Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

2014

Stud Appl Math

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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 multi-soliton 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.

Section: Introductionsupporting

confidence: 76%

Exponential Asymptotics for Solitons in ‐Symmetric Periodic Potentials

Nixon

2014

Stud Appl Math

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“…Combining this result with the finding of this article, we argue that in the 1D NLS equation with a complex potential, PT -symmetry of the potential is both necessary and sufficient for the existence of soliton families (assuming that this potential admits real discrete eigenvalues or real Bloch bands). Soliton families that exist in a 1D PT -symmetric potential are always PT -symmetric, as was shown recently in [37].…”

Section: Summary and Discussionsupporting

confidence: 62%

Necessity of PT symmetry for soliton families in one-dimensional complex potentials

2014

Physics Letters A

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For the one-dimensional nonlinear Schrödinger equation with a complex potential, it is shown that if this potential is not parity-time (PT ) symmetric, then no continuous families of solitons can bifurcate out from linear guided modes, even if the linear spectrum of this potential is all real. Both localized and periodic non-PT -symmetric potentials are considered, and the analytical conclusion is corroborated by explicit examples. Based on this result, it is argued that PT -symmetry of a one-dimensional complex potential is a necessary condition for the existence of soliton families.

“…As a consequence, continuous families of non-PT -symmetric solitons exist in a PT -symmetric potential. This symmetry breaking is surprising because it is forbidden in generic PT -symmetric potentials [31,32]. Analytical understanding of this symmetry breaking is still an open question; however, based on the analysis in this paper, it is hopeful that this symmetry breaking can now be analytically studied.…”

Section: Summary and Discussionmentioning

confidence: 86%

Bifurcation of Soliton Families from Linear Modes in Non‐‐Symmetric Complex Potentials

Nixon

2016

Stud Appl Math

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Continuous families of solitons in the nonlinear Schrödinger equation with non‐scriptPT‐symmetric complex potentials and general forms of nonlinearity are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form g2(x)+ig′(x), where g(x) is an arbitrary real function. Using this constant of motion, the second‐order complex soliton equation is reduced to a new second‐order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons bifurcate out from linear discrete modes in these non‐scriptPT‐symmetric complex potentials. All analytical results are corroborated by numerical examples.