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INVERSE SCATTERING TRANSFORM FOR NEW MIXED SPECTRAL ABLOWITZ-KAUP-NEWELL-SEGUR EQUATIONS

ABSTRACT
The inverse scattering transform plays a very important role in promoting the development of analytical methods to solve non-linear PDE exactly. In this paper, new and more general mixed spectral Ablowitz-Kaup-Newell-Segur equations are derived and solved by embedding a novel time-varying spectral parameter in-to an associated linear problem and the inverse scattering transform. As a result, new exact solutions and n-soliton solutions are obtained. To gain more insights into the embedded spectral parameter and the obtained solutions, some dynamical evolutions, and spatial structures are simulated. It is shown that the derived Ablowitz-Kaup-Newell-Segur equations are Lax integrable and the obtained soliton solutions possess time-varying amplitudes.
KEYWORDS
PAPER SUBMITTED: 2018-04-28
PAPER REVISED: 2019-09-01
PAPER ACCEPTED: 2019-09-08
PUBLISHED ONLINE: 2020-06-21
DOI REFERENCE: https://doi.org/10.2298/TSCI2004437Z
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE 4, PAGES [2437 - 2444]
REFERENCES
  1. Garder, C. S., et al., Method for Solving the Korteweg-de Vries Equation, Physical Review Letters, 19 (1967), 19, pp. 1095-1097
  2. Ablowitz, M. J., et al., The Inverse Scattering Transform-Fourier Analysis for Non-Linear Problems, Studies in Applied Mathematics, 53 (1974), 4, pp. 249-315
  3. Hirota, R., Exact Solution of the sine-Gordon Equation for Multiple Collisions of Solitons, Journal of Physical Society of Japan, 33 (1972), 5, pp. 1459-1463
  4. Zhang, S., et al., Bilinearization and New Multi-Soliton Solutions for the (4+1)-Dimensional Fokas Equation, Pramana-Journal of Physics, 86 (2016), 6, pp. 1259-1267
  5. Zhang, S., Gao, X. D., Exact N-soliton Solutions and Dynamics of a New AKNS Equations with Time-Dependent Coefficients, Nonlinear Dynamics, 83 (2016), 1, 1043-1052
  6. Zhang, S., et al., New Multi-Soliton Solutions of Whitham-Broer-Kaup Shallow-Water-Wave Equations, Thermal Science, 21 (2017), Suppl. 1, pp. S137-S144
  7. Zhang, S., Gao, X. D., Analytical Treatment on a New Generalized Ablowitz-Kaup-Newell-Segur Hier-archy of Thermal and Fluid Equations, Thermal Science, 21 (2017), 4, pp. 1607-1612
  8. Weiss, J., et al., The Painlevé Property for Partial Differential Equations, Journal of Mathematical Phys-ics, 24 (1983), 3, pp. 522-526
  9. Zhang S., et al., Painleve Analysis for a Forced Korteveg-de Vries Equation Arisen in Fluid Dynamics of Internal Solitary Waves, Thermal Science, 19 (2015), 4, pp. 1223-1226
  10. Zhang, S., Chen, M. T., Painlevé Integrability and New Exact Solutions of the (4+1)-Dimensional Fokas Equation, Mathematical Problems in Engineering, 2015 (2015), ID 367425
  11. Wang, M. L., Exact Solutions for a Compound KdV-Burgers Equation, Physics Letters A, 213 (1996), 5-6, pp. 279-287
  12. Zhang, S., Wang, Z. Y., Improved Homogeneous Balance Method for Multi-Soliton Solutions of Gard-ner Equation with Time-Dependent Coefficients, IAENG International Journal of Applied Mathematics, 46 (2016), 4, pp. 592-599
  13. Zhang, S., Xia, T. C., A Generalized Auxiliary Equation Method and its Application to (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Vesselov Equations, Journal of Physics A: Mathematical and Theoretical, 40 (2006), 2, pp. 227-248
  14. Zhang, S., Xia, T. C., A Generalized F-expansion Method and New Exact Solutions of Konopelchenko-Dubrovsky Equations, Applied Mathematics and Computation, 183 (2006), 3, pp. 1190-1200
  15. Zhang, S., Zhang, H. Q., Fractional Sub-Equation Method and its Applications to Non-Linear Fractional PDE, Physics Letters A, 375 (2011), 7, pp. 1069-1073
  16. Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
  17. He, J. H., Homotopy Perturbation Technique, Computer Methods in Applied Mechanics and Engineer-ing, 178 (1999), 3-4, pp. 257-262
  18. He, J. H., Wu, X. H., Exp-Function Method for Non-Linear Wave Equations, Chaos, Solitons and Fractals, 30 (2006), 3, pp. 700-708
