| Abstract: |
| Soliton equations play a central role from an applied perspective because they admit solutions whose shape remains unchanged during propagation and whose amplitude is conserved over time. Such equations frequently appear in applied mathematics, with applications spanning fluid dynamics, nonlinear optics, acoustics, theoretical physics and, more recently, biophysics.
In the present work we consider solutions of the matrix modified Korteweg-de Vries (mKdV) equation. These solutions can properly be termed soliton solutions, since they manifest the characteristic behavior of solitons. In particular, two-soliton solutions of the
$d\\times d$ matrix
mKdV equation are constructed as in [1]. This work builds on a general explicit formula for
$N$-soliton solutions in the infinite-dimensional (operator) case, as studied in [2], while the explicit finite-dimensional (matrix) case is treated in detail in [5].
Additionally, invariance properties of third-order KdV-type equations in non-commutative (matrix) contexts have been investigated more recently in [6].
Solutions of matrix mKdV equation are presented which can be termed soliton solutions since they exhibit the typical behaviour of solitons. Specifically,
two-soliton solutions of the $d \\times d$-matrix modified Korteweg de-Vries equation are obtained in [1].
An explicit formula in the matrix case is studied in [3] while invariance properties of third order KdV-type equations are investigated in [6]. An overview on recent and perspective result is in [7].
[1] S. Carillo, C. Schiebold, Construction of soliton solutions of the matrix Korteweg-de Vries and modified Korteweg-de Vries equations,
in Advances in Nonlinear Dynamics. NODYCON Conference Proceedings Series. Springer, Cham,
ISBN 978-3-030-81169-3,
W. Lacarbonara, et al.
Ed.s, 481--491 (2022).
doi: 10.1007/978-3-030-81170-9_42, arXiv: 2011.12677.
[2]
S.~Carillo and C.~Schiebold,
Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies:
Non-commutative soliton solutions,
J. Math. Phys. 52 (2011), 053507. doi: 10.1063/1.3576185 27-37.
[3] S. Carillo, M. Lo Schiavo, C. Schiebold, $N$-soliton matrix mKdV solutions: Some Special Solutions Revisited, Studies in Applied Mathematics, 2025; 154: e70061.
doi: doi.org/10.1111/sapm.70061.
[4] S.~Carillo and C.~Schiebold,
On the asymptotical description of soliton solutions to the matrix modified Korteweg-de Vries equation, submitted 2023.
Advances in Nonlinear Dynamics - Proc.s of the Third International Nonlinear Dynamics Conference (NODYCON 2023), Vol. 3, W. Lacarbonara, Ed.
565--575 (2024). doi: 10.1007/978-3-031-50635-2_43.
[5]. S. Carillo, M. Lo Schiavo, C. Schiebold, Matrix solitons solutions of the modified Korteweg-de Vries equation,
NODYCON 2019 Springer Proceedings,
B. Balachandran, J. Ma, W. Lacarbonara, G.Quaranta,
J. Machado, G. Stepan, Ed.s, (2020), 75-83. doi: 10.1007/978-3-030-34713-0_8.
[6] S. Carillo, C. Schiebold, Soliton equations: admitted solutions and invariances via B\\cklund transformations, Open Commun. Nonlinear Math. Phys., Episciences, ISSN: 2802-9356, S.I. 1, 1--11 (2024).
doi: 10.46298/ocnmp.12497.
[7] S. Carillo, C. Schiebold, F. Zullo, Nonlinear evolution equations of soliton type:
old and new results, Proceedings, MDPI, 2025; 123(1):9, 2025.
doi: 10.3390/proceedings2025123009. |
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