$N$-spike string in $AdS_3 \times S^1$ with mixed flux

Sigma model in $AdS_3\times S^3$ background supported by both NS-NS and R-R fluxes is one of the most distinguished integrable models. We study a class of classical string solutions for $N$-spike strings moving in $AdS_3 \times S^1$ with angular momentum $J$ in $S^1 \subset S^5$ in the presence of mixed flux. We observe that the addition of angular momentum $J$ or winding number $m$ results in the spikes getting rounded off and not end in cusp. The presence of flux shows no alteration to the rounding-off nature of the spikes. We also consider the large $N$-limit of $N$-spike string in $AdS_3 \times S^1$ in the presence of flux and show that the so-called Energy-Spin dispersion relation is analogous to the solution we get for the periodic-spike in $AdS_3-pp-$wave $\times S^1$ background with flux.