  19. Zhang, S., et al., Multi-Wave Solutions for a Non-Isospectral KdV-Type Equation with Variable Coefficients, Thermal Science, 16 (2012), 5, pp. 1576-1579
  20. Zhang, S., et al., A Direct Algorithm of Exp-Function Method for Non-Linear Evolution Equations in Fluids, Thermal Science, 20 (2016), 3, pp. 881-884
  21. Zhang, S., Hong, S. Y., On a Generalized Ablowitz-Kaup-Newell-Segur Hierarchy in Inhomogeneities of Media: Soliton Solutions and Wave Propagation Influenced from Coefficient Functions and Scattering Data, Waves in Random and Complex Media, 28 (2018), 3, pp. 435-452
  22. Zhang, S., Wang, D., Variable-Coefficient Non-Isospectral Toda Lattice Hierarchy and its Exact Solutions, Pramana-Journal of Physics, 85 (2015), 6, 1143-1156
  23. Zhang, S., Gao, X. D., Exact Solutions and Dynamics of a Generalized AKNS Equations Associated with the Non-Isospectral Depending on Exponential Function, Journal of Nonlinear Sciences and Appli-cations, 19 (2016), 6, pp. 4529-4541
  24. Zhang, S., Li, J. H., On Nonisospectral AKNS System with Infinite Number of Terms and its Exact So-lutions, IAENG International Journal of Applied Mathematics, 47 (2017), 1, pp. 89-96
  25. Zhang, S., Li, J. H., Soliton Solutions and Dynamical Evolutions of a Generalized AKNS System in the Framework of Inverse Scattering Transform, Optik, 137 (2017), 1, pp. 228-237
  26. Gao, X. D., Zhang, S., Inverse Scattering Transform for a New Non-Isospectral Integrable Non-Linear AKNS Model, Thermal Science, 21 (2017), Suppl. 1, pp. S153-S160
  27. Zhang, S., Hong, S. Y., Lax Integrability and Exact Solutions of a Variable-Coefficient and Non-Isospectral AKNS Hierarchy, International Journal of Nonlinear Sciences and Numerical Simulation, 19 (2018), 3-4, pp. 251-262
  28. Zhang, S., et al., Exact Solutions of a KdV Equation Hierarchy with Variable Coefficients, International Journal of Computer Mathematics, 91 (2014), 7, pp. 1601-1616
  29. Zhang, S., Gao, X. D., Mixed Spectral AKNS Hierarchy from Linear Isospectral Problem and its Exact Solutions, Open Physics, 13 (2015), 1, pp. 310-322
  30. Zhang, S., Gao, X. D., A New Variable-Coefficient AKNS Hierarchy and its Exact Solutions via Inverse Scattering Transform, Advances in Computer Science Research, 71 (2017), Jan., pp. 1238-1243
  31. Zhang, S., Hong, S. Y., Lax Integrability and Soliton Solutions for a Non-Isospectral Integro-Differential System, Complexity, 2017 (2017), ID 9457078
  32. Gao, X. D., Zhang, S., Time-Dependent-Coefficient AKNS Hierarchy and its Exact Multi-Soliton Solu-tions, International Journal of Applied Science and Mathematics, 3 (2016), 2, pp. 72-75
  33. Xu, B., Zhang, S., Derivation and Soliton Dynamics of a New Non-Isospectral and Variable-Coefficient System, Thermal Science, 23 (2019), Suppl. 3, pp. S639-S646
  34. Zhang, S. You, C. H., Inverse Scattering Transform for a Supersymmetric Korteweg-de Vries Equation, Thermal Science, 23 (2019), Suppl. 3, pp. S677-S684
  35. He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934
  36. He, J. H., The Simplest Approach to Nonlinear Oscillators, Results in Physics, 15 (2019), 102546
  37. He, J. H., Ji, F. Y. Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
  38. He, J. H., A Simple Approach to One-Dimensional Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), 113565
  39. He, J.H., A Modified Li-He's Variational Principle for Plasma, International Journal of Numerical Methods for Heat and Fluid Flow, On line first, doi.org/10.1108/HFF-06-2019-0523, 2019
  40. He, J. H., Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, On line first, doi.org/10.1108/HFF-07-2019-0577, 2019
  41. He, J. H., Sun, C., A Variational Principle for a Thin Film Equation, Journal of Mathematical Chemistry, 57 (2019), 9, pp. 2075-2081
  42. He, J. H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, J. Appl. Comput. Mech., 6 (2020), 4, pp. 735-740
  43. Zhang, J. J., et al. Some Analytical Methods for Singular Boundary Value Problem in a Fractal Space, Appl. Comput. Math., 18 (2019), 3, pp. 225-235

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