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Reciprocal transformations are introduced for two super Camassa–Holm (CH) equations. Under these transformations and appropriate changes of dependent variables, the super CH equation, proposed by Geng et al. [Stud. Appl. Math. 130, 1 (2013)], is converted to a negative member of the super Korteweg–de Vries (KdV) hierarchy studied by Geng and Wu in 2010 [Appl. Math. Lett. 23, 716 (2010)], while the other super CH equation, due to Zhang and Zuo [J. Math. Phys. 52, 073503 (2011)], is related to a new super KdV hierarchy. In the latter case, algebraic properties of this new super KdV hierarchy are established, including Hamiltonian operators, a recursion operator, and conserved quantities. [ABSTRACT FROM AUTHOR]

A Darboux transformation is constructed for the generalized Hirota-Satsuma coupled KdV system and the result is compared with the recent work of Geng and his collaborators [X. G. Geng et al., Phys. Rev. E 79, 056602 (2009) and X. G. Geng and G. L. He, J. Math. Phys. 51, 033514 (2010)]. It is shown that our Darboux transformation may be applied to three interesting reductions of the general system. In addition, the iteration of this Darboux transformation is worked out, and some solutions to the associated systems are obtained. [ABSTRACT FROM AUTHOR]

The N = 2 supercomformal transformations are employed to study supersymmetric integrable systems. It is proved that two known N = 2 supersymmetric Harry Dym equations are transformed into two N = 2 supersymmetric modified Korteweg-de Vries equations, thus are connected with two N = 2 supersymmetric Korteweg-de Vries equations. [ABSTRACT FROM AUTHOR]

Darboux transformation is constructed for a third-order spectral problem. By proper reduction, a Darboux transformation for a long-short wave model is obtained. Furthermore, a closed multi-soliton solution formula is found for this equation. [ABSTRACT FROM AUTHOR]

The supersymmetric analog of the reciprocal transformation is introduced. This is used to establish a transformation between one of the supersymmetric Harry Dym equations and the supersymmetric modified Korteweg-de Vries equation. The reciprocal transformation, as a Bäcklund-type transformation between these two equations, is adopted to construct a recursion operator for the supersymmetric Harry Dym equation. By proper factorization of the recursion operator, a bi-Hamiltonian structure is found for the supersymmetric Harry Dym equation. Furthermore, a supersymmetric Kawamoto equation is proposed and is associated with the supersymmetric Sawada-Kotera equation. The recursion operator and odd bi-Hamiltonian structure of the supersymmetric Kawamoto equation are also constructed. [ABSTRACT FROM AUTHOR]

The mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Feje´r average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Feje´r average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the above-mentioned points. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Physically interesting exact solutions are constructed for a large class of nonlinear nonintegrable evolution equations. These solutions describe the interaction of traveling waves. They exhibit rich phenomenology including breaking of solitary waves. Generalized conditional symmetries and bilinearity are used to derive these exact results. [ABSTRACT FROM AUTHOR]

A multicomponent KdV system, Kupershmidt’s coupled KdV equation, is considered. The system is shown to be a modification of some system, and it can be modified. [ABSTRACT FROM AUTHOR]

The canonical quantization is a procedure for quantizing a classical theory while preserving the formal algebraic structure among observables in the classical theory to the extent possible. For a system without constraint, we have the so-called fundamental commutation relations (CRs) among positions and momenta, whose algebraic relations are the same as those given by the Poisson brackets in classical mechanics. For the constrained motion on a curved hypersurface, we need more fundamental CRs otherwise neither momentum nor kinetic energy can be properly quantized, and we propose an enlarged canonical quantization scheme with introduction of the second category of fundamental CRs between Hamiltonian and positions, and those between Hamiltonian and momenta, whereas the original ones are categorized into the first. As an N - 1 (N >= 2) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the spherical surface, a long-standing problem in the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N, 1) symmetry for the motion on the sphere is demonstrated. [ABSTRACT FROM AUTHOR]

A linear eigenvalue problem is constructed for the gradients of the eigenvalue of the Wadati-Konno-Ichikawa (WKI) spectral problem. This in turn leads to a compatible pair of Hamiltonian operators, which endows the WKI hierarchy with a bi-Hamiltonian structure. Furthermore, a proper recombination of these two Hamiltonian operators, or the duality of this bi-Hamiltonian structure, generates a coupled system of Camassa-Holm type, which is nothing but a system proposed by Song, Qu, and Qiao [J. Math. Phys. 52, 013503 (2011)] recently. [ABSTRACT FROM AUTHOR]

Provides a corrected reprint to the article 'The Fejer average and the mean value of quantity in a quasiclassical wave packet,' published in the June 2002 issue of the 'Journal of Mathematical Physics.'

We consider the Miura map between the lattice Kadomtsev-Petviashvili hierarchy and the lattice modified KP hierarchy and prove that the map is canonical not only between the first Hamiltonian structures, but also between the second Hamiltonian structures. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]