A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy
Abstract
1. Introduction
- (a)
- The direct sum of the kernel and the image of spans the entire algebra ;
- (b)
- The kernel of forms an abelian (commuting) subalgebra.
2. A 4 × 4 Spectral Problem and Its Integrable Hierarchy
3. Hereditary Recursion Operator and Bi-Hamiltonian Structure
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Ablowitz, M.J.; Segur, H. Solitons and the Inverse Scattering Transform; SIAM: Philadelphia, PA, USA, 1981. [Google Scholar]
- Das, A. Integrable Models; World Scientific: Teaneck, NJ, USA, 1989. [Google Scholar]
- Lax, P.D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 1968, 21, 467–490. [Google Scholar] [CrossRef]
- Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 1974, 53, 249–315. [Google Scholar] [CrossRef]
- Drinfel’d, V.; Sokolov, V.V. Lie algebras and equations of Korteweg–de Vries type. Sov. J. Math. 1985, 30, 1975–2036. [Google Scholar] [CrossRef]
- Tu, G.Z. On Liouville integrability of zero-curvature equations and the Yang hierarchy. J. Phys. A Math. Gen. 1989, 22, 2375–2392. [Google Scholar]
- Liu, C.S. How many first integrals imply integrability in infinite-dimensional Hamilton system. Rep. Math. Phys. 2011, 67, 109–123. [Google Scholar] [CrossRef]
- Antonowicz, M.; Fordy, A.P. Coupled KdV equations with multi-Hamiltonian structures. Phys. D 1987, 28, 345–357. [Google Scholar] [CrossRef]
- Xia, T.C.; Yu, F.J.; Zhang, Y. The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions. Phys. A 2004, 343, 238–246. [Google Scholar] [CrossRef]
- Manukure, S. Finite-dimensional Liouville integrable Hamiltonian systems generated from Lax pairs of a bi-Hamiltonian soliton hierarchy by symmetry constraints. Commun. Nonlinear Sci. Numer. Simul. 2018, 57, 125–135. [Google Scholar] [CrossRef]
- Liu, T.S.; Xia, T.C. Multi-component generalized Gerdjikov-Ivanov integrable hierarchy and its Riemann-Hilbert problem. Nonlinear Anal. Real World Appl. 2022, 68, 103667. [Google Scholar] [CrossRef]
- Wang, H.F.; Zhang, Y.F. Application of Riemann-Hilbert method to an extended coupled nonlinear Schrödinger equations. J. Comput. Appl. Math. 2023, 420, 114812. [Google Scholar] [CrossRef]
- Gerdjikov, V.S. Nonlinear evolution equations related to Kac-Moody algebras Ar(1): Spectral aspects. Turkish J. Math. 2022, 46, 1828–1844. [Google Scholar] [CrossRef]
- Ma, W.X. AKNS type reduced integrable hierarchies with Hamiltonian formulations. Rom. J. Phys. 2023, 68, 116. [Google Scholar] [CrossRef]
- Ma, W.X. A combined integrable hierarchy with four potentials and its recursion operator and bi-Hamiltonian structure. Indian J. Phys. 2025, 99, 1063–1069. [Google Scholar] [CrossRef]
- Takhtajan, L.A. Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 1977, 64, 235–237. [Google Scholar] [CrossRef]
- Kaup, D.J.; Newell, A.C. An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 1978, 19, 798–801. [Google Scholar] [CrossRef]
- Wadati, M.; Konno, K.; Ichikawa, Y.H. New integrable nonlinear evolution equations. J. Phys. Soc. Jpn. 1979, 47, 1698–1700. [Google Scholar] [CrossRef]
- Ma, W.X. Four-component integrable hierarchies and their Hamiltonian structures. Commun. Nonlinear Sci. Numer. Simul. 2023, 126, 107460. [Google Scholar] [CrossRef]
- Zhang, Y.F. A few expanding integrable models, Hamiltonian structures and constrained flows. Commun. Theor. Phys. 2011, 55, 273–290. [Google Scholar] [CrossRef]
- Zhaqilao. A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 2319–2332. [Google Scholar] [CrossRef]
- Ma, W.X. Reduced AKNS spectral problems and associated complex matrix integrable models. Acta Appl. Math. 2023, 187, 17. [Google Scholar] [CrossRef]
- Ma, W.X. Novel Liouville integrable Hamiltonian models with six components and three signs. Chin. J. Phys. 2023, 86, 292–299. [Google Scholar] [CrossRef]
- Ma, W.X. Integrable matrix nonlinear Schrödinger equations with reduced Lax pairs of AKNS type. Appl. Math. Lett. 2025, 168, 109574. [Google Scholar] [CrossRef]
- Ma, W.X. Matrix mKdV integrable hierarchies via two identical group reductions. Mathematics 2025, 13, 1438. [Google Scholar] [CrossRef]
- Magri, F. A simple model of the integrable Hamiltonian equation. J. Math. Phys. 1978, 19, 1156–1162. [Google Scholar] [CrossRef]
