Related papers: Some addition to the generalized Riemann-Hilbert p…
Singularities in $(3+1)$-dimensional Horava-Lifshitz (HL) theory of gravity are studied. These singularities can be divided into scalar, non-scalar curvature, and coordinate singularities. Because of the foliation-preserving diffeomorphisms…
The Landau--Lifshitz--Baryakhtar (LLBar) equation is a generalisation of the Landau--Lifshitz--Gilbert and the Landau--Lifshitz--Bloch equations which takes into account contributions from nonlocal damping and is valid at moderate…
This work addresses the problem of solving the Cahn-Hilliard equation numerically. For that we introduce an abstract formulation for Cahn-Hilliard type equations with dynamic boundary conditions, we conduct the spatial semidiscretization…
As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness…
The paper aims at developing the Riemann-Hilbert problem approach to the modified Camassa-Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at the both infinities of the space variable. In this…
We present a boundary integral method for solving a certain class of Riemann-Hilbert problems known as the general conjugation problem. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann…
In this note, we compute the explicit formula of the monodromy data for a generalized Lam\'{e} equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemman-Hilbert problem.
We consider distributions on $\R^n\setminus{0}$ which satisfy a given set of partial differential equations and provide criteria for the existence of extensions to $\R^n$ that satisfy the same set of equations on $\R^n$. We use the results…
For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…
We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which,…
In this work, we consider the generalized variable-coefficient nonlinear Schr\"{o}dinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an…
Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…
An extension of Horava-Lifshitz gravity was recently proposed in order to address the pathological behavior of the scalar mode all previous versions of the theory exhibit. We show that even in this new extension the strong coupling…
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We show that the solutions of the Riemann-Hilbert problem are in bijection with some…
In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…
In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the…
Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is performed for most general second order differential equation, which involves all physically interesting cases, as Schrodinger and…
The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr\"{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace…