Inverse Scattering Transform for the Fifth-Order Fractional KdV Equation with a Self-Consistent Source
DOI:
https://doi.org/10.56143/ujmcs.v2i3s.6Keywords:
Riesz fractional Korteweg-de Vries equation, self-consistent source, inverse scattering transform method, Sturm-Liouville operatorAbstract
In this paper, we constract and integrate the fifth-order Riesz fractional KdV (fRfKdV) equation with a self-consistent source in the class of rapidly decreasing functions via the inverse scattering transform(IST) method. We show that fifth-order Riesz fractional KdV equation with self-consistent sources is also
an important theoretical model, since it is a completely integrable system. For one soliton cases, explicit formulas for the solutions of the problem under consideration are derived as examples to simulate their spatial structures and analyze their structural properties by selecting different values of fractional orders. The results indicate that both the source and the fractional order significantly influence the velocity of soliton propagation, while no energy dissipation occurs through out the entire motion process. This behavior corresponds to a super-dispersive transport mechanism, which is an experimentally testable prediction of the present theory. Furthermore, the analysis reveals that the absolute value of wave velocity becomes larger as
$\epsilon$ increases. This study contributes valuable insights into nonlinear dynamics and soliton behavior, improving the understanding of multifaceted wave interactions in nonlinear fractional equations.
References
[1] Drazin, P.G.; Johnson, R.S. Solitons: An Introduction; Cambridge University Press: New York, NY, USA, 1989.
[2] Kakutani, T.; Ono, H. Weak non-linear hydromagnetic waves in a cold collision free plasma. J. Phys. Soc. Jpn. 26, 1305–1318 (1969). doi:10.1143/JPSJ.26.1305
[3] Iqbal, N.; Akgul, A.; Shah, R.; Bariq, A.; Mossa Al-Sawalha, M.; Ali, A. On Solutions of Fractional-Order Gas Dynamics Equation by Effective Techniques. J. Funct. Spaces. 1–14 (2022). doi:10.1155/2022/2147805
[4] El-Tantawy, S.A.; Salas Alvaro, H.; Alharthi, M.R. Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation. Chaos Solitons Fractals, 147, 110965 (2021). doi:10.1016/j.chaos.2021.110965
[5] El-Tantawy, S.A.; Salas Alvaro, H.; Alharthi, M.R. On the dissipative extended Kawahara solitons and cnoidal waves in a collisional plasma: Novel analytical and numerical solutions. Phys. Fluids, 33, 106101 (2021). doi:10.1063/5.0065659
[6] Wazwaz, A.-M. New solitary wave solutions to the Kuramoto–Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182, 1642–1650 (2006). doi:10.1016/j.amc.2006.05.002
[7] Wazwaz, A.-M. Two-mode fifth-order KdV equations: Necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. 87, 1685–1691 (2017). doi:10.1007/s11071-016-3174-z
[8] Goswami, A.; Singh, J.; Kumar, D. Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves. Ain Shams Eng. J. 9, 2265–2273 (2018). doi:10.1016/j.asej.2017.03.004
[9] E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H. Schellnhuber, Indication of a universal persistence law governing atmospheric variability, Physical Review Letters, 81, 729–732 (1998). https://doi.org/10.1103/physrevlett.81.729
[10] M. F. Shlesinger, B. J. West, and J. Klafter, Levy dynamics of enhanced diffusion: application to turbulence, Physical Review Letters, 58, 1100–1103 (1987). https://doi.org/10.1103/physrevlett.58.1100
[11] W. Wang, A. G. Cherstvy, A. V. Chechkin, S. Thapa, F. Seno, X. Liu, and R. Metzler, Fractional Brownian motion with random diffusivity: emerging residual nonergodicity below the correlation time, Journal of Physics A: Mathematical and Theoretical, 53, 474001 (2020). https://doi.org/10.1088/1751-8121/aba467
[12] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1–77 (2000). https://doi.org/10.1016/s0370-1573(00)00070-3
[13] B. J. West, P. Grigolini, R. Metzler, and T. F. Nonnenmacher, Fractional diffusion and Levy stable processes, Physical Review E, 55, 99–106 (1997). https://doi.org/10.1103/physreve.55.99
[14] M. J. Ablowitz, J. B. Been, and L. D. Carr, Fractional Integrable Nonlinear Soliton Equations, Physical Review Letters, 128, 184101 (2022). https://doi.org/10.1103/PhysRevLett.128.184101
