In a physics problem I am working on, I need to solve a system of nonlinear algebraic equations arising from truncating Taylor polynomials. I will outline the physical background for context, but my question is more mathematical.
Introduction
When dealing with the propagation of light or sound, one often uses ray theory to predict both the propagation path and the amplitude of the quantity studied. The amplitude is governed by the conservation of energy (or an energy-like quantity) along ray tubes, or narrow bundles of rays. In particular, the amplitude is inversely proportional to the square root of the ray tube area.
There are cases where envelopes of rays, or caustics, can develop. Ray theory may still be used to study the basic geometrical structure of caustics, but because the ray tube area vanishes on caustics, ray theory predicts an infinite amplitude at all points on a caustic.
To describe the rays, we define ray coordinates, $(u,v,\tau)$ where fixed values of $u$ and $v$ identify a particular ray and the distance along any ray is a function of $\tau$. The set of all values of $(u,v,\tau)$ satisfying the ray equations for a particular set of initial conditions then determines the ray family, which has the vector parametric equation \begin{equation} \mathbf{R}=\mathbf{R}(u,v,\tau). \end{equation} For my particular problem, we use a set of local Cartesian coordinates with origin $O$ on the caustic. The $x$ axis is taken to be aligned with the tangent vector of the ray touching the caustic at $O$, and hence is also tangent to the caustic (since, again, it is an envelope of rays). The $z$ axis is normal to the caustic at $O$, and the $y$ axis is mutually perpendicular to each, hence also tangent to the caustic at $O$.
We will therefore decompose the parametric equation for the ray family as \begin{equation}\tag{1}\label{Eq 1} \begin{cases} x=x(u,v,\tau) \\ y=y(u,v,\tau) \\ z=z(u,v,\tau) \end{cases}. \end{equation} It can be shown that the ray tube area is proportional to the Jacobian determinant $D$ of the transformation from ray coordinates to the local caustic coordinates defined by $\mathbf{R}$, which in component form is given by \begin{equation} D(u,v,\tau)=\det\begin{bmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial \tau} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial \tau} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial \tau} \end{bmatrix} \end{equation} so the caustic is equivalently the set of points at which $D=0$.
A simpler problem
Say we want to find the ray(s) passing through a particular point $P$, for which $(x,y,z)=(x_P,y_P,z_P)$. This amounts to solving \begin{equation}\tag{2}\label{Eq 2} \begin{cases} x(u,v,\tau)=x_P, \\ y(u,v,\tau)=y_P, \\ z(u,v,\tau)=z_P, \end{cases} \end{equation} for $u$, $v$, and $\tau$. We can assume that $P$ is near $O$ but not on the caustic and that $O$ is the origin of both the ray and caustic coordinate systems.
In one reference, I have seen the simpler problem of solving for a point $(0,0,z_P)$. That is, for a point along the caustic normal. To do so, the author takes $O$ to be the origin of both the ray coordinates and the local caustic coordinates and Taylor expands \eqref{Eq 2} about $O$, resulting in an equation of the form \begin{equation} \tag{3}\label{Eq 3} \begin{cases} \begin{matrix} c_0\tau& +& \text{a}u^2& & & +& \text{c}\tau^2& & & & & +& \text{f}\tau u& +& \text{g} v^3& +\cdots=0, \\ \delta u& +& \text{a}_1 u^2& +& \text{b}_1 v^2& +& \text{c}_1\tau^2& +& \text{d}_1 uv& & & +& \text{f}_1\tau u& +& \text{g}_1 v^3& +\cdots=0, \\ & & \text{a}_2 u^2& +& \text{b}_2 v^2& +& \text{c}_2 \tau^2& +& \text{d}_2 uv& +& \text{e}_2 v\tau& +& \text{f}_2\tau u& +& \text{g}_2 v^3& +\cdots=z_P, \end{matrix} \end{cases} \end{equation} where the coefficients of $u$, $v$, and $\tau$ are the appropriate Taylor coefficients. The gaps indicate terms that end up being zero due to the choice of coordinate system and properties