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Existence and stability of localized breather modes in a Heisenberg helimagnet under biquadratic exchange interactions

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Abstract

Discrete bright and dark type breather modes in a one dimensional Heisenberg helimagnet with biquadratic exchange interactions is studied under semiclassical treatment. The dynamics governed by a nonlinear partial differential equation is solved by applying multiple scales combined with quasi-discreteness approximation and reduced to a Nonlinear Schr\(\ddot{o}\)dinger Equation (NLSE). Based on the solution of NLSE derived by Inverse Scattering Technique (IST), the impact of exchange interaction on the breather modes are analysed. The Modulational Instability (MI) features of the present system is also investigated via linear stability analysis.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Author’s comment: The paper contents are purely theoretical and did not need any data].

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Author notes

  1. J. Akhalya

    Present address: Department of Physics, Women’s Christian College, Nagercoil, 629 001, India

Authors and Affiliations

  1. Department of Physics, Women’s Christian College, Nagercoil, 629 001, India

    M. M. Latha

  2. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli, 627 012, Tamilnadu, India

    J. Akhalya

Authors
  1. J. Akhalya
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Appendices

Appendix A

$$\begin{aligned} G_{1}= & {} 2\mu _{n-1}^{2}+4\mu _{n-1}+2\mu _{n}\mu _{n-2}^{2}+4\mu _{n}\mu _{n-1}\\{} & {} +2\mu _{n+1}^{2} 4\mu _{n+1}+2\mu _{n}\mu _{n+1}^{2}+4\mu _{n}\mu _{n+1},\\ G_{2}= & {} 6\mu _{n+1}^{2}+12\mu _{n+1}+8\mu _{n}+6\mu _{n}\mu _{n+1}+12\mu _{n}\mu _{n+1}\\{} & {} \quad +6\mu _{n-1}^{2}+12\mu _{n-1}+6\mu _{n}\mu _{n-1}^{2}\\{} & {} +12\mu _{n}\mu _{n-1}+12\mu _{n}^{2}+4\mu _{n}\mu _{n}^{2},\\ G_{3}= & {} 2\mu _{n-1}^{2}+4\mu _{n-1}+\mu _{n}\mu _{n-1}^{2}+4\mu _{n}\mu _{n-1}+2\mu _{n+1}^{2}\\{} & {} +4\mu _{n+1}+2\mu _{n}\mu _{n+1}^{2}+4\mu _{n}\mu _{n+1},\\ G_{4}= & {} 6\phi _{0}^{2}\mu _{n}^{2}+4\phi _{0}^{2}\mu _{n}+2\phi _{0}^{2}\mu _{n}^{3},\\ G_{5}= & {} 2J\phi _{0}^{2}\mu _{n}-2JS\mu _{n}-4J_{1}S^{3}\mu _{n}+20J_{1}S^{2}\phi _{0}^{2}\mu _{n}\\{} & {} +JS\mu _{n+1}-JS\mu _{n-1}-\frac{J\phi _{0}^{2}}{2}[3\mu _{n-1}^{2}\\{} & {} +4\mu _{n-1}+\mu _{n-1}^{3}+2\mu _{n}{2}+4\mu _{n}+\mu _{n+1}\mu _{n}^{2}+2\mu _{n+1}\mu _{n}\\{} & {} +4\mu _{n+1}+\mu _{n+1}^{3}+3\mu _{n+1}^{2}+\mu _{n-1}\mu _{n}^{2}\\{} & {} +2\mu _{n-1}\mu _{n}]+J_{1}S^{3}[\mu _{n+1}+\mu _{n-1}]-J_{1}S^{2}\phi _{0}^{2}[3\mu _{n-1}^{3}\\{} & {} +4\mu _{n-1}\mu _{n}^{2}+8\mu _{n}\mu _{n-1}\\{} & {} +14\mu _{n}^{2}+3\mu _{n-1}\mu _{n}^{2}+6\mu _{n-1}\mu _{n}+3\mu _{n+1}^{3}+7\mu _{n+1}\mu _{n}^{2}\\{} & {} +14\mu _{n}\mu _{n+1}+16\mu _{n-1}+28\mu _{n}\\{} & {} +9\mu _{n+1}^{2}+9\mu _{n-1}^{2}+16\mu _{n+1}-\frac{2S\Gamma q}{i}[\mu _{n+1}-\mu _{n-1}]\\{} & {} -\frac{\Gamma q}{i}\phi _{0}^{2}[\mu _{n-1}^{2}+2\mu _{n-1}+\mu _{n-1}^{3}+2\mu _{n-1}^{2}\\{} & {} +\mu _{n+1}\mu _{n}^{2}+2\mu _{n}\mu _{n+1}-\mu _{n-1}\mu _{n}^{2}-2\mu _{n-1}\mu _{n}+9\mu _{n+1}^{2}\\{} & {} +9\mu _{n-1}^{2}+16\mu _{n+1},\\ G_{6}= & {} JS[\mu _{n+1}-\mu _{n-1}]+\frac{J\phi _{0}^{2}}{2}[3\mu _{n-1}^{2}+2\mu _{n-1}\\{} & {} +\mu _{n-1}^{3}+\mu _{n+1}\mu _{n}^{2}+2\mu _{n+1}\mu _{n}\\{} & {} -3\mu _{n+1}^{2}-2\mu _{n+1}-\mu _{n+1}^{3}-\mu _{n-1}\mu _{n}^{2}-2\mu _{n-1}\mu _{n}]\\{} & {} +2J_{1}S^{3}[\mu _{n+1}-\mu _{n-1}]-J_{1}S^{2}\phi _{0}^{2}[3\mu _{n-1}^{2}+3\mu _{n-1}^{3}\\{} & {} +6\mu _{n-1}^{2}+\mu _{n-1}\mu _{n}^{2}+2\mu _{n}\mu _{n-1}-3\mu _{n+1}^{3}-\mu _{n+1}\mu _{n}^{2}\\{} & {} -2\mu _{n}\mu _{n+1}-10\mu _{n+1}\\{} & {} -9\mu _{n+1}^{2}+10\mu _{n-1}]-\frac{2S\Gamma q}{i}[\mu _{n+1}+\mu _{n-1}-2\mu _{n}]\\{} & {} +\frac{\Gamma q}{i}\phi _{0}^{2}[\mu _{n-1}^{3}+3\mu _{n-1}^{2}+2\mu _{n}^{2}+\mu _{n+1}\mu _{n}^{2}\\{} & {} +2\mu _{n}\mu _{n+1}+4\mu _{n-1}+\mu _{n-1}\mu _{n}^{2}+2\mu _{n-1}\mu _{n}]\\ {}{} & {} +4\mu _{n+1}+\mu _{n+1}^{3}+3\mu _{n+1}^{2},\\ G_{7}= & {} J_{1}S^{2}\phi _{0}^{2}[2\mu _{n-1}^{2}+4\mu _{n-1}+2\mu _{n}\mu _{n-1}^{2}\\{} & {} +4\mu _{n}\mu _{n-1}+2\mu _{n+1}^{2}+2\mu _{n+1}^{2}+4\mu _{n+1}+2\mu _{n}\mu _{n+1}^{2}\\{} & {} +4\mu _{n}\mu _{n+1}]-\Gamma S^{2}\phi _{0}^{2}[2\mu _{n-1}^{2}\\{} & {} +4\mu _{n-1}+2\mu _{n}\mu _{n-1}^{2}+4\mu _{n}\mu _{n-1}+2\mu _{n+1}^{2}+4\mu _{n+1}\\{} & {} +2\mu _{n}\mu _{n+1}^{2}-4\mu _{n}\mu _{n+1},\\ G_{8}= & {} J_{1}S^{2}\phi _{0}^{2}[2\mu _{n+1}^{2}+4\mu _{n+1}+2\mu _{n}\mu _{n+1}^{2}+4\mu _{n}\mu _{n+1}\\{} & {} -2\mu _{n-1}^{2}-4\mu _{n-1}^{2}-2\mu _{n}\mu _{n-1}^{2}-4\mu _{n}\mu _{n-1}]\\{} & {} +\Gamma S^{2}\phi _{0}^{2}[2\mu _{n-1}^{2}+4\mu _{n-1}\\{} & {} +2\mu _{n}\mu _{n-1}^{2}+4\mu _{n}\mu _{n-1}-2\mu _{n+1}^{2}-4\mu _{n+1}\\{} & {} -2\mu _{n}\mu _{n+1}^{2}-4\mu _{n}\mu _{n+1},\\ \end{aligned}$$

