Construction of soliton solutions of the matrix modified Korteweg-de Vries equation.

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado, G. Stepan (Springer, Cham, 2020), pp. 75-83). In fact, the solutions in Ref.7 are part of a subclass studied in detail by the authors in a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties.

[1]  Yi Zhang,et al.  The N-soliton solutions for the matrix modified Korteweg–de Vries equation via the Riemann–Hilbert approach , 2020, The European Physical Journal Plus.

[2]  S. Carillo,et al.  Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation , 2019, Nonlinear Dynamics of Structures, Systems and Devices.

[3]  S. Carillo,et al.  Abelian versus non-Abelian Bäcklund charts: Some remarks , 2018, Evolution Equations & Control Theory.

[4]  S. Carillo.  KdV-type equations linked via Bäcklund transformations: Remarks and perspectives , 2017, Applied Numerical Mathematics.

[5]  N. Euler.  Matrix solutions for equations of the AKNS system , 2018, Nonlinear Systems and Their Remarkable Mathematical Structures.

[6]  N. Euler.  Nonlinear Systems and Their Remarkable Mathematical Structures , 2018 .

[7]  Mauro Lo Schiavo,et al.  Recursion operators admitted by non-Abelian Burgers equations: Some remarks , 2016, Math. Comput. Simul..

[8]  S. Carillo,et al.  A novel noncommutative KdV-type equation, its recursion operator, and solitons , 2017, 1704.03208.

[9]  S. Carillo,et al.  Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart , 2015, 1512.02386.

[10]  L. Sakhnovich,et al.  Inverse Problems and Nonlinear Evolution Equations: Solutions, Darboux Matrices and Weyl-Titchmarsh Functions , 2013 .

[11]  Francesco Calogero,et al.  Spectral Transform and Solitons , 2012 .

[12]  S. Carillo,et al.  Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions , 2011 .

[13]  C. Schiebold.  Noncommutative AKNS systems and multisolitonsolutions to the matrix sine-gordon equation , 2009 .

[14]  S. Carillo,et al.  Noncommutative Korteweg–de Vries and modified Korteweg–de Vries hierarchies via recursion methods , 2009 .

[15]  Zixiang Zhou,et al.  Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry , 2005 .

[16]  Robert McNaughton,et al.  Some Remarks , 1891, Mathematical systems theory.

[17]  Wolfgang K. Schief,et al.  Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory , 2002 .

[18]  C. Rogers,et al.  Bäcklund and Darboux Transformations: Frontmatter , 2002 .

[19]  V. Goncharenko.  Multisoliton Solutions of the Matrix KdV Equation , 2001 .

[20]  H. Blohm.  Solution of nonlinear equations by trace methods , 2000 .

[21]  B. Carl,et al.  Ein direkter Ansatz zur Untersuchung von Solitonengleichungen , 2000 .

[22]  B. Fuchssteiner,et al.  Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions , 1989 .

[23]  Antonio Degasperis,et al.  Spectral Transform and Solitons: How to Solve and Investigate Nonlinear Evolution Equations , 1988 .

[24]  V. Marchenko.  Nonlinear Equations and Operator Algebras , 1987 .

[25]  D. Levi,et al.  Continuous and discrete matrix Burgers’ hierarchies , 1983 .

[26]  Colin Rogers,et al.  Bäcklund transformations and their applications , 1982 .

[27]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .