Dressing for a Vector Modified KdV
Hierarchy
Panagiota Adamopoulou and Georgios Papamikos
Abstract We present the hierarchy and soliton solutions associated with a multicomponent generalisation of the modified Korteweg–de Vries equation. A recursive
formula for obtaining the Lax operators associated with the higher flows of the
hierarchy is provided. Using the method of rational dressing and the symmetries
of the Lax operators we obtain the one-soliton solution. We also derive the general
rank one-breather solution and express it in terms of determinants. We present the
simplest case of the one-breather solution, which is parameterised by two constant
unit vectors normal to each other. Finally, we obtain the associated Bäcklund
transformation for the hierarchy.
Keywords Multi-component integrable hierarchies · Soliton solutions ·
Darboux-dressing transformations · Lax representation
1 Introduction
Since the establishment of the modern theory of integrable systems there have
appeared many generalisations of soliton equations in both independent and dependent variables, such as extensions in 2 + 1 variables, systems of equations, and
equations with non-commutative variables. Multi-component integrable equations
and their hierarchies have also attracted much attention due to their rich mathematical structure and appearance in applications. In the current paper we are concerned
with the hierarchy associated with the following multi-component generalisation of
the modified KdV (mKdV) equation:
P. Adamopoulou ()
Heriot-Watt University, Edinburgh, UK
e-mail: p.adamopoulou@hw.ac.uk
G. Papamikos
University of Essex, Colchester, UK
e-mail: g.papamikos@essex.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_48
513
Hierarchy
Panagiota Adamopoulou and Georgios Papamikos
Abstract We present the hierarchy and soliton solutions associated with a multicomponent generalisation of the modified Korteweg–de Vries equation. A recursive
formula for obtaining the Lax operators associated with the higher flows of the
hierarchy is provided. Using the method of rational dressing and the symmetries
of the Lax operators we obtain the one-soliton solution. We also derive the general
rank one-breather solution and express it in terms of determinants. We present the
simplest case of the one-breather solution, which is parameterised by two constant
unit vectors normal to each other. Finally, we obtain the associated Bäcklund
transformation for the hierarchy.
Keywords Multi-component integrable hierarchies · Soliton solutions ·
Darboux-dressing transformations · Lax representation
1 Introduction
Since the establishment of the modern theory of integrable systems there have
appeared many generalisations of soliton equations in both independent and dependent variables, such as extensions in 2 + 1 variables, systems of equations, and
equations with non-commutative variables. Multi-component integrable equations
and their hierarchies have also attracted much attention due to their rich mathematical structure and appearance in applications. In the current paper we are concerned
with the hierarchy associated with the following multi-component generalisation of
the modified KdV (mKdV) equation:
P. Adamopoulou ()
Heriot-Watt University, Edinburgh, UK
e-mail: p.adamopoulou@hw.ac.uk
G. Papamikos
University of Essex, Colchester, UK
e-mail: g.papamikos@essex.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_48
513