## Darboux transformations and rogue wave solutions of a generalized AB system for the geophysical flows.(English)Zbl 1448.76087

Summary: In this paper, we investigate a generalized AB system, which is used to describe certain baroclinic instability processes in the geophysical flows. For the two short waves and mean flow, we derive out the Darboux and generalized Darboux transformations, both relevant to the coefficient of the nonlinear term and coefficient related to the shear. When the coefficient of the nonlinear term is positive, with the generalized Darboux transformation, we present the algorithm to derive the $$N$$th-order $$(N = 1, 2, \ldots)$$ rogue wave solutions. The first- and second-order rogue wave solutions are shown, where our first-order rogue waves are different from those in the existing literatures. The two short waves and mean flow are related to the coefficient of the nonlinear term under certain conditions; the coefficient related to the shear has a linear effect on the mean flow while has no effect on the two short waves. The $$N$$th-order rogue wave solutions turn to be singular when the coefficient of the nonlinear term is negative.

### MSC:

 76E20 Stability and instability of geophysical and astrophysical flows 35Q35 PDEs in connection with fluid mechanics 35A22 Transform methods (e.g., integral transforms) applied to PDEs
Full Text:

### References:

 [1] Sun, W. R.; Sun, W. R.; Wang, L.; Sun, W. R.; Liu, D. Y.; Xie, X. Y., Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers, Ann. Phys., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Chaos, 27, 1, (2017) [2] Wang, Y. F.; Guo, B. L.; Liu, N.; Zhao, X. H.; Tian, B.; Chai, J.; Wu, X. Y.; Guo, Y. J.; Yuan, Y. Q.; Tian, B.; Liu, L.; Sun, Y., Bright-dark solitons for a set of the general coupled nonlinear schrodinger equations in a birefringent fiber, Appl. Math. Lett., Eur. Phys. J. Plus, EPL, 120, 30001, (2017) [3] Hasegawa, A.; Tappert, F.; Gao, X. Y.; Lan, Z. Z., Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation, Appl. Phys. Lett., Appl. Math. Lett., Appl. Math. Lett., 86, 243, (2018) [4] Pelinovsky, E.; Kharif, C.; Wu, X. Y.; Tian, B.; Chai, H. P.; Du, Z., Rogue waves for a discrete (2+1)-dimensional Ablowitz-Ladik equation in the nonlinear optics and Bose-Einstein condensation, Superlattices Microstruct., 115, 130, (2018), Springer Berlin [5] Osborne, A. R.; Liu, L.; Tian, B.; Yuan, Y. Q.; Du, Z.; Du, Z.; Tian, B.; Qu, Q. X.; Chai, H. P.; Wu, X. Y., Semirational rogue waves for the three-coupled fourth-order nonlinear schrodinger equations in an alpha helical protein, Phys. Rev. E, Superlattices Microstruct., 112, 362, (2017), Elsevier New York [6] Akhmediev, N.; Soto-Crespo, J. M.; Ankiewicz, A.; Du, Z.; Tian, B.; Chai, H. P.; Sun, Y.; Zhao, X. H.; Wu, X. Y.; Tian, B.; Liu, L.; Sun, Y., Rogue waves for a variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics, Phys. Lett. A, Chaos, Solitons Fract., Comput. Math. Appl., 72, 215, (2018) [7] Peregrine, D. H., Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. B, 25, 16, (1983) · Zbl 0526.76018 [8] Ohta, Y.; Yang, J. K., General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468, 1716, (2012) · Zbl 1364.76033 [9] Guo, B. L.; Ling, L. M.; Liu, Q. P., Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85, 026607, (2012) [10] Hirota, R., Exact envelop-soliton solutions of a nonlinear wave-equation, J. Math. Phys., 14, 805, (1973) · Zbl 0257.35052 [11] Porsezian, K.; Daniel, M.; Lakshmanan, M., On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain, J. Math. Phys., 33, 1807, (1992) · Zbl 1112.82309 [12] Wu, C. F.; Grimshaw, R. H.J.; Chow, K. W.; Chan, H. N., A coupled “AB” system: rogue waves and modulation instabilities, Chaos, 25, 103113, (2015) · Zbl 1374.35391 [13] Tan, B.; Boyd, J. P., Envelope solitary waves and periodic waves in the AB equations, Stud. Appl. Math., 109, 67, (2002) · Zbl 1114.76315 [14] Yu, G. F.; Xu, Z. W.; Hu, J.; Zhao, H. Q., Bright and dark soliton solutions to the AB system and its multi-component generalization, Commun. Nonlinear Sci. Numer. Simul., 47, 178, (2017) [15] Moroz, I. M.; Brindley, J., Evolution of baroclinic wave packets in a flow with continuous shear and stratification, Proc. R. Soc. Lond., 377, 379, (1981) · Zbl 0481.76119 [16] Gibbon, J. D.; James, I. N.; Moroz, I. M., An example of soliton behavior in a rotating baroclinic fluid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367, 219, (1979) · Zbl 0423.76016 [17] Yuan, Y. Q.; Tian, B.; Liu, L.; Wu, X. Y.; Sun, Y.; Gao, X. Y.; Zhao, X. H.; Tian, B.; Xie, X. Y.; Wu, X. Y.; Sun, Y.; Guo, Y. J., Solitons, backlund transformation and Lax pair for a (2+1)-dimensional Davey-Stewartson system on surface waves of finite depth, J. Math. Anal. Appl., Ocean Eng., Wave. Random Complex, 28, 356, (2018) [18] Su, C. Q.; Gao, Y. T.; Yu, X.; Xue, L.; Shen, Y. J., Exterior differential expression of the (1+1)-dimensional nonlinear evolution equation with Lax integrability, J. Math. Anal. Appl., 435, 735, (2016) · Zbl 1360.37166 [19] Sasaki, R., Geometric approach to soliton equations, Proc. R. Soc. Lond., 373, 373, (1980) · Zbl 0451.35080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.