| {"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": "$$ \\boxed{ u(x, t) = \\frac{1}{2} \\left[ 1 - \\tanh\\left( \\frac{x - t/\\sqrt{2}}{2\\sqrt{2}} \\right) \\right] } $$", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 0} | |
| {"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": "$\\boxed{y(x,t) = (3/4)^{1/4} x^{-\\frac{1}{2}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 1} | |
| {"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": "$\\boxed{u(x, t) = \\frac{1}{1 + e^{0.5(x - 0.5 t)}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 2} | |
| {"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": "$$ \\boxed{ u(x, t) = \\frac{1}{1 + e^{x - 0.5t}} } $$", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 3} | |
| {"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)} $", "parameters": "$x; t;$", "type": "nonlinear_pde", "index": 4} | |
| {"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": "$\\boxed{u(x, y) = sech(\\sqrt{\\frac{1}{2}} (x-\\frac{y}{2}))}$", "parameters": "$x; y$", "type": "nonlinear_pde", "index": 5} | |
| {"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": "$\\boxed{u(x, t) = \\frac{1}{4\\cosh^2(\\frac{x-\\sqrt{\\frac{3}{2}}t}{2\\sqrt{2}})}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 6} | |
| {"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": "$\\boxed{u(x,t) = \\frac{225}{256} \\sech^4( \\frac{\\sqrt{2}}{8} (x - 0.5t) )}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 7} | |
| {"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": "$\\boxed{u(x,t) = e^{\\tanh\\left(\\frac{x}{\\sqrt{2}}\\right)} ; u(x,t) = e^{-\\tanh\\left(\\frac{x}{\\sqrt{2}}\\right)}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 8} | |
| {"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": "$$ \\boxed{u(x,t) \\approx \\frac{\\pi}{2} - \\sqrt{2(0.1-t)} - 5 \\frac{x^2}{\\sqrt{0.1-t}}} $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 9} | |
| {"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.", "solution": "$$ \\boxed{u(x,t) \\approx \\frac{\\pi}{2} - \\sqrt{2(0.1-t)} - 5 \\frac{x^2}{\\sqrt{0.1-t}}} $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 10} | |
| {"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": "$$\\boxed{u(x, t \\rightarrow \\infty) \\approx \\left(1 + \\frac{\\sqrt{5}}{4}\\right) e^{-\\sqrt{5}x} - \\frac{\\sqrt{5}}{4} e^{-3\\sqrt{5}x}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 11} | |
| {"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": "$$\\boxed{u(x, t) = 2(1 + e^{-\\sqrt{2}(x + 3\\sqrt{2}t)})^{-1}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 12} | |
| {"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": "$$\\boxed{u(x, t) = 2(e^{\\sqrt{2}(x - (1/\\sqrt{2})t)} + e^{-\\sqrt{2}(x - (1/\\sqrt{2})t)})^{-1}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 13} | |
| {"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": "$\\boxed{u(x, t) = \\tanh\\left( \\frac{x }{\\sqrt{2}} \\right)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 14} | |
| {"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": "$x; t; \\delta$", "type": "nonlinear_pde", "index": 15} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{\\left(1 + \\exp\\left[\\sqrt{\\frac{5}{6}}\\left(x + \\frac{10}{\\sqrt{30}} t \\right)\\right] \\right)^2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 16} | |
| {"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) }$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 17} | |
| {"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": "$\\boxed{u(x, t) = -\\ln\\left(1 - \\frac{t^3}{3} + x^2 t - x t^2\\right)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 18} | |
| {"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": "$\\boxed{u(x,t) =\\frac{1}{1 + e^{\\frac{\\sqrt{5}}{2}(x - 2\\sqrt{5}\\, t)}}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 19} | |
| {"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": "$\\boxed{ u(x,t) = \\tanh(-\\cosh(x) + x^2/t)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 20} | |
| {"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": "$x; t; \\epsilon$", "type": "nonlinear_pde", "index": 21} | |
| {"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": "$\\boxed{u(x,t) = \\tanh(\\sqrt{2} x)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 22} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{(4t)^{1/4}} ( \\frac{12}{\\pi} - (\\frac{x}{(4t)^{1/4}})^2 )^{1/2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 23} | |
