Dressing for a Vector Modified KdV Hierarchy
521
u k =
−
4
Δ Re
i cos ξ(F ∗ + H − G)
, k = j ,
0 , k = j .
(31)
The denominator in the above expression can be written in the form
Δ = −
1
r 2
tan θ sin
2 A +
cosh
2 B
tan θ
2
,
(32)
where r = |μ|, θ = arg(μ), A = Re(ξ ) and B = Im(ξ ). Finally, for the particular
choice of C given above, we obtain the following expression for the rank one onebreather solution:
u j = 4r
sin θ sin A sinh B − cos θ cos A cosh B
tan θ sin
2 A +
cosh 2 B
tan θ
,
(33)
which, as expected, is a breather solution of the scalar mKdV.
4 Conclusions
In this paper we presented a vectorial generalisation of the well-known mKdV
equation, namely Eq. (1). It is important to note that this equation is not the third
flow of the vector NLS hierarchy, since its Lax pair is characterised by relations (4),
and additionally J in (5) is not a regular element of the underlying Lie algebra,
contrary to the Drinfel’d–Sokolov construction [4]. For Eq. (1) we presented its
recursion operator which can be used to construct all higher flows of the integrable
hierarchy, and also a recursive formula that produces the Lax operator associated
with each member of the hierarchy. We used the structure of the Lax operators to
construct two Darboux transformations and we obtained the one-soliton solution
and the general rank one-breather solution. The breather solution, in the simplest
case, is a breather solution of the scalar mKdV equation. We leave the study of the
more general breather solutions, as well as the n-soliton and n-breather solution
for future work. As part of this goal, we aim to study Hirota’s direct method as
well as the general scattering problem and its inversion through a Riemann–Hilbert
problem.
References
1. P. Adamopoulou, G. Papamikos, Drinfel’d-Sokolov construction and exact solutions of vector
modified KdV hierarchy. Nucl. Phys. B 952, 114933 (2020)
2. S.C. Anco, Hamiltonian flows of curves in G/SO(N ) and vector soliton equations of mKdV
and Sine-Gordon type. Symmetry Integr. Geom. Methods Appl. 2, 044 (2006)
521
u k =
−
4
Δ Re
i cos ξ(F ∗ + H − G)
, k = j ,
0 , k = j .
(31)
The denominator in the above expression can be written in the form
Δ = −
1
r 2
tan θ sin
2 A +
cosh
2 B
tan θ
2
,
(32)
where r = |μ|, θ = arg(μ), A = Re(ξ ) and B = Im(ξ ). Finally, for the particular
choice of C given above, we obtain the following expression for the rank one onebreather solution:
u j = 4r
sin θ sin A sinh B − cos θ cos A cosh B
tan θ sin
2 A +
cosh 2 B
tan θ
,
(33)
which, as expected, is a breather solution of the scalar mKdV.
4 Conclusions
In this paper we presented a vectorial generalisation of the well-known mKdV
equation, namely Eq. (1). It is important to note that this equation is not the third
flow of the vector NLS hierarchy, since its Lax pair is characterised by relations (4),
and additionally J in (5) is not a regular element of the underlying Lie algebra,
contrary to the Drinfel’d–Sokolov construction [4]. For Eq. (1) we presented its
recursion operator which can be used to construct all higher flows of the integrable
hierarchy, and also a recursive formula that produces the Lax operator associated
with each member of the hierarchy. We used the structure of the Lax operators to
construct two Darboux transformations and we obtained the one-soliton solution
and the general rank one-breather solution. The breather solution, in the simplest
case, is a breather solution of the scalar mKdV equation. We leave the study of the
more general breather solutions, as well as the n-soliton and n-breather solution
for future work. As part of this goal, we aim to study Hirota’s direct method as
well as the general scattering problem and its inversion through a Riemann–Hilbert
problem.
References
1. P. Adamopoulou, G. Papamikos, Drinfel’d-Sokolov construction and exact solutions of vector
modified KdV hierarchy. Nucl. Phys. B 952, 114933 (2020)
2. S.C. Anco, Hamiltonian flows of curves in G/SO(N ) and vector soliton equations of mKdV
and Sine-Gordon type. Symmetry Integr. Geom. Methods Appl. 2, 044 (2006)