Mixed hierarchy of soliton equations

Abstract

The mixed hierarchy of soliton equations in (1+1) dimensions is introduced. It contains nonholonomic deformations of soliton equations such the KdV6 equation and the Kupershmidt deformations of soliton equations as special members. Based on the commutator representation method, a recipe for constructing zero curvature representations of mixed hierarchy is proposed. As applications, we obtain the mixed hierarchies and their zero curvature representations for the Korteweg–de Vries hierarchy, the Ablowitz–Kaup–Newell–Segur hierarchy, the modified Korteweg–de Vries hierarchy, the Toda lattice hierarchy, and the Volterra lattice hierarchy.

Quasi-periodic solutions of mixed AKNS equations

“…The AKNS equations are very important because they can be reduced to some well-known nonlinear evolution equations such as the KdV, the mKdV, the nonlinear Schrödinger, and the sine-Gordon equations, and others, which have many applications in physics and other nonlinear sciences. Various methods have been developed for obtaining explicit solutions of the AKNS equations, for instance, the inverse scattering transformation [1][2][3][4], the Bäcklund transformation [5], the Darboux transformation [6], the algebraic-geometrical approach (see, e.g., [7] and references therein), and others [8][9][10][11][12][13][14][15][16][17]. It has been shown that the AKNS equations are completely integrable in the Liouville sense and possess Hamiltonian structures.…”

Применение Обобщенной Деформации Купершмидта Для Построения Новых Дискретных Интегрируемых Систем

“…В работе [14] путем комбинации уравнений отрицательных и положительных по-рядков введены смешанные иерархии солитонных уравнений и показано, что систе-мы, полученные в результате деформации Купершмидта, являются частными пред-ставителями смешанных солитонных иерархий. К смешанной иерархии солитонных уравнений из работы [14], имеющих один неголономный член, относятся некоторые частные случаи неголономных деформированных уравнений.…”

“…К смешанной иерархии солитонных уравнений из работы [14], имеющих один неголономный член, относятся некоторые частные случаи неголономных деформированных уравнений. Наша ОДК допускает N неголономных членов.…”

The generalized Kupershmidt deformation for constructing new discrete integrable systems

“…Zhou introduced a concept of mixed hierarchy of soliton equations combining the negative order equations to the positive order equations and showed that the Kupershmidt deformed systems are just special members in the mixed soliton hierarchies [14]. The mixed hierarchy of soliton equations [14], which has one nonholonomic term, are some special nonholonomic deformed equations.…”

“…The mixed hierarchy of soliton equations [14], which has one nonholonomic term, are some special nonholonomic deformed equations. Our GKD admit N nonholonomic terms.…”