[15] M. J. Ablowitz, J. B. Been, and L. D. Carr, Fractional integrable and related discrete nonlinear Schrodinger equations, Physics Letters A, 452, 128459 (2022). https://doi.org/10.1016/j.physleta.2022.128459
[16] M. J. Ablowitz, J. B. Been, and L. D. Carr, Integrable fractional modified Korteweg–de Vries, sine-Gordon, and sinh-Gordon equations, Journal of Physics A: Mathematical and Theoretical, 55, 384010 (2022). https://doi.org/10.1088/1751-8121/ac8844
[17] M. Zhang, W. Weng, and Z. Yan, Interactions of fractional N-solitons with anomalous dispersions for the integrable combined fractional higher-order mKdV hierarchy, Physica D: Nonlinear Phenomena, 444, 133614 (2022). https://doi.org/10.1016/j.physd.2022.133614
[18] W. Weng, M. Zhang, G. Zhang, and Z. Yan, Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrodinger equations, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32, 123110 (2022). https://doi.org/10.1063/5.0101921
[19] L. An, L. Ling, and X. Zhang, Inverse scattering transform for the integrable fractional derivative nonlinear Schrodinger equation, Physica D: Nonlinear Phenomena, 458, 133888 (2023). https://doi.org/10.1016/j.physd.2023.133888
[20] L. An, L. Ling, and X. Zhang, Nondegenerate solitons in the integrable fractional coupled Hirota equation, Physics Letters A, 460, 128629 (2023). https://doi.org/10.1016/j.physleta.2023.128629
[21] S. Zhang, H. Li, and B. Xu, Inverse scattering integrability and fractional soliton solutions of a variable-coefficient fractional-order KdV-type equation, Fractal and Fractional, 8, 520 (2024). https://doi.org/10.3390/fractalfract8090520
[22] B. Babajanov and F. Abdikarimov, Integration of the fractional modified Korteweg–de Vries–sine-Gordon equation by the inverse scattering method, Results in Applied Mathematics, 26, 100586 (2025). https://doi.org/10.1016/j.rinam.2025.100586
[23] V. Melnikov, Exact solutions of the Korteweg–de Vries equation with a self-consistent source, Physics Letters A, 128, 488–492 (1988). https://doi.org/10.1016/0375-9601(88)90881-x
[24] V. Melnikov, Integration method of the Korteweg–de Vries equation with a self-consistent source, Physics Letters A, 133, 493–496 (1988). https://doi.org/10.1016/0375-9601(88)90522-1
[25] V. K. Melnikov, Integration of the Korteweg–de Vries equation with a source, Inverse Problems, 6, 233–246 (1990). https://doi.org/10.1088/0266- 5611/6/2/007
[26] V. K. Melnikov, Integration of the nonlinear Schroedinger equation with a self-consistent source, Communications in Mathematical Physics, 137, 359–381 (1991). https://doi.org/10.1007/bf02431884
[27] J. Leon and A. Latifi, Solution of an initial-boundary value problem for coupled nonlinear waves, Journal of Physics A: Mathematical and General, 23, 1385–1403 (1990). https://doi.org/10.1088/0305-4470/23/8/013
[28] C. Claude, A. Latifi, and J. Leon, Nonlinear resonant scattering and plasma instability: an integrable model, Journal of Mathematical Physics, 32, 3321–3330 (1991). https://doi.org/10.1063/1.529443
[29] D. Zhang and D. Chen, The N-soliton solutions of the sine-Gordon equation with self-consistent sources, Physica A: Statistical Mechanics and Its Applications, 321, 467–481 (2003). https://doi.org/10.1016/s0378-4371(02)01742-9
[30] Y. Zeng, Y. Shao, and W. Xue, Negaton and positon solutions of the soliton equation with self-consistent sources, Journal of Physics A: Mathematical and General, 36, 5035–5043 (2003). https://doi.org/10.1088/0305-4470/36/18/308
[31] P. G. Grinevich and I. A. Taimanov, Spectral conservation laws for periodic nonlinear equations of the Melnikov type, Translations of the American Mathematical Society, 224, 125–138 (2008). https://doi.org/10.1090/trans2/224/05
[32] B. Babajanov and F. Abdikarimov, New exact soliton and periodic wave solutions of the nonlinear fractional evolution equations with additional term, Partial Differential Equations in Applied Mathematics, 8, 100567 (2023). https://doi.org/10.1016/j.padiff.2023.100567
