Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials

• Main Authors: Nixon, Sean D. (Author), Yang, Jianke (Author), Akylas, Triantaphyllos R. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mechanical Engineering (Contributor)
• Format: Article
• Language: English
• Published: Wiley Blackwell, 2015-06-08T15:36:50Z.
• Subjects: Article
• Online Access: Get fulltext

 As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one-dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence-relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch-band edge; and for each rational slope, two line-soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline-soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.