respectively. Using these forms of $A(y)$ and $\xi(y)$, we determine the solutions of $\zeta_{L,R}(y)$ as follows : $$\begin{aligned}
   \zeta_{L,R}(y) = \sqrt{\frac{k}{e^{(\frac{p}{k}+\frac{q}{2k})}\big[e^{2k\pi r_c}-1}\big]} e^{2ky}
   \exp{\bigg[\frac{p}{2k}e^{(2+\nu)ky} - \frac{q}{4k}e^{-2(2+\nu)ky}}\bigg]
   \label{solution1}
  \end{aligned}$$ for $m_5 = 0$; and

$$\begin{aligned}
   \zeta_{L,R}(y)&=&\sqrt{\frac{k \pm m_5}{e^{(\frac{p}{k}+\frac{q}{2k})}\big[e^{(2k\pm m_5)\pi r_c}-1}\big]} e^{(2k \pm m_5)y}*\nonumber\\ 
   &\exp&{\bigg[\frac{p}{2k}e^{(2+\nu)ky} - \frac{q}{4k}e^{-2(2+\nu)ky}}\bigg]
   \label{solution2}
  \end{aligned}$$

for $m_5 \neq 0$.

with $p$ and $q$ have following expressions : $$\begin{aligned}
   p = \frac{2}{\sqrt{3}}k\kappa \bigg[v_v e^{-(2+\nu)kr_c\pi} + e^{-4kr_c\pi}\frac{\kappa v_h k^{3/2}}{2\sqrt{3}}
   \frac{[2\sqrt{9 - 40\kappa^2\alpha\Lambda} + 3]}{\sqrt{9 - 40\kappa^2\alpha\Lambda}}\bigg] 
   \label{p}
  \end{aligned}$$

and $$\begin{aligned}
 q = \frac{2}{3} k\kappa^2 v_h^2 e^{-8k\pi r_c}
 \label{q}
 \end{aligned}$$

where we use the explicit form of $m_{\Phi}^2$ (see eqn.(\[mass\_phi\])). Moreover, the overall normalization constants in eqn. (\[solution1\]) and eqn. (\[solution2\]) are determined by using the normalization condition presented earlier in eqn. (\[norm1\]). It may be noticed that left and right chiral modes have the same solution when $m_5=0$, but the degeneracy between the two chiral modes are lifted in the presence of non-zero bulk fermionic mass term.\
It is worthwhile to study how the localization scenario depends on the higher curvature parameter ($\alpha$) as well as the bulk mass parameter ($m_5$).

### Effect of higher curvature parameter {#effect-of-higher-curvature-parameter .unnumbered}

From eqn.(\[solution1\]) and using the expressions of $p$ and $q$, we 