I'm trying to solve a modified Poisson-Boltzmann equation given by
$\frac{d^{2}\phi(z)}{dz^{2}}=2k_{1}\sinh(\phi(z))-k_{2}$,
where $k_{1}$ and $k_{2}$ are constants, and I'm not sure of how to solve it.
I know that if $k_{2}=0$ then I can multiply both sides by $\frac{d\phi(z)}{dz}$, leading to
$\frac{1}{2}\frac{d}{dz}(\frac{d\phi(z)}{dz})^{2}=2k_{1}\sinh(\phi(z))\frac{d\phi(z)}{dz}$,
and integrating once,
$\frac{1}{2}(\frac{d\phi(z)}{dz})^{2}-2k_{1}\cosh(\phi(z))=c$, where $c$ is a constant related to the boundary conditions of the equation.
Then, I define $\psi=\sqrt{4k_{1}\cosh(\phi)+2c}$ and solve $\frac{dz}{d\phi}=\frac{1}{\psi}$.
I know that $\frac{dz}{d\psi}=\frac{dz}{d\phi}\frac{d\phi}{d\psi}$, the first term is $1/\psi$ and the other one is $\psi/2k_{1}\sinh(\phi)$, and as $\sinh(\phi)=\sqrt{(\frac{\psi^{2}-2c}{4k_{1}})^{2}-1}$, I get that
$dz=\frac{d\psi}{2k_{1}\sqrt{(\frac{\psi^{2}-2c}{4k_{1}})^{2}-1}}$.
I can probably integrate this, but it looks like it will be nasty.
However, the part that has been bothering me the most is for $k_{2}\neq 0$.
Following the same procedure, once I multiply both sides by $d\phi/dz$ and integrate, I get $k_{2}\phi$.
Then, I define $\psi=\sqrt{4k_{1}\cosh(\phi)+2c+2k_{2}\phi}$ and
$\frac{dz}{d\psi}=\frac{1}{2k_{1}\sinh(\psi)+2k_{2}}$. Now, $\sinh(\phi(z))=\sqrt{(\frac{\psi^{2}-2k_{2}\phi-2c}{4k_{1}})^{2}-1}$ and therefore I get $\frac{dz}{d\psi}=\frac{1}{2k_{1}\sqrt{(\frac{\psi^{2}-2c-2k_{2}\phi}{4k_{1}})^{2}-1}+2k_{2}}$.
And I'm horribly stuck, as I don't like having $\phi$ down there... Any suggestions are welcome!!!
The $k_2=0$ has been discussed in the literature. You do not specify boundary conditions, but the case $\phi(z=0)=\Phi$ and $\phi(z\to\infty)=0$ was famously solved by Gouy in 1910 (SUR LA CONSTITUTION DE LA CHARGE ÉLECTRIQUE A LA SURFACE D’UN ÉLECTROLYTE, yes, in French) and Chapman in 1913 (A contribution to Electrocapilarity). The same solution in modern notation can be found in Eq. 12 of this paper by Chow and coworkers.
Corkill and Rosenhead (1939, link) considered a case wherein $\partial_z \phi(z=L)=0$ for some finite length $L$; this corresponds physically to two like-charged plates separated by a distance $2L$. Their implicit solution can be found in Eq. 3.7 of that paper. Verweij and Overbeek's reproduce these results in chapter IV of their famous 1942 book on colloid stability.
I cannot help you now with the $k_2\neq0$ case, but suggest you to have a look at the above mentioned paper by Chow et al.