Journal of Nonlinear Mathematical Physics

We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup–Kupershmidt equation derived by E. G. Reyes in his paper titled "Nonlocal symmetries and the Kaup–Kupershmidt equation" [J. Math. Phys.46 (2005) 073507, 19 pp.]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup–Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialization of the Kupershmidt equation.

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.

Billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrability condition of a line congruence. A new notion of the double reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properties and several examples are presented. Corresponding Yang–Baxter maps, associated with pencils of quadrics are defined and discussed.

The infinitesimal deformations of the embedding of the Lie superalgebra of contact vector fields on the supercircle S^1|4 into the Poisson superalgebra of symbols of pseudodifferential operators on S^1|2 are explicitly calculated.

We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol.48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol.44 (1982) 749–760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation that can be derived from them, even in the case where those authors failed such as the host-parasite model. Also we show that the Lagrangians of certain second-order ordinary differential equations derived by Volterra in [V. Volterra, Calculus of variations and the logistic curve, Hum. Biol.11 (1939) 173–178] are particular cases of the Lagrangians that can be obtained by means of the Jacobi Last Multiplier. Actually we provide more than one Lagrangian for those Volterra's equations.

This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the extended AKNS hierarchy. As an additional result we obtain the functional representation of the Landau–Lifshitz hierarchy.

We consider the Recursion Operator approach to the soliton equations related to a auxiliary linear system introduced recently by Gerdjikov, Mikhailov and Valchev (GMV system) and their interpretation as dual of Nijenhuis tensors on the manifold of potentials.

We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated μCH equation.

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x^2M + αx^M-1 + (λ² - 1/4)/x². After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form of the zeta functions can be simplified. An immediate application of this work is to derive functional relations and identities involving hypergeometric series, allowing for known identities to be found as instances of more general results. Our work is then extended to a class of related -symmetric eigenvalue problems. Using the fused quantum Wronskian, we give a simple method for indirectly calculating the associated spectral zeta functions. This method is then applied to calculate the nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.