On the full Kostant–Toda system and the discrete Korteweg–de Vries equations

On the existence of Darboux transformations for banded matrices

“…The case p ¼ 2 was analyzed in [3]. However, another condition over a 1 -0 was given to obtain the factorization (12), what we prove that is equivalent to (8).…”

“…; p, are defined as in (2), (3). This fact has been proved in [3] for p ¼ 2, but the existence and uniqueness of such transformations for an arbitrary p 2 N only was conjectured. Our main goal in the present work is to established some conditions under which the above problem can be solved for an arbitrary p 2 N. With this purpose, we set the triangular matrices of kind…”

“…. ; p, are defined as in (2), (3). This fact has been proved in [3] for p ¼ 2, but the existence and uniqueness of such transformations for an arbitrary p 2 N only was conjectured.…”

“…819 in [3]). For p 2 N, let B ¼ b i;j À Á ; i; j 2 N, be a lower Hessenberg banded matrix, We remark that, in the present paper, the Darboux factorization and the Darboux transformation are studied for Hessenberg matrices for which the existence of its LU factorization is assumed.…”

“…This study can be extended to the case of high order systems using more general banded matrices. In fact, the concept of Darboux transformations was generalized in [3] for Hessenberg matrices ða i;j Þ 1 i;j¼1 such that a i;j ¼ 0 for i > j þ p; p 2 N. We include here the following summary for a more independent reading.…”