| {"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.", "solution": "$\\boxed{(\\ln((x-t)^2))^{1/2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 24} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{2} \\text{ sech}^2(\\frac{1}{\\sqrt{2}} (x - \\sqrt{3}t)); u(x,t) = \\frac{1}{2} \\text{ sech}^2(\\frac{1}{\\sqrt{2}} (x + \\sqrt{3}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 25} | |
| {"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": "$\\boxed{u(x,t) = \\frac{\\sqrt{3}}{3} \\text{ sech}(\\frac{1}{\\sqrt{3}} (x - \\frac{2}{\\sqrt{3}}t)); u(x,t) = \\frac{\\sqrt{3}}{3} \\text{ sech}(\\frac{1}{\\sqrt{3}} (x + \\frac{2}{\\sqrt{3}}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 26} | |
| {"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": "$\\boxed{u(x,t)= \\sqrt{6} \\, \\mathrm{sech}(x-t)}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 27} | |
| {"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": "$x; t;\\alpha;\\beta; z$", "type": "nonlinear_pde", "index": 28} | |
| {"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": "$\\boxed{u(x,t) = 2+\\tanh(\\frac{1}{6}(x+\\frac{10}{9}t))}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 29} | |
| {"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": "$$\\boxed{u(x,t) = \\sqrt{2} (x-t) (\\ln(|x-t|))^{\\frac{1}{4}}}$$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 30} | |
| {"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": "$\\boxed{ u(x,t) = \\exp(\\sin^2(x) t^3 \\log(1 + xt)) }$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 31} | |
| {"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": "$\\boxed{e^{\\cos(x)^2 t - \\log(1 + t^2)}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 32} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{1+\\exp{\\left(\\frac{x}{\\sqrt{2}}-\\frac{t}{4}\\right)}}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 33} | |
| {"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": "$\\boxed{u(x, t) \\sim \\frac{599}{600}e^{-2x} + \\frac{1}{600}e^{-4x}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 34} | |
| {"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": "$\\boxed{u(x,t) \\sim \\frac{1}{\\sqrt{2}} (x-t)^2}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 35} | |
| {"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": "$\\boxed{\\frac{2}{\\cosh^2(2\\sqrt{\\frac{2}{3}}(x-\\sqrt{\\frac{35}{3}}t))}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 36} | |
| {"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": "$\\boxed{u(x,t)=\\frac{1}{\\cosh(\\frac{3}{\\sqrt{2}}(x+\\sqrt{\\frac{11}{2}}t))}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 37} | |
| {"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": "$\\boxed{u(x,t) = 1 + (x-t)^{-1/2}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 38} | |
| {"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": "$\\boxed{\\left(2\\ln\\left(x-\\frac{3}{2}t\\right)\\right)^{1/3}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 39} | |
| {"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": "$\\boxed{u(x,t)=-\\frac{1}{254 (x-2t)^2}-\\frac{2}{16129 (x-2t)}+\\frac{4 \\log (x-2t)}{2048383}-\\frac{4 \\log (127-2 (x-2t))}{2048383}+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 40} | |
| {"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": "$\\boxed{u(x,t)=-2(x-2t)^{-1/2}/(2+5^3)+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 41} | |
| {"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": "$\\boxed{u(x,t) = 2+\\tanh(\\frac{3}{2}(x-9t))}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 42} | |
| {"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", "solution": "$\\boxed{u(x,t)=-\\frac{1}{254 (x-2t)^2}-\\frac{2}{16129 (x-2t)}+\\frac{4 \\log (x-2t)}{2048383}-\\frac{4 \\log (127-2 (x-2t))}{2048383}+5}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 43} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{\\sqrt{t}} e^{-x/\\sqrt{t}}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 44} | |
| {"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": "$\\boxed{u(x,t) = \\frac{1}{1 + 2.5x} + 0.611 e^{-0.5 x} - \\frac{0.509}{1 + 2.5x} e^{-0.5 x}}$", "parameters": "$x;t$", "type": "nonlinear_pde", "index": 45} | |
