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This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. The following PDE is the corrected version (June 13, 2017) from the original post (February 25, 2015), which had a sign error. This PDE is the result of 47 years of work for me, so any help would be appreciated since I am not a professional mathematician or physicist. $$u_{rr}+u_{zz}+u_{r}/r-u/r^2+u(u_r ^2+u_z ^2+uu_r/r)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\lt\infty\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. I do not think there is a Lagrangian for this PDE. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated! P.S. I have since found that this PDE is not the simplest equation, which is instead found at the question Solve PDE system almost equal to the Ernst equation of general relativity.

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. $$u_{rr}+u_{zz}+\left(u_{r}/r\right)\left(1-u^2\right)-u\left(u_{r}^2+u_{z}^2+1/r^2\right)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\leq 1\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated! P.S. I have since found that this PDE is not the simplest equation, which is instead found at the question Solve PDE system almost equal to the Ernst equation of general relativity.