Appendix B

$$\begin{aligned} H_{1}= & {} -2JS-4J_{1}S^{3}+2JS \cos (Q)\\{} & {} -6J\phi _{0}^{2}\beta \beta ^{*}-4J\phi _{0}^{2} \cos (Q)\\{} & {} +4J_{1}S^{3}\phi _{0}^{2} \cos (Q)-60J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*} \cos (Q)\\{} & {} -8J_{1}S^{2}\phi _{0}^{2}-32J_{1}S^{2}\\{} & {} \phi _{0}^{2} \cos (Q)+4\Gamma Sqsin (Q)-4\Gamma q \phi _{0}^{2}\sin (Q)\\{} & {} -4\Gamma \phi _{0}^{2}\beta \beta ^{*}\sin (Q),\\ H_{2}= & {} 2JS \sin (Q)-2J\phi _{0}^{2}\sin (Q)-2J\phi _{0}^{2}\beta \beta ^{*} \sin (Q)-4J_{1}S^{3}\\{} & {} \sin (Q)-20J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*}\sin (Q)-20J_{1}S^{2}\phi _{0}^{2} \sin (Q)\\{} & {} +4\Gamma Sq-4S\Gamma q \cos (Q)+8\Gamma q\phi _{0}^{2} \cos (Q) \\{} & {} +12\Gamma q\phi _{0}^{2}\beta \beta ^{*}\cos (Q),\\ H_{3}= & {} 8J_{1}S^{2}\phi _{0}^{2} \cos (Q)+4J_{1}S^{2}\phi _{0}^{2} \beta \beta ^{*} \cos (2Q)+8J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*}\\{} & {} -8\Gamma S^{2}\phi _{0}^{2} \cos (Q)-4\Gamma S^{2} \phi _{0}^{2}\beta \beta ^{*} \cos (2Q)-8\Gamma S^{2}\phi _{0}^{2} \beta \beta ^{*},\\ H_{4}= & {} 8J_{1}S^{2} \phi _{0}^{2}\sin (Q)+4J_{1}S^{2}\phi _{0}^{2} \beta \beta ^{*}\sin (2Q)-8\Gamma S^{2}\phi _{0}^{2}\\{} & {} \sin (Qa)-4\Gamma S^{2} \phi _{0}^{2}\beta \beta ^{*} \sin (2Q),\\ H_{5}= & {} -2JS-4J_{1}S^{3}+2JS \cos (Q)-6J\phi _{0}^{2}\beta \beta ^{*} \cos (Q)\\{} & {} -4J\phi _{0}^{2} \cos (Q)+4J_{1}S^{3} \cos (Q)\\{} & {} -60J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*} \cos (Q)-8J_{1}S^{2}\\{} & {} \phi _{0}^{2}-32J_{1}S^{2}\phi _{0}^{2} \cos (Q)-4\Gamma Sqsin (Q)\\{} & {} +4\Gamma q \phi _{0}^{2}\sin (Q)+4\Gamma q\phi _{0}^{2} \beta \beta ^{*}\sin (Q), \\ H_{6}= & {} -2JS \sin (Q)+2J\phi _{0}^{2}\sin (Q)+2J\phi _{0}^{2}\beta \beta ^{*} \sin (Q)\\{} & {} -4J_{1}S^{3} \sin (Q)+20J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*}\sin (Q)\\{} & {} +20J_{1}S^{2}\phi _{0}^{2} \sin (Q)+4\Gamma Sq-4S\Gamma q \cos (Q)\\{} & {} +8\Gamma q\phi _{0}^{2} \cos (Q) +12\Gamma q\phi _{0}^{2}\beta \beta ^{*}\cos (Q),\\ H_{7}= & {} 8J_{1}S^{2}\phi _{0}^{2} \cos (Q)+4J_{1}S^{2}\phi _{0}^{2} \beta \beta ^{*} \cos (2Q)\\{} & {} +8J_{1}S^{2}\phi _{0}^{2}\beta \beta ^{*}\\{} & {} -8\Gamma S^{2}\phi _{0}^{2} \cos (Q)-4\Gamma S^{2} \phi _{0}^{2}\beta \beta ^{*} \cos (2Q)-8\Gamma S^{2}\phi _{0}^{2} \beta \beta ^{*},\\ H_{8}= & {} -8J_{1}S^{2} \phi _{0}^{2}\sin (Q)-4J_{1}S^{2}\phi _{0}^{2} \beta \beta ^{*}\sin (2Q)+8\Gamma S^{2}\phi _{0}^{2} \\{} & {} \sin (Q)+4\Gamma S^{2} \phi _{0}^{2}\beta \beta ^{*} \sin (2Q)], \end{aligned}$$

