Abstract
AbstractThe main purpose of this work is solving a generalized (2 + 1)-dimensional nonlinear wave equation via$$\bar{\partial }$$∂¯-dressing method. The key to this process is to establish connection between characteristic functions and$$\bar{\partial }$$∂¯-problem. With use of Fourier transformation and Fourier inverse transformation, we obtain explicit expressions of Green’s function and give two characteristic functions corresponding to general potential. Further, the$$\bar{\partial }$$∂¯-problem is constructed by calculating$$\bar{\partial }$$∂¯derivative of characteristic function. The solution of$$\bar{\partial }$$∂¯-problem can be shown by Cauchy–Green formula, and after determining time evolution of scatter data, we can give solutions of the (2 + 1)-dimensional equation.
Funder
National Natural Science Foundation of China
K. C. Wong Magna Fund in Ningbo University
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference32 articles.
1. Zhu, J.Y., Geng, X.G.: A hierarchy of coupled evolution equations with self-consistent sources and the dressing method. J. Phys. A Math. Theor. 46(3), 35204–35204 (2013)
2. Zhu, J.Y., Geng, X.G.: The $$\bar{\partial }$$-dressing method for the Sasa–Satsuma equation with self-consistent sources. Chin. Phys. Lett. 30(8), 080204 (2013)
3. Bogdanov, L.V., Manakov, S.V.: The non-local delta problem and (2 + 1)-dimensional soliton equations. J. Phys. A Math. Theor. 21(10), L537 (1999)
4. Sun, S.F., Li, B.: A $$\bar{\partial }$$-dressing method for the mixed Chen–Lee–Liu derivative nonlinear Schödinger equation. J. Nonlinear Math. Phys. 30(1), 201–214 (2023)
5. Chai, X.D., Zhang, Y.F., Chen, Y., et al.: The $$\bar{\partial }$$-dressing method for the (2 + 1)-dimensional Jimbo–Miwa equation. American Mathematical Society, Providence (2022)