Broken and unbroken $$\varvec{\mathcal {PT}}$$-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

The nonlinear dynamics of coupled P T -symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.

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“…According to the general definition [27][28][29][30][31][32][33][34][35], the linear matrix transformation…”

Section: Crucial Ideas Of the Darboux-bäcklund Integration Methodsmentioning

confidence: 99%

Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

Vakhnenko

Verchenko

2021

Proc. R. Soc. A.

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“…Nonlocal problems in integrable equations have recently been the subject of intensive investigation [12][13][14][15][16][17][18][19][20], where they occur due to parity-time (PT ) symmetry. Bender and Boettcher showed that large amounts of non-Herimitan Hamiltonians, the so called PT -symmetric Hamiltonians, possess entirely real and positive spectrum [21].…”

Section: Introductionmentioning

confidence: 99%

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

2022

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In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.

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“…However, it is easy to find that most research results of nonlocal equations are the study of continuous nonlocal integrable equations, while the revelent research of discrete NLEs is not sufficient. [9,[33][34][35][36][37] Especially, the study of nonlocal discrete equations is not systematic, [38][39][40][41] so focusing on discrete nonlocal NLEs motivates us.…”

Section: Introductionmentioning

confidence: 99%

“…Some methods for solving local equations have been extended to nonlocal equations such as the inverse scattering transform method, [15,24,30] Hirota's bilinear method, [25,26,31,32,[42][43][44] the function expansion method, [27] the Riemann-Hilbert method, [28] and the DT method. [9,12,16,29,33,[35][36][37][38][39][40][41][45][46][47][48][49][50][51] Among them, DT is a simple algebraic technique to generate a family of infinite nontrivial solutions of a linear equation from a known trivial solution which is widely used in soliton theory. [9,45,46] It needs to be noted that the N-fold DT technique is a very effective method to derive N-soliton solutions compared to iterative DT because there is no iterative process.…”

Section: Introductionmentioning

confidence: 99%

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type*

Yuan¹,

Wen²

2021

Chinese Phys. B

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We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

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Broken and unbroken $$\varvec{\mathcal {PT}}$$-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

Cited by 26 publications

References 30 publications

Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type*

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