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Holographic s-wave condensate with non-linear electrodynamics: A nontrivial boundary value problem

In this paper, considering the probe limit, we analytically study the onset of holographic s-wave condensate in the planar Schwarzschild-AdS background. Inspired by various low energy features of string theory, in the present work we replace the conventional Maxwell action by a (non-linear) Born-Infeld (BI) action which essentially corresponds to the higher derivative corrections of the gauge fields. Based on a variational method, which is commonly known as the Sturm-Liouville (SL) eigenvalue problem and considering a non-trivial asymptotic solution for the scalar field, we compute the critical temperature for the s-wave condensation. The results thus obtained analytically agree well with the numerical findings\cite{hs19}. As a next step, we extend our perturbative technique to compute the order parameter for the condensation. Interestingly our analytic results are found to be of the same order as the numerical values obtained earlier.