The ∂̄-dressing method for the (2+1)-dimensional Jimbo-Miwa equation

Chai, Xuedong; Zhang, Yufeng; Chen, Yong; Zhao, Shiyin

doi:10.1090/proc/15716

The $\bar {{\partial }}$-dressing method for the (2+1)-dimensional Jimbo-Miwa equation
HTML articles powered by AMS MathViewer

by Xuedong Chai, Yufeng Zhang, Yong Chen and Shiyin Zhao PDF

Proc. Amer. Math. Soc. 150 (2022), 2879-2887 Request permission

Abstract:

The (2+1)-dimensional Jimbo-Miwa equation is analyzed by means of the $\bar {{\partial }}$-dressing method. By means of the characteristic function and Green’s function of the Lax representation, the problem has been transformed into a new $\bar {{\partial }}$ problem. A solution is constructed based on solving the $\bar {{\partial }}$ problem with the help of Cauchy-Green formula and choosing the proper spectral transformation. Furthermore, we can obtain the solution formally of the Jimbo-Miwa equation when the time evolution of the spectral data is determined.

References

M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991. MR 1149378, DOI 10.1017/CBO9780511623998
M. J. Ablowitz, D. Bar Yaacov, and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 69 (1983), no. 2, 135–143. MR 715426, DOI 10.1002/sapm1983692135
A. Bekir, Painlevé test for some $(2+1)$-dimensional nonlinear equations, Chaos Solitons Fractals 32 (2007), no. 2, 449–455. MR 2280095, DOI 10.1016/j.chaos.2006.06.047
A. M. Bruckner and J. B. Bruckner, Darboux transformations, Trans. Amer. Math. Soc. 128 (1967), 103–111. MR 218500, DOI 10.1090/S0002-9947-1967-0218500-1
Yong Chen, Zhenya Yan, and Wenjun Liu, Impact of near-PT symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model, Optics express 26 (2018), no. 25, 33022–33034.
Yong Chen, Zhenya Yan, and Dumitru Mihalache, Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity, Phys. Rev. E 102 (2020), no. 1, 012216, 11. MR 4137469, DOI 10.1103/PhysRevE.102.012216
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
Evgeny V. Doktorov and Sergey B. Leble, A dressing method in mathematical physics, Mathematical Physics Studies, vol. 28, Springer, Dordrecht, 2007. MR 2345237, DOI 10.1007/1-4020-6140-4
V. G. Dubrovsky, The application of the $\overline \partial$-dressing method to some integrable $(2+1)$-dimensional nonlinear equations, J. Phys. A 29 (1996), no. 13, 3617–3630. MR 1400166, DOI 10.1088/0305-4470/29/13/027
V. G. Dubrovsky and A. V. Topovsky, Multi-lump solutions of KP equation with integrable boundary via $\overline \partial$-dressing method, Phys. D 414 (2020), 132740, 11. MR 4156154, DOI 10.1016/j.physd.2020.132740
A. S. Fokas and V. E. Zakharov, The dressing method and nonlocal Riemann-Hilbert problems, J. Nonlinear Sci. 2 (1992), no. 1, 109–134. MR 1158355, DOI 10.1007/BF02429853
Hirota and Ryogo, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), no. 18, 1456–1458.
Ryogo Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Mathematical Phys. 14 (1973), 805–809. MR 338587, DOI 10.1063/1.1666399
Ryogo Hirota, The direct method in soliton theory, 2004.
Michio Jimbo and Tetsuji Miwa, Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 943–1001. MR 723457, DOI 10.2977/prims/1195182017
B. G. Konopel′chenko, Solitons in multidimensions, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. Inverse spectral transform method. MR 1249273, DOI 10.1142/1982
B. G. Konopel′chenko, Introduction to multidimensional integrable equations: the inverse spectral transform in 2+1 dimensions, Springer Science & Business Media, 2013.
Yonghui Kuang and Junyi Zhu, The higher-order soliton solutions for the coupled Sasa-Satsuma system via the $\overline {\partial }$-dressing method, Appl. Math. Lett. 66 (2017), 47–53. MR 3583858, DOI 10.1016/j.aml.2016.11.008
C. Rogers and W. F. Shadwick, Bäcklund transformations and their applications, Mathematics in Science and Engineering, vol. 161, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 658491
Manjit Singh, Multi-soliton solutions, bilinear Bäcklund transformations and Lax pair of nonlinear evolution equation in $(2+1)$-dimension, Comput. Methods Differ. Equ. 3 (2015), no. 2, 134–146. MR 3551571
John Weiss, M. Tabor, and George Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. 24 (1983), no. 3, 522–526. MR 692140, DOI 10.1063/1.525721
Jian Xu and Engui Fan, Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: without solitons, J. Differential Equations 259 (2015), no. 3, 1098–1148. MR 3342409, DOI 10.1016/j.jde.2015.02.046
V.E. Zakharov, Nonlinear and turbulent processes in Physics Proc. 3rd Int, Workshop, vol. 1, 1988.
V. E. Zakharov and S. V. Manakov, Construction of multidimensional nonlinear integrable systems and their solutions, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 11–25, 96 (Russian). MR 800917
Junyi Zhu and Xianguo Geng, The generalized dressing method with applications to variable-coefficient coupled Kadomtsev-Petviashvili equations, Chaos Solitons Fractals 31 (2007), no. 5, 1143–1148. MR 2261481, DOI 10.1016/j.chaos.2005.04.096
Junyi Zhu and Xianguo Geng, A hierarchy of coupled evolution equations with self-consistent sources and the dressing method, J. Phys. A 46 (2013), no. 3, 035204, 18. MR 3007513, DOI 10.1088/1751-8113/46/3/035204
Junyi Zhu and Yonghui Kuang, Cusp solitons to the long-short waves equation and the $\overline \partial$-dressing method, Rep. Math. Phys. 75 (2015), no. 2, 199–211. MR 3343239, DOI 10.1016/S0034-4877(15)30003-3