[33] B. A. Babajanov, A. K. Babadjanova, and A. S. Azamatov, Integration of the differential-difference sine-Gordon equation with a self-consistent source, Theoretical and Mathematical Physics, 210, 327–336 (2022). https://doi.org/10.1134/s0040577922030035
[34] A. Baev, Direct and inverse problems for Korteweg–de Vries family equations in the Fermi–Pasta–Ulam–Tsingou model for an inhomogeneous structure of lattice, Journal of Inverse and Ill-Posed Problems, 33, 665–681 (2025). https://doi.org/10.1515/jiip-2025-0018
[35] A. V. Baev, On an inverse problem for the KdV equation with variable coefficient, Mathematical Notes, 106, 837–841 (2019). https://doi.org/10.1134/s0001434619110166
[36] B. A. Babajanov, A. S. Azamatov, and R. B. Atajanova, Integration of the Kaup–Boussinesq system with time-dependent coefficients, Theoretical and Mathematical Physics, 216, 961–972 (2023). https://doi.org/10.1134/s004057792307005x
[37] B. A. Babajanov and D. O. Atajonov, Integration of the generalized Camassa–Holm equation in the class of periodic functions, Theoretical and Mathematical Physics, 223, 624–635 (2025). https://doi.org/10.1134/s0040577925040075
[38] A. B. Khasanov and M. M. Matyakubov, Integration of the nonlinear Korteweg–de Vries equation with an additional term, Theoretical and Mathematical Physics, 203, 596–607 (2020). https://doi.org/10.1134/s0040577920050037
[39] G. Urazboev, I. Baltaeva, and O. Ismoilov, Integration of the negative order Korteweg–de Vries equation by the inverse scattering method, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 33, 523–533 (2023). https://doi.org/10.35634/vm230309
[40] G. T. Bekova, G. N. Shaikhova, K. R. Yesmakhanova, and R. Myrzakulov, Lax representation and soliton solutions for the (2+1)- dimensional two-component complex modified Korteweg–de Vries equations, Journal of Physics: Conference Series, 804, 012004 (2017). https://doi.org/10.1088/1742-6596/804/1/012004
[41] M. Zhassybayeva, K. Yesmakhanova, and Z. Myrzakulova, Modified Fokas–Lenells equation: self-consistent sources and soliton solutions of the spin and (2+1)-dimensional models, Symmetry, 17, 1961 (2025). https://doi.org/10.3390/sym17111961
[42] Q. Li, H. Huang, and Q. Duan, Solutions of three nonlocal equations with self-consistent sources by the inverse scattering transform and reductions, Theoretical and Mathematical Physics, 222, 198–210 (2025). https://doi.org/10.1134/s0040577925020023
[43] U. A. Hoitmetov, Integration of the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions, Siberian Advances in Mathematics, 32, 102–114 (2022). https://doi.org/10.1134/s1055134422020043
[44] U. B. Muminov and A. B. Khasanov, Integration of a defocusing nonlinear Schrodinger equation with additional terms, Theoretical and Mathematical Physics, 211, 514–531 (2022). https://doi.org/10.1134/s0040577922040067
[45] A. B. Khasanov, K. N. Normurodov, and U. O. Khudaerov, Integrating the modified Korteweg–de Vries–sine-Gordon equation in the class of periodic infinite-gap functions, Theoretical and Mathematical Physics, 214, 170–182 (2023). https://doi.org/10.1134/s0040577923020022
[46] A. B. Khasanov and A. B. Yakhshimuratov, The Korteweg–de Vries equation with a self-consistent source in the class of periodic functions, Theoretical and Mathematical Physics, 164, 1008–1015 (2010). https://doi.org/10.1007/s11232-010-0081-8
[47] A. B. Khasanov and A. A. Reyimberganov, On the Hirota equation with a self-consistent source, Theoretical and Mathematical Physics, 221, 1852–1866 (2024). https://doi.org/10.1134/s0040577924110059
[48] M. Riesz, L’integrale de Riemann–Liouville et le probleme de Cauchy, Acta Mathematica 81, 1–222 (1949). https://doi.org/10.1007/bf02395016
[49] B. M. Levitan, Inverse Sturm–Liouville Problems, (Nauka, Moscow, 1984).
[50] M. J. Ablowitz, Nonlinear Dispersive Waves, (Cambridge University Press, Cambridge, 2011). https://doi.org/10.1017/cbo9780511998324
[51] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315 (1974). https://doi.org/10.1002/sapm1974534249
[52] R. L. Sachs, Completeness of derivatives of squared Schrodinger eigenfunctions and explicit solutions of the linearized KdV equation, SIAM Journal on Mathematical Analysis, 14, 674–683 (1983). https://doi.org/10.1137/0514051
