In this paper, we study two constrained cases of the Kadomtsev-Petviashvili hierarchy, named the generalized constrained Kadomtsev-Petviashvili and the generalized constrained modified Kadomtsev-Petviashvili hierarchies. The additional symmetries and gauge transformations for the generalized constrained Kadomtsev-Petviashvili hierarchy are given. These flows form a subalgebra of the Virasoro algebra. We give explicit formulas for the action of the additional flows for the generalized constrained Kadomtsev-Petviashvili and the generalized constrained modified Kadomtsev-Petviashvili hierarchies on tau functions. The additional flows for the generalized constrained Kadomtsev-Petviashvili hierarchy commuted with gauge transformations preserve their original form, up to shifting of the additional flows by the sum of two squared eigenfunction symmetry flows.
We classify surjective maps that preserve both order and spectrum on positive unbounded self-adjoint operators. As a consequence, we provide a comprehensive description of such maps on the set of all lower semi-bounded self-adjoint operators.
We present a method of finding non-Liouvillian first integrals of rational two-dimensional differential systems. The method is based on the existence of two independent invariants that satisfy a linear second-order ordinary differential equation with respect to one of the variables. We call systems with this property R-integrable. These invariants are not necessarily polynomial; they can be expressed via special functions. We demonstrate that if there exists a rational transformation that maps some Riccati system to a given rational system, then the latter is always R-integrable. We prove that the famous Van der Pol systems and their generalizations are not R-integrable. We find novel integrable polynomial Liénard differential systems without invariant algebraic curves. The related first integrals are expressible via the Airy functions. We prove that the Kudashev system arising in the theory of the Korteweg–de Vries equation is R-integrable.
In the first part, we study different geometric properties of some well-known families of polyharmonic mappings due to Li et al. (J. Math. Anal. Appl., 434, 1462–1473 [28]). In particular, the sufficient coefficient conditions for the starlikeness and convexity, the conditions on the q^th - integral mean, and the bounds on arclength and Bloch constant of a polyharmonic mapping are obtained. The Bohr phenomenon for harmonic and polyharmonic mappings is studied in the next part. We have refined two results of Kayumov et al. (Math. Nachr., 291(11-12), 1757–1768 [25]) and found their polyharmonic analogue. The article concludes with an improved Bohr phenomenon of K-quasiconformal sense-preserving harmonic mappings.
This paper investigates a novel class of generalized fractional Musielak-Sobolev spaces denoted by ℋ^s/2;Ψ _ξ ,y_ϕ . Key properties such as separability and reflexivity are analyzed in detail. Furthermore, continuous and compact embeddings are rigorously established, together with additional functional-analytic properties. As an application, the framework is used to study the existence and multiplicity of solutions to a class of nonlocal generalized problems, focusing in particular on the nonlocal Schrödinger problem with discontinuous nonlinearity.
