Spatially discrete Hirota equation: rational and breather solution, gauge equivalence, and continuous limit. (English) Zbl 1509.35283
Summary: It is well known that the nonlinear Schrödinger equation, the Hirota equation, and their integrable discrete versions are very important integrable equations. These integrable equations not only have deep integrability theory, but also have wide physical applications. In this paper, we focus on an integrable spatially discrete Hirota equation. The new Lax pair, Darboux transformation, rational wave solution, rogue wave solution, breather solutions and gauge equivalent structure of the spatially discrete Hirota equation are investigated. The continuous limit of the rogue wave solution and breather solution of the discrete Hirota equation yields the counterparts of the Hirota equation, which shows that the spatially discrete Hirota equation is a very good model for the numerical analysis for considering the Cauchy problem with a general initial date of the Hirota equation. Besides, we prove that this spatially discrete Hirota equation is gauge equivalent to a discrete integrable generalized Heisenberg spin model under the discrete gauge transformation. In this performance, we see that utilizing the new Lax pair, an essential properties of integrability, of the spatially discrete Hirota equation plays a significant role in the investigation of continuous limit of Lax pair of discrete integrable generalized Heisenberg spin model.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37J70 | Completely integrable discrete dynamical systems |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 82D40 | Statistical mechanics of magnetic materials |
Keywords:
discrete Hirota equation; rogue wave; breather solution; continuous limit; gauge equivalenceReferences:
| [1] | Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, J Math Phys, 14, 7, 805-809 (1973) · Zbl 0257.35052 |
| [2] | Mollenauer, L. F.; Stolen, R. H.; Gordon, J. P., Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys Rev Lett, 45, 13, 1095-1098 (1980) |
| [3] | Lamb, G. L., Elements of soliton theory (1980), Wiley: Wiley New York, NY · Zbl 0445.35001 |
| [4] | Lakshmanan, M.; Ganesan, S., Equivalent forms of a generalized Hirota’s equation with linear inhomogeneities, J Phys Soc Japan, 52, 12, 4031-4033 (1983) · Zbl 0711.35128 |
| [5] | Akhmediev, N.; Korneev, V. I.; Mitskevich, N. V., Modulation instability of a continuous signal in an optical fiber taking into account third-order dispersion, Izv Vyssh Uchebn Zaved Radiofiz, 33, 1, 95-100 (1990) |
| [6] | Ankiewicz, A.; Soto-Crespo J. M. Akhmediev, N., Rogue waves and rational solutions of the Hirota equation, Phys Rev E, 81, 4, Article 046602 pp. (2010) |
| [7] | Tao, Y. S.; He, J. S., Multisolitons, breathers, and rogue waves for the Hirota equation generated by darboux transformation, Phys Rev E, 85, 2, Article 026601 pp. (2012) |
| [8] | Karpman, V. I.; Rasmussen, J. J.; Shagalov, A. G., Dynamics of solitons and quasisolitons of the cubic third-order nonlinear Schrödinger equation, Phys Rev E, 64, 2, Article 026614 pp. (2001) |
| [9] | Zhang, D. G.; Liu, J., A higher-order deformed heisenberg spin equation as an exactly solvable dynamical equation, J Phys A: Math Gen, 22, 2, L53-L54 (1989) · Zbl 0696.35163 |
| [10] | Ablowitz, M. J.; Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis, J Math Phys, 17, 6, 1011 (1976) · Zbl 0322.42014 |
| [11] | Porsezian, K.; Lakshmanan, M., Discretised Hirota equation, equivalent spin chain and backlund transformations, Inverse Problems, 5, 2, L15-L19 (1989) · Zbl 0696.35182 |
| [12] | Narita, K., Soliton solution for discrete Hirota equation, J Phys Soc Japan, 59, 10, 3528-3530 (1990) |
| [13] | Narita, K., Soliton solution for discrete Hirota equation II, J Phys Soc Japan, 60, 5, 1497-1500 (1991) · Zbl 1514.37095 |
| [14] | Ankiewicz, A.; Akhmediev, N.; Soto-Crespo, J. M., Discrete rogue waves of the ablowitz-ladik and Hirota equations, Phys Rev E, 82, 2, Article 026602 pp. (2010) |
| [15] | Ohta, Y.; Yang, J. K., General rogue waves in the focusing and defocusing Ablowitz-Ladik equations, J Phys A, 47, 25, Article 255201 pp. (2014) · Zbl 1294.35121 |
| [16] | Pickering, A.; Zhao, H. Q.; Zhu, Z. N., On the continuum limit for a semidiscrete Hirota equation, Proc R Soc Lond Ser A Math Phys Eng Sci, 472, 2195, Article 20160628 pp. (2016) · Zbl 1371.35275 |
| [17] | Li, M.; Li, M. H.; He, J. S., Degenerate solutions for the spatial discrete Hirota equation, Nonlinear Dyn (2020) |
| [18] | Faddeev, L. D.; Takhtajan, L. A., Hamiltonian methods in the theory of soliton (1987), Spring: Spring Berlin · Zbl 0632.58004 |
| [19] | Lakshmanan, M., Continuum spin system as an exactly solvable dynamical system, Phys Lett A, 61, 1, 53-54 (1977) |
| [20] | Zakharov, V. E.; Takhtajan, L. A., Equivalence of the nonlinear Schrödinger equation and the equation of a heisenberg ferromagnet, Theor Math Phys, 38, 1, 17-23 (1979) |
| [21] | Ding, Q., A note on the NLS and the Schrödinger flow of maps, Phys Lett A, 248, 1, 49-56 (1998) · Zbl 1115.35368 |
| [22] | Ishimori, Y., An integrable classical spin chain, J Phys Soc Japan, 51, 11, 3417-3418 (1982) |
| [23] | Ding, Q., On the gauge equivalent structure of the discrete nonlinear Schrödinger equation, Phys Lett A, 266, 2-3, 146-154 (2000) · Zbl 0949.37048 |
| [24] | Kundu, A., Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equation, J Math Phys, 25, 12, 3433-3438 (1984) |
| [25] | Kundu, A., Gauge equivalence of \(\sigma\) models with non-compact Grassmannian manifolds, J Phys A: Math Gen, 19, 8, 1303-1313 (1986) · Zbl 0654.35084 |
| [26] | Myrzakulov, R.; Vijayalakshmi, S.; Syzdykova, R. N.; Lakshmanan, M., On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schrödinger equations, J Math Phys, 39, 4, 2122-2140 (1998) · Zbl 1001.37057 |
| [27] | Cheng, Y.; Li, Y. S.; Tang, G. X., The gauge equivalence of the Davey-Stewartson equation and (2+1)-dimensional continuous Heisenberg ferromagnetic model, J Phys A: Math Gen, 23, 10, L473-L477 (1990) · Zbl 0711.35133 |
| [28] | Ma, L. Y.; Shen, S. F.; Zhu, Z. N., Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation, J Math Phys, 58, 10, Article 103501 pp. (2017) · Zbl 1380.37132 |
| [29] | Ma, L. Y.; Zhao, H. Q.; Gu, H., Integrability and gauge equivalence of the reverse space-time Sasa-Satsuma equation, Nonlinear Dyn, 91, 3, 1909-1920 (2018) · Zbl 1390.35335 |
| [30] | Yang, J.; Zhu, Z. N., Higher-order rogue wave solutions to a spatial discrete Hirota equation, Chin Phys Lett, 35, 9, Article 090201 pp. (2018) |
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