\(\bar{\partial}\)-dressing method for the \((2+1)\)-dimensional Korteweg-de Vries equation. (English) Zbl 1519.35280
Summary: In this paper, we investigate the \((2+1)\)-dimensional Korteweg-de Vries equation by applying the \(\bar{\partial}\)-dressing method. A new \(\bar{\partial}\) problem is obtained through the characteristic functions and Green’s functions by Lax representation. A solution of \(\bar{\partial}\) problem is constructed by applying Cauchy-Green formula and selecting appropriate spectral transformation. Finally, we can give the solution of the \((2+1)\) dimensional Korteweg-de Vries equation after determining the time evolution of the spectral data.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 34K35 | Control problems for functional-differential equations |
| 35P25 | Scattering theory for PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
\(\bar{\partial}\)-dressing method; Green’s function; characteristic function; \((2+1)\)-dimensional KdV equation; inverse spectral transformationReferences:
| [1] | Boiti, M.; Leon, J. J.P.; Manna, M.; Pempinelli, F., On the spectral transform of a Korteweg-devries equation in 2 spatial dimensions, Inverse Problems, 2, 271-279 (1986) · Zbl 0617.35119 |
| [2] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving Korteweg-devries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520 |
| [3] | Lou, S. Y.; Hu, X. B., Infinitely many Lax pairs and symmetry constraints of the KP equation, J. Math. Phys., 38, 6401-6427 (1997) · Zbl 0898.58029 |
| [4] | Estevez, P. G.; Leble, S., A wave equation in 2+1: painleve analysis and solutions, Inverse Problems, 11, 925 (1995) · Zbl 0834.35102 |
| [5] | Leble, S. B.; Ustinov, N. V., Third order spectral problems: reductions and Darboux transformations, Inverse Problems, 10, 617 (1994) · Zbl 0806.35170 |
| [6] | Hirota, R.; Satsuma, J., N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan, 40, 611 (2007) · Zbl 1334.76016 |
| [7] | Lou, S. Y.; Ruan, H. Y., Revisitation of the localized excitations of the (2+1)-dimensional KdV equation, J. Phys. A: Math. Gen., 34, 305-316 (2001) · Zbl 0979.37036 |
| [8] | Tang, X. Y.; Lou, S. Y., A variable separation approach to solve the integrable and nonintegrable models: Coheren structures of the (2+1)-dimensional KdV equation, Commun. Theor. Phys., 38, 1-8 (2002) · Zbl 1267.35206 |
| [9] | Liu, J. G.; Du, J. Q.; Zeng, Z. F.; Ai, G. P., Exact periodic cross-kink wave solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics, Chaos, 26, Article 103114 pp. (2016) · Zbl 1378.35270 |
| [10] | Xiang, C.; Wang, H., Travelling solitary wave solutions to higher order Korteweg-de Vries equation, Open J. Appl. Sci., 9, 354-360 (2019) |
| [11] | Zakharov, V. E.; Shabat, A. B., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, J. Funct. Anal. Appl., 8, 226-235 (1974) · Zbl 0303.35024 |
| [12] | Manakov, S. V., The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation, Physica D, 3, 1-2, 420-427 (1981) · Zbl 1194.35507 |
| [13] | Beals, R.; Coifman, R. R., The dbar approach to inverse scattering and nonlinear equations, Physica D, 18, 242-249 (1986) · Zbl 0619.35090 |
| [14] | Manakov, S. V., Scattering transformations spectrales et equations d’evolution nonlineare. I. Seminaire Goulaouic-Meyer-Schwartz, Physica D, 13, 420-427 (1981) · Zbl 1194.35507 |
| [15] | Zakharov, V. E.; Manakov, S. V., The construction of multidimensional nonlinear integrable systemsand their solutions, Funct. Anal. Appl., 19, 89 (1985) · Zbl 0597.35115 |
| [16] | Ablowitz, M. J.; Bar Yaacov, D.; Fokas, A. S., On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math., 69, 135-142 (1983) · Zbl 0527.35080 |
| [17] | Fokas, A. S.; Santini, P. M., Dromions and a boundary value problem for the Davey-Stewartson I equation, Physica D, 44, 99-130 (1990) · Zbl 0707.35144 |
| [18] | Doktorov, E. V.; Leble, S. B., A Dressing Method in Mathematical Physics (2007), Springer · Zbl 1142.35002 |
| [19] | Fokas, A. S.; Zakharov, V. E., The dressing method and nonlocal Riemann-Hilbert problem, J. Nonlinear Sci., 2, 109-134 (1992) · Zbl 0872.58032 |
| [20] | Konopelchenko, B. G.; Matkarimov, B. T., Inverse spectral transform for the nonlinear evolution equations generating the Davey-Stewartson and Ishimori equations, Stud. Appl. Math., 82, 319-359 (1990) · Zbl 0705.35133 |
| [21] | Bogdanov, L. V.; Manakov, S. V., The nonlocal Dbar problem and (2+1)-dimensional soliton equations, J. Phys. A, 21, L537 (1988) · Zbl 0662.35112 |
| [22] | Zhang, Z. C.; Fan, E. G., Inverse scattering transform for the Gerdjikov-Ivanov equation with nonzero boundary conditions, Z. Angew. Math. Phys., 71, 149 (2020) · Zbl 1448.35439 |
| [23] | Luo, J. H.; Fan, E. G., Dbar-dressing method for the Gerdjikov-Ivanov equation with nonzero boundary conditions, Appl. Math. Lett., 120, Article 107297 pp. (2021) · Zbl 1479.35809 |
| [24] | Zhu, J.; Kuang, Y., Solitons to the long-short waves equation and the \(\overline{\partial} \)-dressing method, Rep. Math. Phys., 75, 199-211 (2015) · Zbl 1321.35215 |
| [25] | Kuang, Y.; Zhu, J., The higher-order soliton solutions for the coupled Sasa-Satsuma system via the \(\overline{\partial} \)-dressing method, Appl. Math. Lett., 66, 47-53 (2017) · Zbl 1356.35084 |
| [26] | Chai, X. D.; Zhang, Y. F., The \(\overline{\partial} \)-dressing method for the (2+1)-dimensional Konopelchenko-Dubrovsky equation, Appl. Math. Lett., 134, Article 108378 pp. (2022) · Zbl 1498.35176 |
| [27] | Cao, Y.; Hu, P. Y.; Cheng, Y.; He, J. S., Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation, Chin. Phys. B, 30, Article 030503 pp. (2021) |
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