Giant Magnons under NS-NS and Melvin Fields

We analyze the energy and angular momenta of the two-spin spiky string moving on the $AdS_3\times S^1$ with NS-NS field. Due to the infinite boundary of the AdS spacetime the solution will extend to infinity and it appears the divergences. After adding the counter terms we then obtain the dispersion relation of the corresponding giant magnon and see how the background field produces the correction terms. We also investigate the energy and angular momentum of the one-spin spiky string moving on the Melvin-field deformed $R \times S^2$. The dispersion relation of the corresponding deformed giant magnon is also obtained. We discuss the property of the correction term and its relation with the spin chain