Related papers: Progresses on some open problems related to infini…
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such…
In this thesis, we investigate various integral submodels and generalize them. In part I, we study the submodel of the nonlinear $\mathbf{C}P^1$-model and the related submodels in $(1+2)$ dimensions. In part II, we construct integrable…
We find new families of shape invariant potentials depending on n>=1 parameters subject to translation by the inclusion of non-trivial invariants. New dependencies of the spectra are found, and it opens the door to the engineering of…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode…
The existence of an unbounded sequence of solutions to a conformally invariant elliptic equation having nonlocal critical-power nonlinearity is established. The primary obstacle to establishing existence of solutions is the failure of…
We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the…
A general algebraic approach, incorporating both invariance groups and dynamic symmetry algebras, is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum many-body physics models due to symmetries…
A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean…
We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain…
The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that…
Context: Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics. Aims: We aim to show how PT and anti-PT symmetries determine the…
Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the…
Nonlinear initial-boundary value problem on deep-water gravity waves of finite amplitude is solved approximately (up to small terms of higher order) assuming that the waves are generated by an initial disturbance to the water and the…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
Using time as an additional design parameter in electromagnetism, photonics, and wave physics is attracting considerable research interest, motivated by the possibility to explore physical phenomena and engineering opportunities beyond the…
We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these…
In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra…
Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary…
It is shown that the emergence of obstacles to asymptotic integrability in the analysis of perturbed evolution equations may, often, be a consequence of the manner, in which the freedom in the ex-pansion is exploited in the derivation of…