Inverse scattering transform in two spatial dimensions for the N-wave interaction problem with a dispersive term

References

[1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981. 10.1137/1.9781611970883Search in Google Scholar

[2] A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys 74 (2019), no. 3, 373–386. 10.1070/RM9867Search in Google Scholar

[3] R. Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the ∂ ¯ -method, Inverse Problems 5 (1989), no. 2, 87–130. 10.1088/0266-5611/5/2/002Search in Google Scholar

[4] D. J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math. 56 (1976/77), no. 1, 81–94. 10.1002/sapm197756181Search in Google Scholar

[5] M. Boiti, B. G. Konopelchenko and F. Pempinelli, Bäcklund transformations via gauge transformations in 2 + 1 dimensions, Inverse Problems 1 (1985), no. 1, 33–56. 10.1088/0266-5611/1/1/005Search in Google Scholar

[6] O. I. Chvartatskyi and Y. M. Sydorenko, A new bidirectional generalization of ( 2 + 1 ) -dimensional matrix k-constrained Kadomtsev–Petviashvili hierarchy, J. Math. Phys. 54 (2013), no. 11, Article ID 113508. 10.1063/1.4830025Search in Google Scholar

[7] O. I. Chvartatskyi and Y. M. Sydorenko, Darboux transformations for ( 2 + 1 ) -dimensional extensions of the KP hierarchy, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper No. 028. 10.3842/SIGMA.2015.028Search in Google Scholar

[8] H. Cornille, Solutions of the generalized nonlinear Schrödinger equation in two spatial dimensions, J. Math. Phys. 20 (1979), no. 1, 199–209. 10.1063/1.523942Search in Google Scholar

[9] A. S. Fokas and M. J. Ablowitz, Method of solution for a class of multidimensional nonlinear evolution equations, Phys. Rev. Lett. 51 (1983), no. 1, 7–10. 10.1103/PhysRevLett.51.7Search in Google Scholar

[10] A. S. Fokas and M. J. Ablowitz, On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev–Petviashvili equation, Stud. Appl. Math. 69 (1983), no. 3, 211–228. 10.1002/sapm1983693211Search in Google Scholar

[11] A. S. Fokas and M. J. Ablowitz, On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane, J. Math. Phys. 25 (1984), no. 8, 2494–2505. 10.1063/1.526471Search in Google Scholar

[12] A. S. Fokas and L.-Y. Sung, On the solvability of the N-wave, Davey–Stewartson and Kadomtsev–Petviashvili equations, Inverse Problems 8 (1992), no. 5, 673–708. 10.1088/0266-5611/8/5/002Search in Google Scholar

[13] A. S. Fokas and V. E. Zakharov, The dressing method and nonlocal Riemann–Hilbert problems, J. Nonlinear Sci. 2 (1992), no. 1, 109–134. 10.1007/BF02429853Search in Google Scholar

[14] B. K. Harrison, On methods of finding Bäcklund transformations in systems with more than two independent variables, J. Nonlinear Math. Phys. 2 (1995), no. 3–4, 201–215. 10.2991/jnmp.1995.2.3-4.1Search in Google Scholar

[15] N. S. Iskenderov and M. I. Ismailov, On the inverse scattering transform of a nonlinear evolution equation with 2 + 1 dimensions related to nonstrict hyperbolic systems, Nonlinearity 25 (2012), no. 7, 1967–1979. 10.1088/0951-7715/25/7/1967Search in Google Scholar

[16] M. I. Ismailov, Inverse scattering problem for nonstationary Dirac-type systems on the plane, J. Math. Anal. Appl. 365 (2010), no. 2, 498–509. 10.1016/j.jmaa.2009.10.068Search in Google Scholar

[17] M. I. Ismailov, Integration of nonlinear system of four waves with two velocities in ( 2 + 1 ) dimensions by the inverse scattering transform method, J. Math. Phys. 52 (2011), no. 3, Article ID 033504. 10.1063/1.3560476Search in Google Scholar

[18] B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations. The Inverse Spectral Transform in 2 + 1 Dimensions, Plenum Press, New York, 1992. 10.1007/978-1-4899-1170-4Search in Google Scholar

[19] Y. K. Liu and B. Li, Rogue waves in the ( 2 + 1 ) -dimensional nonlinear schrödinger equation with a parity-time-symmetric potential, Chin. Phys. Lett. 34 (2017), Article ID 010202. 10.1088/0256-307X/34/1/010202Search in Google Scholar

[20] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetig waves, Sov. Phys. JETP 38 (1974), 248–253. Search in Google Scholar

[21] L. P. Nižnik, An inverse problem of nonstationary scattering for the Dirac equations, Ukrainian Math. J. 24 (1972), 112–115. 10.1007/BF01089129Search in Google Scholar

[22] L. P. Nižnik, Integration of multidimensional nonlinear equations by the inverse problem method, Soviet Phys. Dokl. 25 (1980), no. 9, 706–708. Search in Google Scholar

[23] L. P. Nizhnik, The inverse scattering problems for the hyperbolic equations and their application to nonlinear integrable systems, Rep. Math. Phys. 26 (1988), no. 2, 261–283. 10.1016/0034-4877(88)90028-6Search in Google Scholar

[24] L. P. Nizhnik and M. D. Pochinaĭko, Integration of a spatially two-dimensional nonlinear Schrödinger equation by the inverse problem method, Funct. Anal. Appl. 16 (1982), no. 1, 80–82. 10.1007/BF01081819Search in Google Scholar

[25] R. G. Novikov, Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system, Selecta Math. (N. S.) 3 (1997), no. 2, 245–302. 10.1007/s000290050011Search in Google Scholar

[26] A. Sakhnovich, Dirac type system on the axis: Explicit formulae for matrix potentials with singularities and soliton-positon interactions, Inverse Problems 19 (2003), no. 4, 845–854. 10.1088/0266-5611/19/4/304Search in Google Scholar

[27] L.-Y. Sung and A. S. Fokas, Inverse problem for N × N hyperbolic systems on the plane and the N-wave interactions, Comm. Pure Appl. Math. 44 (1991), no. 5, 535–571. 10.1002/cpa.3160440503Search in Google Scholar

[28] P. L. Vu, The inverse scattering problem for the perturbed string equation and its application to integration of the two-dimensional generalization from Korteweg–de Vries equation, Acta Math. Vietnam. 45 (2020), no. 4, 807–831. 10.1201/9780429328459-8Search in Google Scholar

[29] V. E. Zakharov and A. B. Shabat, The scheme of integration of nonlinear equations of mathematical physics by inverse scattering method. I, Funct. Anal. Appl. 8 (1974), no. 3, 226–235. 10.1007/BF01075696Search in Google Scholar

[30] V. E. Zakharov and A. B. Shabat, Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem. II, Funct. Anal. Appl. 13 (1979), no. 3, 166–174. 10.1007/BF01077483Search in Google Scholar

[31] Y. Zhang and Y. Liu, Darboux transformation and explicit solutions for 2 + 1 -dimensional nonlocal Schrödinger equation, Appl. Math. Lett. 92 (2019), 29–34. 10.1016/j.aml.2018.12.014Search in Google Scholar