THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Xudong Gao

,

Sheng Zhang

INVERSE SCATTERING TRANSFORM FOR A NEW NON-ISOSPECTRAL INTEGRABLE NON-LINEAR AKNS MODEL

ABSTRACT
Constructing integrable systems and solving non-linear partial differential equations are important and interesting in non-linear science. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS)'s linear isospectral problem and its accompanied time evolution equation are first generalized by embedding a new non-isospectral parameter whose varying with time obeys an arbitrary smooth enough function of the spectral parameter. Based on the generalized AKNS linear problem and its evolution equation, a new non-isospectral Lax integrable non-linear AKNS model is then derived. Furthermore, exact solutions of the derived AKNS model is obtained by extending the inverse scattering transformation method with new time-varying spectral parameter. In the case of reflectinless potentials, explicit n-soliton solutions are finally formulated through the obtained exact solutions.
KEYWORDS
non-isospectral integrable non-linear AKNS model, AKNS linear problem, Lax integrability, inverse scattering transformation method, soliton solution
PAPER SUBMITTED: 2017-04-12
PAPER REVISED: 2017-05-16
PAPER ACCEPTED: 2017-05-26
PUBLISHED ONLINE: 2017-12-02
DOI REFERENCE: https://doi.org/10.2298/TSCI17S1153G
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Supplement 1, PAGES [S153 - S160]
REFERENCES
  1. Chen, D. Y., Introduction to Soliton (in Chinese), Science Press, Beijing, 2006
  2. Ablowitz, M. J., Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, Mass., USA, 1991
  3. Zhang, S., Gao, X. D., Exact Solutions and Dynamics of Generalized AKNS Equations Associated with the Non-Isospectral Depending on Exponential Function, Journal of Nonlinear Science and Applications, 19 (2016), 6, pp. 4529-4541
  4. Zhang, S., Li, J. H., On Non-Isospectral AKNS System with Infinite Number of Terms and its Exact Solutions, IAENG International Journal of Applied Mathematics, 47 (2017), 1, pp. 89-96
  5. Zhang, S., Li, J. H., Soliton Solutions and Dynamical Evolutions of a Generalized AKNS System in the Framework of Inverse Scattering Transform, Optik-International Journal for Light and Electron Optics, 137 (2017), 1, pp. 228-237
  6. Calogreo, F., Degasperis, A., Exact Solution via the Spectral Transform of a Generalization with Linearly x-Dependent Coefficients of the Modified Korteweg-de Vries Equation, Letter al Nuovo Cimento, 22 (1978), 7, pp. 270-273
  7. Zhang, S., Wang, D., Variable-Coefficient Non-Isospectral Toda Lattice Hierarchy and its Exact Solutions, Pramana-Journal of Physics, 85 (2015), 6, pp. 1143-1156
  8. 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
  9. Lou, S. Y., Tang, X. Y., Method of Non-Linear Mathematical Physics (in Chinese), Science Press, Beijing, 2006
  10. Gardner, C. S., et al., Method for Solving the Korteweg-de Vries Equation, Physical Review Letters, 19 (1967), 19, pp. 1095-1197
  11. Zhang, S., et al., Variable Separation for Time Fractional Advection-Dispersion Equation with Initial and Boundary Conditions, Thermal Science, 20 (2016), 3, pp. 789-792
  12. Yang, X. J., et al., On Exact Traveling-Wave Solutions for Local Fractional Korteweg-de Vries Equation, Chaos, 26 (2016), 8, ID 084312
  13. Yang, X. J., et al., Exact Travelling Wave Solutions for the Local Fractional Two-Dimensional Burgers- Type Equations, Computers and Mathematics with Applications, 73 (2017), 2, pp. 203-210
  14. Yang, X. J., et al., On a Fractal LC-Electric Circuit Modeled by Local Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 47 (2017) 6, pp. 200-206
  15. Fujioka, J., et al., Fractional Optical Solitons, Physics Letters A, 374 (2010), 9, pp. 1126-1134
  16. Yang, X. J., et al., Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation, Abstract and Applied Analysis, 2014 (2014), ID 278672
  17. Chen, H. H., Liu, C. S., Solitons in Nonuniform Media, Physical Review Letters, 37 (1976) , 11, pp. 693-697
  18. Serkin, V. N., et al., Nonautonomous Solitons in External Potentials, Physical Review Letters, 98 (2007), 7, ID 074102
PDF VERSION [DOWNLOAD]
Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence