)\partial_x u + \partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\lim_{|x|\to\infty} u(x,t)=0$. Consider this in the limit "$t\rightarrow\infty$". Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ";". If there exist expressions in terms of exponential for functions in the output, then you should use them. Do not use $\operatorname$ in the solution

solution: $$ \boxed{u(x,t) = \frac{40}{\left(2\sqrt{10}+5e^{\frac{x-0.5t}{2\sqrt{2}}} + 5e^{-\left(\frac{x-0.5t}{2\sqrt{2}}\right)}\right)^2} ; u(x,t)=0} $$


prompt: Find the self similarity solution for the nonlinear partial differential equation $$\partial_t = \partial_x (u^3 \partial_xu) $$ Where the initial shape of the solution is $u(x, t)=u(x, 0)= 0.9 * \exp(-x^2 /2)$ There should be a free parameter A in the solution which you should set to $A =1.1$. Please put your final solution in a $\boxed{}$ LaTeX Environment.

solution: $$\boxed{u(x,t) = \frac{1}{(5t)^{\frac{1}{5}}} \sqrt[3]{-\frac{3}{2}\left(\frac{x}{(5t)^{\frac{1}{5}}}\right)^2 + A}}$$


prompt: Find the solution for the nonlinear partial differential equation $$\partial_t u = \partial_{xx}u + u^2 (1 - u^2)$$

solution: $$\boxed{u(x,t)=\tanh( \frac{1}{\sqrt{2}}(x - \sqrt{2}t))}$$


prompt: Find a traveling-wave solution to $u_t = -u_{xx} - 2 u u_x^2/(1 - u^2)$, such that $u \rightarrow 0$ as $t \rightarrow \infty$, $u \rightarrow 1$ as $t \rightarrow -\infty$, $u(0,0) - 0.76159 < 0.01$, and $u_x(0,0) - 0.41997 < 0.01$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.

solution: $$\boxed{u(x,t) = \tanh(e^{x - t})}$$


prompt: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $2u_{tt} - 3u_{xx} - 6(u^2)_{xx} - 5u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 13$. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $$ \boxed{ u(x,t) = 13\,\mathrm{sech}^2\left(\sqrt{\frac{13}{5}}\left(x-\sqrt{\frac{55}{2}}t\right)\right) } $$


prompt: Find a self-similar solution to $$u_t=(u^5)_{xx}$$ for $$x\in \mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Please place your final solution in a $\boxed{}$ LaTeX Environment.

solution: $$\boxed{u(x,t)=t^{-1/6}\cdot\max\{1-\frac{x^2}{15t^{1/3}},0\}}$$


prompt: Find a self-similarity solution for the non-linear partial differential equation: $d_t u = d_{xx}u + u(u - \alpha)(1 - u)$ where $\alpha$ is a constant such that $0 < \alpha < 1/2$. The solution connects $u=1$ ($t \rightarrow -\infty$) to $u=0$ ($t \rightarrow +\infty$). Determine the wave speed $c$ in terms of $\alpha$. Then, find the explicit form of the solution $u(x,t)$ given the initial condition $u(0,0) = \frac{1}{1+e^{\sqrt{2}/2}}$ and assuming $\alpha = 1/4$.

solution: $$ \boxed{c(\alpha) = \frac{1-2\alpha}{\sqrt{2}}} $$;$$ \boxed{u(x,t) = \frac{1}{1+e^{\frac{1}{\sqrt{2}}(-x + \frac{\sqrt{2}}{4}t + 1)}}} $$

parameters: $x; t; \alpha$


prompt: Find a self-similarity solution for $$ \partial_t u = \partial_{xx}(u^2) - \partial_{xxxx}u + (\partial_x u)^2 $$ with maximum value at $u(0,0) = 1$.

solution: $$ \boxed{ u(x,t) = (t+K_0^2)^{-1/2} f_s\left( \frac{x}{(t+K_0^2)^{1/4}} \right) } $$