A Study of Nonholonomic Deformations of Nonlocal Integrable Systems Belonging to the Nonlinear Schrödinger Family
2019, Vol. 15, no. 3, pp. 293-307
Author(s): Mukherjee I., Guha P.
Download File PDF, 257.61 Kb |
References |
|
[1] |
Das, A., Integrable Models, World Sci. Lecture Notes Phys., 30, World Sci., Teaneck, N.J., 1989, xiv+342 pp. ![]() ![]() ![]() ![]() |
[2] |
Lax, P. D., “Integrals of Nonlinear Equations of Evolution and Solitary Waves”, Comm. Pure Appl. Math., 21 (1968), ![]() ![]() ![]() |
[3] |
Ablowitz, M. J. and Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., 149, Cambridge Univ. Press, Cambridge, 1991, xii+516 pp. ![]() ![]() |
[4] |
Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H., “The Inverse Scattering Transform: Fourier Analysis for Nonlinear Problems”, Stud. Appl. Math., 53:4 (1974), ![]() ![]() ![]() |
[5] |
Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987, x+592 pp. ![]() ![]() |
[6] |
Magri, F., “A Simple Model of the Integrable Hamiltonian Equation”, J. Math. Phys., 19:5 (1978), ![]() ![]() ![]() ![]() |
[7] |
Malomed, B., “Nonlinear Schrödinger Equations”, Encyclopedia of Nonlinear Science, ed. A. Scott, Routledge, New York, 2005, ![]() |
[8] |
Zhidkov, P. E., Korteweg – de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math., 1756, Springer, Berlin, 2001, vi+147 pp. ![]() |
[9] |
Rogers, C. and Schief, W., Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge Texts Appl. Math., 30, Cambridge Univ. Press, Cambridge, 2002, 432 pp. ![]() |
[10] |
Ablowitz, M. and Musslimani, Z., “Integrable Nonlocal Nonlinear Schrödinger Equation”, Phys. Rev. Lett., 110:6 (2013), 064105, 5 pp. ![]() ![]() |
[11] |
Guo, A., Salamo, G. J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G. A., and Christodoulides, D. N., “Observation of PT-Symmetry Breaking in Complex Optical Potentials”, Phys. Rev. Lett., 103:9 (2009), 093902, 4 pp. ![]() ![]() |
[12] |
Regensburger, A., Bersch, C., Miri, M. A., Onishchukov, G., Christodoulides, D. N., and Peschel, U., “Parity-Time Synthetic Photonic Lattices”, Nature, 488:7410 (2012), ![]() ![]() ![]() |
[13] |
Rüter, Ch. E., Makris, K. G., El-Ganainy, R., Christodoulides, D. N., Segev, M., and Kip, D., “Observation of Parity-Time Symmetry in Optics”, Nature Phys., 6:3 (2010), ![]() ![]() ![]() |
[14] |
Fokas, A. S., Its, A. R., and Sung, L.-Y., “The Nonlinear Schrödinger Equation on the Half-Line”, Nonlinearity, 18:4 (2005), ![]() ![]() ![]() ![]() ![]() |
[15] |
Fokas, A. S., “Integrable Nonlinear Evolution Equations on the Half-Line”, Comm. Math. Phys., 230:1 (2002), ![]() ![]() ![]() ![]() |
[16] |
Fokas, A. S. and Its, A. R., “The Nonlinear Schrödinger Equation on the Interval”, J. Phys. A, 37:23 (2004), ![]() ![]() ![]() ![]() |
[17] |
Fokas, A. S. and Its, A. R., “The Linearization of the Initial-Boundary Value Problem of the Nonlinear Schrödinger Equation”, SIAM J. Math. Anal., 27:3 (1996), ![]() ![]() ![]() |
[18] |
Its, A. R., “Asymptotic Behavior of the Solutions to the Nonlinear Schrödinger Equation, and Isomonodromic Deformations of Systems of Linear Differential Equations”, Soviet Math. Dokl., 24:3 (1981), ![]() ![]() ![]() ![]() ![]() |
[19] |
Ablowitz, M. J., Fokas, A. S., and Musslimani, Z. H., “On a New Nonlocal Formulation of Water Waves”, J. Fluid Mech., 562 (2006), ![]() ![]() ![]() ![]() |
[20] |
Fokas, A. S., “Integrable Multidimensional Versions of the Nonlocal Nonlinear Schrödinger Equation”, Nonlinearity, 29:2 (2016), ![]() ![]() ![]() ![]() |
[21] |
Ablowitz, M. J. and Musslimani, Z. H., “Integrable Nonlocal Nonlinear Equations”, Stud. Appl. Math., 139:1 (2017), ![]() ![]() ![]() ![]() |
[22] |
Ablowitz, M. J. and Musslimani, Z. H., “Integrable Discrete PT Symmetric Model”, Phys. Rev. E, 90:3 (2014), 032912, 5 pp. ![]() ![]() ![]() ![]() |
[23] |
Ablowitz, M. J. and Musslimani, Z. H., “Inverse Scattering Transform for the Integrable Nonlocal Nonlinear Schrödinger Equation”, Nonlinearity, 29:3 (2016), ![]() ![]() ![]() ![]() |
[24] |
Ablowitz, M. J., Luo, X.-D., and Musslimani, Z. H., “Inverse Scattering Transform for the Nonlocal Nonlinear Schrödinger Equation with Nonzero Boundary Conditions”, J. Math. Phys., 59:1 (2018), 011501, 42 pp. ![]() ![]() ![]() ![]() |