of rays, but this isn't particularly important to the question. They then apply the implicit function theorem to justify solving the first two equations for $u$ and $\tau$ as functions of $v$. This is subsequently done by writing $u$ and $\tau$ as power series in $v$, substituting them into the first two equations, and equating the resulting coefficients of $v$ to zero to find the series coefficients. Substituting the result into the third equation gives an equation of the form \begin{equation} z_P=b_2v^2+\mathcal{O}(v^3), \end{equation} which can be solved to get $v$ as a function of $z_p$ by expressing $v$ as a power series in $\pm\sqrt{z_P}$, one for each branch of the square root function. Finally, this result is substituted back into the expressions for $u$ and $\tau$, so that $u$, $v$, and $\tau$ are now expressed in terms of $z_P$ (and the Taylor coefficients) as required. The main quantity of interest to me here is $\tau$, which in this case has the form \begin{equation}\tag{4}\label{Eq 4} \tau=\pm\frac{g}{c_0} \biggl(\frac{z_P}{b_2}\biggr)^{3/2}+\mathcal{O}(z_P^2). \end{equation}
The general problem (my attempt)
My primary question is how this argument should be extended to the more general case $(x,y,z)$=$(x_P,y_P,z_P)$. My confusion stems in part from the fact that $D=0$ on the caustic. This makes it seem suspect to expand about $O$ in the first place, since $D=0$ implies that the transformation defined by $\mathbf{R}$ is not invertible at $O$.
Ignoring that for the moment, my thought would be to use the result of the simpler case to suppose we can express $u$, $v$, and $\tau$ as series of the form \begin{equation} \begin{gathered} u(x_P,y_P,z_P)=\pm\sum\limits_{i,j,k} a_{ijk} x^i y^j z^{k/2}, \\ v(x_P,y_P,z_P)=\pm\sum\limits_{i,j,k} b_{ijk} x^i y^j z^{k/2}, \\ \tau(x_P,y_P,z_P)=\pm\sum\limits_{i,j,k} c_{ijk} x^i y^j z^{k/2}, \end{gathered} \end{equation} then substituting these into the expansions for $x$, $y$, and $z$, and equating coefficients of the various powers of $x_P$, $y_P$, and $z_P$. The algebra is pretty painful, but with the help of Mathematica I found, for the positive branch, \begin{equation}\tag{5} \label{Eq 5} \tau(x_P,y_P,z_P)=\frac{g}{c_0}\biggl(\frac{z_P}{b_2}\biggr)^{3/2}+\frac{x_P}{c_0}\biggl(1-\frac{f}{c_0\delta}\biggl[y_P-\frac{b_1}{b_2}z_P\biggr]\biggr)+\cdots \end{equation} The agreement of the first term with \eqref{Eq 4} is promising, and the new terms seem reasonable. However, when matching coefficients, I ran into the following system for the coefficients of $x_P y_P$: \begin{equation} \begin{cases} \frac{f}{c_0 \delta}+\frac{1}{c_0}c_{110}=0, \\ \frac{1}{c_0\delta}\biggl(f_1+\frac{e_2(b_1 d_2-b_2 d_1)}{b_2^2}\biggr)+\delta a_{110}=0, \\ \frac{1}{c_0 \delta}\biggl( f_2-\frac{d_2 e_2}{2 b_2}\biggr)=0. \end{cases} \end{equation} The first two equations can readily be solved for the coefficients, but the last one does not contain any undetermined coefficients, and instead requires that \begin{equation}\tag{6}\label{Eq 6} f_2=d_2 e_2/2 b_2 \end{equation} identically. I assumed this to be true to find higher-order terms in \eqref{Eq 5}, but haven't been able to prove or disprove it yet. I am not asking for a proof, but am curious if in general, the appearance of a constraint like this signals a problem in how I have carried out the expansion. I have, for instance, also tried an expansion in terms of $x_P\sqrt{y_P}\sqrt{z_P}$ and the end result is the same.
Questions
In summary, my questions are:
- Is the expansion procedure I have attempted reasonable?
- What is the significance of $D=0$ on the caustic as far as the expansions performed (both in my attempt and the simpler problem)?
- In general, when matching coefficients for a series expansion, is running into a constraint like \eqref{Eq 6} where no expansion coefficients appear a sign that there is something wrong with the assumed form of the expansion?
Thanks for any help you might be able to offer!