NDSolve PDE, not enough boundary condition?

The PDE that I want to solve is: $$\frac{\partial f}{\partial t} + \frac{1}{m} \left( p_x \frac{\partial f}{\partial x} + p_y \frac{\partial f}{\partial y} + p_z \frac{\partial f}{\partial z} \right) - \frac{\partial V}{\partial x} \frac{\partial f}{\partial p_x} - \frac{\partial V}{\partial y} \frac{\partial f}{\partial p_y} - \frac{\partial V}{\partial z} \frac{\partial f}{\partial p_z} = 0$$

where $$V(x,y,z) = c \sqrt{\frac{x^2 + y^2}{4 a^2} + \frac{z^2}{a^2}}$$

the boundary/initial condition I have is: $$f(x,y,z,p_x,p_y,p_z,t=0) = exp \left(- \frac{p_x^2 + p_y^2 + p_z^2}{2 m} - (x^2 + y^2 + z^2) \right)$$

potentialCartesian[x_, y_, z_, a_, c_] := c Sqrt[(x^2 + y^2)/(4 a^2) + z^2/a^2]

eqn = D[f[x, y, z, px, py, pz, t], t] + 1/m (px D[f[x, y, z, px, py, pz, t], x] + py D[f[x, y, z, px, py, pz, t], y] + pz D[f[x, y, z, px, py, pz, t], z]) - D[potentialCartesian[x, y, z, a, c], x] D[f[x, y, z, px, py, pz, t], px] -D[potentialCartesian[x, y, z, a, c], y] D[f[x, y, z, px, py, pz, t], py] - D[potentialCartesian[x, y, z, a, c], z] D[f[x, y, z, px, py, pz, t], pz] == 0;

ic = f[x, y, z, px, py, pz, 0] ==  Exp[-((px^2 + py^2 + pz^2)/(2 m)) - (x^2 + y^2 + z^2)];

sol = NDSolve[
{
eqn /. {m -> 1, a -> 1, c -> 1},
ic /. {m -> 1, a -> 1, c -> 1}
},
f,
{x, -5, 5},
{y, -5, 5},
{z, -5, 5},
{px, -5, 5},
{py, -5, 5},
{pz, -5, 5},
{t, 0, 5}
]

When I run the NDSolve command, I get the following warnings:

NDSolve::nerres: The maximum number of spatial points (10) for independent variable x allowed by MaxPoints -> Automatic or MinStepSize -> Automatic is too few to compute a spatial error estimate. >>
NDSolve::nerres: The maximum number of spatial points (10) for independent variable y allowed by MaxPoints -> Automatic or MinStepSize -> Automatic is too few to compute a spatial error estimate. >>
NDSolve::nerres: The maximum number of spatial points (10) for independent variable z allowed by MaxPoints -> Automatic or MinStepSize -> Automatic is too few to compute a spatial error estimate. >>
General::stop: Further output of NDSolve::nerres will be suppressed during this calculation. >>
NDSolve::bcart: Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable x. Artificial boundary effects may be present in the solution. >>
NDSolve::bcart: Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable y. Artificial boundary effects may be present in the solution. >>
NDSolve::bcart: Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable z. Artificial boundary effects may be present in the solution. >>
General::stop: Further output of NDSolve::bcart will be suppressed during this calculation.

Is this because I don't have enough boundary/initial conditions? The problem I am trying to solve only has a single initial condition, do I need to use other methods to solve it?

• Maybe adding some background of the PDE will attract more attention? – xzczd Jul 26 '14 at 4:17