- Ma, W.X. The algebraic structure of zero curvature representations and application to coupled KdV systems. J. Phys. A Math. Gen. 1993, 26, 2573–2582. [Google Scholar] [CrossRef]
- Fuchssteiner, B.; Fokas, A.S. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 1981, 4, 47–66. [Google Scholar] [CrossRef]
- Novikov, S.P.; Manakov, S.V.; Pitaevskii, L.P.; Zakharov, V.E. Theory of Solitons: The Inverse Scattering Method; Consultantn Bureau: New York, NY, USA, 1984. [Google Scholar]
- Zhang, X.F.; Tian, S.F. Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions. J. Math. Phys. 2023, 64, 051503. [Google Scholar] [CrossRef]
- Matveev, V.B.; Salle, M.A. Darboux Transformations and Solitons; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
- Geng, X.G.; Li, R.M.; Xue, B. A vector general nonlinear Schrödinger equation with (m + n) components. J. Nonlinear Sci. 2020, 30, 991–1013. [Google Scholar] [CrossRef]
- Ye, R.S.; Zhang, Y. A vectorial Darboux transformation for the Fokas-Lenells system. Chaos Solitons Fractals 2023, 169, 113233. [Google Scholar] [CrossRef]
- Doktorov, E.V.; Leble, S.B. A Dressing Method in Mathematical Physics; Springer: Dordrecht, The Netherlands, 2007. [Google Scholar]
- Aktosun, T.; Busse, T.; Demontis, F.; van der Mee, C. Symmetries for exact solutions to the nonlinear Schrödinger equation. J. Phys. A Math. Theoret. 2010, 43, 025202. [Google Scholar] [CrossRef]
- Cheng, L.; Zhang, Y. Grammian-type determinant solutions to generalized KP and BKP equations. Comput. Math. Appl. 2017, 74, 727–735. [Google Scholar] [CrossRef]
- Manukure, S.; Chowdhury, A.; Zhou, Y. Complexiton solutions to the asymmetric Nizhnik-Novikov-Veselov equation. Internat. J. Modern Phys. B 2019, 33, 1950098. [Google Scholar] [CrossRef]
- Wang, Y.; Lü, X.; Ma, W.X. Integrability characteristics and exact solutions of an extended (3+1)-dimensional variable-coefficient shallow water wave model. Nonlinear Dyn. 2025, 113, 21725–21741. [Google Scholar] [CrossRef]
- Sulaiman, T.A.; Yusuf, A.; Abdeljabbar, A.; Alquran, M. Dynamics of lump collision phenomena to the (3+1)-dimensional nonlinear evolution equation. J. Geom. Phys. 2021, 169, 104347. [Google Scholar] [CrossRef]
- Geng, X.G.; Liu, W.; Xue, B. Finite genus solutions to the coupled Burgers hierarchy. Results Math. 2019, 74, 11. [Google Scholar] [CrossRef]
- Chu, J.Y.; Liu, Y.Q.; Ma, W.X. Integrability and multiple-rogue and multi-soliton wave solutions of the 3+1-dimensional Hirota–Satsuma–Ito equation. Mod. Phys. Lett. B 2025, 39, 2550060. [Google Scholar] [CrossRef]
- Yusuf, A.; Sulaiman, T.A.; Abdeljabbar, A.; Alquran, M. Breathem waves, analytical solutions and conservation lawn using Lie–Bäcklund symmetries to the (2+1)-dimensional Chaffee-Infante equation. J. Ocean Eng. Sci. 2023, 8, 145–151. [Google Scholar] [CrossRef]
- Ma, W.X. Lump waves and their dynamics of a spatial symmetric generalized KP model. Rom. Rep. Phys. 2024, 76, 108. [Google Scholar] [CrossRef]
- Zhou, Y.; Manukure, S.; McAnally, M. Lump and rogue wave solutions to a (2+1)-dimensional Boussinesq type equation. J. Geom. Phys. 2021, 167, 104275. [Google Scholar] [CrossRef]
- Manukure, S.; Zhou, Y. A study of lump and line rogue wave solutions to a (2+1)-dimensional nonlinear equation. J. Geom. Phys. 2021, 167, 104274. [Google Scholar] [CrossRef]
- Yang, S.X.; Wang, Y.F.; Zhang, X. Conservation laws, Darboux transformation and localized waves for the N-coupled nonautonomous Gross–Pitaevskii equations in the Bose–Einstein condensates. Chaos Solitons Fractals 2023, 169, 113272. [Google Scholar] [CrossRef]
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Ma, W.-X.; Zhong, Y.-D. A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy. Axioms 2025, 14, 594. https://doi.org/10.3390/axioms14080594
Ma W-X, Zhong Y-D. A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy. Axioms. 2025; 14(8):594. https://doi.org/10.3390/axioms14080594
Chicago/Turabian StyleMa, Wen-Xiu, and Ya-Dong Zhong. 2025. "A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy" Axioms 14, no. 8: 594. https://doi.org/10.3390/axioms14080594
APA StyleMa, W.-X., & Zhong, Y.-D. (2025). A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy. Axioms, 14(8), 594. https://doi.org/10.3390/axioms14080594