Appendix C

$$\begin{aligned} F_{1}= & {} 2\mid \phi _{n,n-1}^{(1)}\mid ^{2}\phi _{n,n}^{(1)}\\{} & {} +2\mid \phi _{n,n+1}^{(1)}\mid ^{2}\phi _{n,n}^{(1)}-\mid \phi _{n,n-1}^{(1)}\mid ^{2}\phi _{n,n-1}^{(1)}\\{} & {} - \phi _{n,n+1}^{(1)*}{\phi _{n,n}^{(1)}}^{2}-\mid \phi _{n,n+1}^{(1)}\mid ^{2}\phi _{n,n+1}^{(1)}- \phi _{n,n-1}^{(1)*}{\phi _{n,n}^{(1)}}^{2},\\ F_{2}= & {} 2\mid \phi _{n,n-1}^{(1)}\mid ^{2}\phi _{n,n}^{(1)*}+6\mid \phi _{n,n+1}^{(1)}\mid ^{2}\phi _{n,n}^{(1)} \\{} & {} +6\mid \phi _{n,n-1}^{(1)}\mid ^{2}\phi _{n,n}^{(1)}+2 {\phi _{n,n+1}^{(1)}}^{2}\phi _{n,n}^{(1)*}\\{} & {} -3\mid \phi _{n,n-1}^{(1)}\mid ^{2}\phi _{n,n-1}^{(1)}-3{\phi _{n,n}^{(1)}}^{2}\phi _{n,n+1}^{(1)*}\\{} & {} -4\mid \phi _{n,n}^{(1)}\mid ^{2}\phi _{n,n-1}^{(1)}\\{} & {} -3{\phi _{n,n}^{(1)}}^{2}\phi _{n,n-1}^{(1)*}-3\mid \phi _{n,n+1}^{(1)}\mid ^{2}\phi _{n,n+1}^{(1)}\\{} & {} -4 \mid \phi _{n,n}^{(1)}\mid ^{2}\phi _{n,n+1}^{(1)}+4\mid \phi _{n,n}^{(1)}\mid ^{2}\phi _{n,n}^{(1)},\\ F_{3}= & {} 2\mid \phi _{n,n-1}^{(1)}\mid ^{2}u_{n,n}^{(1)}+2\mid u_{n,n+1}^{(1)}\mid ^{2}u_{n,n}^{(1)}\\{} & {} - {u_{n,n-1}^{(1)}}^{2}u_{n,n}^{(1)*}+ {u_{n,n+1}^{(1)}}^{2}u_{n,n}^{(1)*},\\ F_{4}= & {} {u_{n,n}^{(1)}}^{2} u_{n,n+1}^{(1)*}+\mid u_{n,n-1}^{(1)}\mid ^{2}u_{n,n-1}^{(1)}\\{} & {} -u_{n,n-1}^{(1)*}{u_{n,n}^{(1)}}^{2}-\mid u_{n,n+1}^{(1)}\mid ^{2} u_{n,n+1}^{(1)},\\ F_{5}= & {} \mid u_{n,n}^{(1)}\mid ^{2}u_{n,n}^{(1)}. \end{aligned}$$

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Akhalya, J., Latha, M.M. Existence and stability of localized breather modes in a Heisenberg helimagnet under biquadratic exchange interactions. Eur. Phys. J. B 97, 23 (2024). https://doi.org/10.1140/epjb/s10051-024-00653-z

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