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Register a new userContinuous and discrete Schrodinger systems with PT-symmetric nonlinearities
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Added by
Amarendra Kumar Sarma Dr.
Publication date
2013
fields
Physics
and research's language is
English
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We investigate the dynamical behavior of continuous and discrete Schrodinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schrodinger counterparts. In particular, the PT-symmetric nonlinear Schrodinger equation can simultaneously support both bright and dark soliton solutions. In addition, we study a two-element discretized version of this PT nonlinear Schrodinger equation. By obtaining the underlying invariants, we show that this system is fully integrable and we identify the PT-symmetry breaking conditions. This arrangement is unique in the sense that the exceptional points are fully dictated by the nonlinearity itself.
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We prove existence of discrete solitons in infinite parity-time (PT-) symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum limit the solitons are constructed from elementary PT-symmetric blocks such as dimers, quadrimers, or more general oligomers. We consider in detail a chain of coupled dimers, analyze bifurcations of discrete solitons from the anticontinuum limit and show that the solitons are stable in a sufficiently large region of the lattice parameters. The generalization of the approach is illustrated on two examples of networks of quadrimers, for which stable discrete solitons are also found.
By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schrodinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.
In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlinear Schrodinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrodinger system with PT-symmetric nonlinearity also admits Peregrine Soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we observe the appearance of low-intense Kuznetsov-Ma (KM) soliton trains in the absence of transverse shift (unbroken PT-symmetry) and well-localized high-intense Peregrine Rogue waves in the presence of transverse shift (broken PT-symmetry) in a definite parametric regime.
In this letter, for the discrete parity-time-symmetric nonlocal nonlinear Schr{o}dinger equation, we construct the Darboux transformation, which provides an algebraic iterative algorithm to obtain a series of analytic solutions from a known one. To illustrate, the breathing-soliton solutions, periodic-wave solutions and localized rational soliton solutions are derived with the zero and plane-wave solutions as the seeds. The properties of those solutions are also discussed, and particularly the asymptotic analysis reveals all possible cases of the interaction between the discrete rational dark and antidark solitons.
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schrodinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly non-trivial solutions such as periodic (breathers), resonant or quasiperiodically oscillating solitons. Some implications to the field of matter-waves are also discussed.
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