[25] |
Valchev, T., “On a Nonlocal Nonlinear Schrödinger Equation”, Mathematics in Industry, ed. A. Slavova, Cambridge Scholars Publ., Cambridge, 2014, |
[26] |
Gürses, M. and Pekcan, A., “Nonlocal Nonlinear Schrödinger Equations and Their Soliton Solutions”, J. Math. Phys., 59:5 (2018), 051501, 17 pp. ![]() ![]() ![]() ![]() |
[27] |
Gerdjikov, V. S. and Saxena, A., “Complete Integrability of Nonlocal Nonlinear Schrödinger Equation”, J. Math. Phys., 58:1 (2017), 013502, 33 pp. ![]() ![]() ![]() ![]() ![]() |
[28] |
Bender, C. M., PT Symmetry in Quantum and Classical Physics, World Sci., Hackensack, N.J., 2019, xxii+446 pp. ![]() ![]() |
[29] | Bender, C. M., Introduction to PT-Symmetric Quantum Theory, 2005, 23 pp., arXiv: quant-ph/0501052 |
[30] |
Bender, C. M., “PT Symmetric Quantum Theory”, J. Phys. Conf. Ser., 631 (2015), 012002, 12 pp. ![]() ![]() |
[31] |
Karasu-Kalkanli, A., Karasu, A., Sakovich, A., Sakovich, S., and Turhan, R., “A New Integrable Generalization of the Korteweg – de Vries Equation”, J. Math. Phys., 49:7 (2008), 073516, 10 pp. ![]() ![]() ![]() ![]() ![]() |
[32] |
Kuperschmidt, B. A., “KdV6: An Integrable System”, Phys. Lett. A, 372:15 (2008), ![]() ![]() ![]() |
[33] |
Kundu, A., “Exact Accelerating Solitons in Nonholonomic Deformation of the KdV Equation with Two-Fold Integrable Hierarchy”, J. Phys. A, 41:49 (2008), 495201, 7 pp. ![]() ![]() ![]() |
[34] |
Kundu, A., Sahadevan, R., and Nalinidevi, L., “Nonholonomic Deformation of KdV and mKdV Equations and Their Symmetries, Hierarchies and Integrability”, J. Phys. A, 42:11 (2009), 115213, 13 pp. ![]() ![]() ![]() ![]() ![]() |
[35] |
Kundu, A., “Two-Fold Integrable Hierarchy of Nonholonomic Deformation of the Derivative Nonlinear Schrödinger and the Lenells – Fokas Equation”, J. Math. Phys., 51:2 (2010), 022901, 17 pp. ![]() ![]() ![]() ![]() ![]() |
[36] |
Guha, P., “Nonholonomic Deformation of Generalized KdV-Type Equations”, J. Phys. A, 42:34 (2009), 345201, 17 pp. ![]() ![]() ![]() ![]() |
[37] |
Guha, P., “Nonholonomic Deformation of Coupled and Supersymmetric KdV Equations and Euler – Poincaré – Suslov Method”, Rev. Math. Phys., 27:4 (2015), ID 1550011-25 ![]() ![]() ![]() ![]() |
[38] |
Abhinav, K., Guha, P., and Mukherjee, I., “Study of Quasi-Integrable and Non-Holonomic Deformation of Equations in the NLS and DNLS Hierarchy”, J. Math. Phys., 59:10 (2018), 101507, 18 pp. ![]() ![]() ![]() ![]() |
[39] |
Fokas, A. S. and Pelloni, B., “A Transform Method for Evolution PDEs on the Interval”, IMA J. Appll. Math., 70:4 (2005), ![]() ![]() ![]() |
[40] |
Fokas, A. S. and Pelloni, B., Unified Transform for Boundary Value Problems: Applications and Advances, SIAM, Philadelphia, Pa., 2014, 305 pp. ![]() |
[41] |
Krupková, O., “Mechanical Systems with Nonholonomic Constraints”, J. Math. Phys., 38:10 (1997), ![]() ![]() ![]() ![]() |
[42] |
Ferreira, L. A. and Zakrzewski, W. J., “The Concept of Quasi-Integrability: A Concrete Example”, J. High Energy Phys., 2011, no. 5, 130, 39 pp. ![]() ![]() ![]() ![]() |
[43] |
Blas, H. and Zambrano, M., “Quasi-Integrability in the Modified Defocusing Non-Linear Schrödinger Model and Dark Solitons”, J. High Energy Phys., 2016, no. 3, 005, front matter + , 47 pp. ![]() ![]() |
[44] |
Ablowitz, M. J., Chakravarty, S., and Takhtajan, L. A., “A Self-Dual Yang – Mills Hierarchy and Its Reductions to Integrable Systems in ![]() ![]() ![]() ![]() |
[45] |
Ablowitz, M. J., Chakravarty, S., and Halburd, R. G., “Integrable Systems and Reductions of the Self-Dual Yang – Mills Equations: Integrability, Topological Solitons and Beyond”, J. Math. Phys., 44:8 (2003), ![]() ![]() ![]() ![]() |
[46] |
Ablowitz, M. J. and Musslimani, Z. H., Integrable Nonlocal Asymptotic Reductions of Physically Significant Nonlinear Equations, 2019, 9 pp., arXiv: 1903.06752 [nlin.SI] ![]() ![]() |
[47] | Mukherjee, I., Guha, P., and Abhinav, K., Studies on Different Aspects of Nonlocal Integrable Systems: Soliton Solutions, Deformations and Reductions from SDYM Equations (to appear) |

This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License