| {"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}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 46} | |
| {"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": "$\\boxed{u(x,t) = \\cos(x) e^{-t^2}}$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 47} | |
| {"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": "$\\boxed{u(x,t) = \\frac{16}{\\left(6 + e^{\\frac{x-0.5t}{2\\sqrt{2}}} + e^{-\\left(\\frac{x-0.5t}{2\\sqrt{2}}\\right)}\\right)^2}}$", "parameters": "$x; t; t_0$", "type": "nonlinear_pde", "index": 48} | |
| {"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": "$$\\boxed{u(x,t)=t^{-1/3}\\cdot\\max\\{1-\\frac{x^2}{12t^{2/3}},0\\}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 49} | |
| {"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": "$$\\boxed{u(x,t) = \\frac{1}{2}\\cdot\\text{sech}^2\\left(\\frac{1}{2}\\cdot(x - t)\\right)}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 50} | |
| {"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": "$$\\boxed{u(x,t) = 6 \\sech{\\left( \\sqrt{6} (x - 6t) \\right)}}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 51} | |
| {"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": "$$\\boxed{u(x,t) = \\frac{225}{256} \\sech^4\\left( \\frac{\\sqrt{3}}{12}(x - 0.5t) \\right)}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 52} | |
| {"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": "$$\\boxed{u(x,t) = ( \\frac{35}{16} )^{2/3} \\sech^{4/3}( \\frac{3\\sqrt{2}}{8}(x - 0.5t) )}$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 53} | |
| {"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": "$$\\boxed{u(x,t) = \\frac{225}{1024} \\sech^4\\left( \\frac{\\sqrt{2}}{8}(x - 0.5t) \\right) }$$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 54} | |
| {"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": "$$ \\boxed{U(x)=\\frac{1}{1+\\frac{1}{2}x}} $$", "parameters": "$x$", "type": "nonlinear_pde", "index": 55} | |
| {"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": "$$ \\boxed{U(x)=\\frac{1}{(1+0.5\\,x)^{2}}} $$", "parameters": "$x$", "type": "nonlinear_pde", "index": 56} | |
| {"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": "$$\\boxed{\\frac{0.6}{\\sqrt{4 \\pi t}} e^{-t} e^{-\\frac{x^2}{4t}}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 57} | |
| {"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": "$$\\boxed{\\frac{1}{\\sqrt{t+0.1}} e^{-\\frac{(x-1.5t)^2}{4(t+0.1)} + 0.2t}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 58} | |
| {"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": "$$ \\boxed{u(x,t) = \\frac{40}{\\left(2\\sqrt{10}+5e^{\\frac{x-0.5t}{2\\sqrt{2}}} + 5e^{-\\left(\\frac{x-0.5t}{2\\sqrt{2}}\\right)}\\right)^2} ; u(x,t)=0} $$", "parameters": "$x; t; v$", "type": "nonlinear_pde", "index": 59} | |
| {"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": "$$\\boxed{u(x,t) = \\frac{1}{(5t)^{\\frac{1}{5}}} \\sqrt[3]{-\\frac{3}{2}\\left(\\frac{x}{(5t)^{\\frac{1}{5}}}\\right)^2 + A}}$$", "parameters": "$x; t; A$", "type": "nonlinear_pde", "index": 60} | |
| {"prompt": "Find the solution for the nonlinear partial differential equation $$\\partial_t u = \\partial_{xx}u + u^2 (1 - u^2)$$", "solution": "$$\\boxed{u(x,t)=\\tanh( \\frac{1}{\\sqrt{2}}(x - \\sqrt{2}t))}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 61} | |
| {"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": "$$\\boxed{u(x,t) = \\tanh(e^{x - t})}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 62} | |
| {"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": "$$ \\boxed{ u(x,t) = 13\\,\\mathrm{sech}^2\\left(\\sqrt{\\frac{13}{5}}\\left(x-\\sqrt{\\frac{55}{2}}t\\right)\\right) } $$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 63} | |
| {"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": "$$\\boxed{u(x,t)=t^{-1/6}\\cdot\\max\\{1-\\frac{x^2}{15t^{1/3}},0\\}}$$", "parameters": "$x; t$", "type": "nonlinear_pde", "index": 64} | |
| {"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": "$x; t; \\alpha$", "type": "nonlinear_pde", "index": 65} | |
| {"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": "$$ \\boxed{ u(x,t) = (t+K_0^2)^{-1/2} f_s\\left( \\frac{x}{(t+K_0^2)^{1/4}} \\right) } $$", "parameters": "$x; t;K_0;f_s$", "type": "nonlinear_pde", "index": 66} | |