Center-localized oscillating modes with exponential decay tails, solved from coupled ODE

Question

Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$:

$$ -a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) U(r)+ B(r) (\partial_r-\frac{n}{r}\big) V(r)=n\; k \; U(r) $$ $$ -B(r) (\partial_r+\frac{n}{r}\big) U(r) + a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) V(r) =n\; k \; V(r), $$ We like to solve $U(r)$ and $V(r)$ (say, better analytically and exactly). What are the exact solutions of $U(r)$ and $V(r)$? (or approximate solutions?)

This is a more general set of equations extending from the post here

Here are the conditions of $B(r),a,c,k$.

The B(r) is given such that $B(r)$ is a nice smooth differentiable function, with $$B(0)=0$$ $$\lim_{r \to 0} B(r)=0$$ $$\lim_{r \to \infty} B(r)=b=\text{constant} >0,$$ and $B(r) \geq 0$ is monotonically increasing along $r \in [0, \infty)$, also $$a=\text{constant} >0.$$ $$c=\text{constant} \geq 0.$$

$$n \in \mathbb{Z}^+=\text{integer natural number constant} >0.$$

$$k=\text{constant} >0.$$ For your convenience, here $k$ can be chosen of our convenience such that the equations can be solved more easily. My suggestion and trial finds that $$ k=\frac{b^2 }{a\; c} $$ or at least the leading order $$ k=\frac{b^2 }{a\; c} + \cdots $$ look more promising for exact solutions.

Both $a$, $b$ and $c$ are finite values.

I have done some analysis myself. My expected analysis find that $U(r)$ and $V(r)$ have exponential decay tails that look like $$\exp[-\int_0^r B(r')^{\#} dr']$$ The ${\#}$ means some tentative power. And both $U(r)$ and $V(r)$ likely contain Bessel functions $J_0(r),J_1(r), ...,etc$.

I suppose that their positive-valued peaks are localized near $r=0$, but oscillating modes along $r$, where $U(0)$ and $V(0)$ are nearly in their maximum, with exponential decay tails $\lim_{r \to 0} U(r)=\lim_{r \to 0} V(r)=0.$ But I could be wrong.

If exact analytic solutions are NOT possible, please give arguments, and please feel free to take approximations. Personally, I believe that it can be solved analytically exactly by some Bessel type functions.

It will be OK, for your answer, just for focus on the simple cases that $n=1$ or $n=2$. Even though one may consider generic $n \in \mathbb{Z}^+$ in general, but it is not necessary.

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The special cases of my questions boil down to here and here.

(p.s. This is some fun trial analysis done by myself.)

Answer 1

Due to the lack of solutions, I thought it will still be nice to consider a particualr case --- Here are the solutions for the $n=0$ case. Define $U(r) \pm V(r)=U_\pm(r)$, and $\partial_r U_{\pm}\equiv U'_{\pm}$, the $\pm$ linear combinations give

$$ -a \partial_r (U'_+(r))+ (B(r) -\frac{a}{r}) (U'_+(r))=0 $$ $$ -a\partial_r (U'_-(r))- (B(r) +\frac{a}{r}) (U'_-(r))=0, $$

Thus $$ U_+(r)= c_+ \int_{0}^r dw \cdot \exp(\frac{1}{a}\int_{0}^w ds (B(s) -\frac{a}{s}) $$ $$ U_-(r)= c_- \int_{0}^r dw \cdot \exp(\frac{-1}{a}\int_{0}^w ds (B(s) +\frac{a}{s}) . $$ In the end, we can plug in to solve. $$ U(r)=\frac{1}{2}( U_+(r)+ U_-(r)) $$ $$ V(r)=\frac{1}{2}( U_+(r)- U_-(r)) $$ As for appropriate boundary conditions, one can choose the values of $c_+$, $c_-$ which are some constants to fit the boundary conditions.

Maybe there is a way to deal with the general $n$ from here.