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Exponential Asymptotics for Solitons in PT ‐Symmetric Periodic Potentials
Author(s) -
Nixon Sean D.,
Yang Jianke
Publication year - 2014
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12057
Subject(s) - eigenvalues and eigenvectors , exponential function , soliton , mathematics , nonlinear system , exponential stability , mathematical analysis , instability , exponential growth , exponential decay , mathematical physics , physics , quantum mechanics
Solitons in one‐dimensional parity‐time ( PT )‐symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex‐valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch‐band edge for either self‐focusing or self‐defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT ‐symmetric multisoliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.