## Multiple-pole solutions to a semidiscrete modified Korteweg-de Vries equation.(English)Zbl 1421.35328

Summary: Multiple-pole soliton solutions to a semidiscrete modified Korteweg-de Vries equation are derived by virtue of the Riemann-Hilbert problem with higher-order zeros. A different symmetry condition is introduced to build the nonregular Riemann-Hilbert problem. The simplest multiple-pole soliton solution is presented. The dynamics of the solitons are studied.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q15 Riemann-Hilbert problems in context of PDEs 35C08 Soliton solutions
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### References:

 [1] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. - JETP, 34, 62-69, (1972) [2] Gagnon, L.; Stiévenart, N., N-soliton interaction in optical fibers: The multiple-pole case, Optics Expresss, 19, 9, 619-621, (1994) [3] Wadati, M.; Ohkuma, K., Multiple-pole solutions of the modified Korteweg-de Vries equation, Journal of the Physical Society of Japan, 51, 6, 2029-2035, (1982) [4] Tsuru, H.; Wadati, M., The multiple pole solutions of the sine-Gordon equation, Journal of the Physical Society of Japan, 53, 9, 2908-2921, (1984) [5] Villarroel, J.; Ablowitz, M. J., On the discrete spectrum of the nonstationary Schrödinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation, Communications in Mathematical Physics, 207, 1, 1-42, (1999) · Zbl 0947.35145 [6] Ablowitz, M. J.; Chakravarty, S.; Trubatch, A. D.; Villarroel, J., A novel class of solutions of the non-stationary Schrodinger and the Kadomtsev-Petviashvili I equations, Physics Letters A, 267, 2-3, 132-146, (2000) · Zbl 0947.35129 [7] Shchesnovich, V. S.; Yang, J., Higher-order solitons in the N-wave system, Studies in Applied Mathematics, 110, 4, 297-332, (2003) · Zbl 1141.35442 [8] Shchesnovich, V. S.; Yang, J., General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations, Journal of Mathematical Physics, 44, 10, 4604-4639, (2003) · Zbl 1062.37083 [9] Zhang, D. J.; Zhao, S. L.; Sun, Y.; Zhou, J., Solutions to the modified Korteweg–de Vries equation, Reviews in Mathematical Physics, 26, 07, 1430006, (2014) · Zbl 1341.37049 [10] Lai, D. W. C.; Chow, K. W.; Nakkeeran, K., Multiple-Pole soliton interactions in optical fibres with higher-order effects, Journal of Modern Optics, 51, 3, 455-460, (2004) [11] He, J. S.; Zhang, H. R.; Wang, L. H.; Porsezian, K.; Fokas, A. S., Generating mechanism for higher-order rogue waves, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 87, 5, (2013) [12] Wang, X.; Yang, B.; Chen, Y.; Yang, Y. Q., Higher-order rogue wave solutions of the Kundu-Eckhaus equation, Physica Scripta, 89, 9, (2014) [13] Ablowitz, M. J.; Ladik, J. F., A nonlinear difference scheme and inverse scattering, Studies in Applied Mathematics, 55, 3, 213-229, (1976) · Zbl 0338.35002 [14] Ablowitz, M. J.; Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis, Journal of Mathematical Physics, 17, 6, 1011-1018, (1976) · Zbl 0322.42014 [15] Ohta, Y.; Yang, J., General rogue waves in the focusing and defocusing Ablowitz-Ladik equations, Journal of Physics A: Mathematical and General, 47, 25, (2014) · Zbl 1294.35121 [16] Geng, X.; Gong, D., Quasi-periodic solutions of the discrete mKdV hierarchy, International Journal of Geometric Methods in Modern Physics, 10, 3, (2013) · Zbl 1282.35331
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