Need help in attempting to solve an MHD eigenvalue problem
Background
I am attempting to numerically solve the ideal MHD equations in normal mode form for a <PERSON> current sheet. The linearized perturbed MHD equations can be written in a normal mode form: $$ -\rho^{}_{m0} \omega_n^2 \boldsymbol{\xi}_n = \mathbf{F}(\boldsymbol{\xi}_n) \tag{1} $$
where $\mathbf{F}(\boldsymbol{\xi}_n)$ is the linear force operator given by: $$ \mathbf{F}(\boldsymbol{\xi}_n) = \frac{1}{\mu_0}(\nabla \times \mathbf{B}_0) \times [\nabla \times (\boldsymbol{\xi}_n \times \mathbf{B}_0)] + \frac{1}{\mu_0} \Big{\nabla \times \big[\nabla \times (\boldsymbol{\xi}_n \times \mathbf{B}_0)\big] \Big}\times \mathbf{B}_0+ \nabla \Big[\boldsymbol{\xi}_n \cdot \nabla P_0 + \gamma P_0 \nabla \cdot \boldsymbol{\xi}_n \Big] $$
Here, $\mathbf{B}_0$ is the equilibrium magnetic field given by: $$ \mathbf{B}_0 = B_0 \tanh\left(\frac{y}{a}\right) \hat{\mathbf{x}}+B_T \hat{\mathbf{z}} $$
where $B_0$ and $B_T$ are constants. $\boldsymbol{\xi}_n$ is the plasma displacement vector in two dimensions (the focus of our problem is in 2D), hence: $$ \boldsymbol{\xi}_n = \xi_x \hat{\mathbf{x}} + \xi_y \hat{\mathbf{y}} $$
and finally, $\gamma$ is the polytrope index and $P_0$ is the equilibrium pressure which can be shown to be, using the momentum equation: $$ P_0 = - \frac{B_0^2}{2\mu_0}\tanh^2\left(\frac{y}{a}\right) + C $$
where $C$ is a constant. Furthermore, we assume the plasma is incompressible, that is $\nabla \cdot \boldsymbol{\xi}_n = 0$.
Actual Problem
After a super long and tedious calculation, equation (1) takes on the simple form (I dropped the subscript $n$ for simplicity): $$ \frac{B_0^2}{\mu_0} \tanh^2 \left(\frac{y}{a}\right) (\partial_{xx} + \partial_{yy}) \xi_y + \frac{2B_0^2}{\mu_0a} \text{sech}^2\left(\frac{y}{a}\right)\tanh\left(\frac{y}{a}\right) \partial_y \xi_y = -\rho^{}_{m0} \omega_n^2 \xi_y \tag{2} $$
and $\xi_x = 0$. 955