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9 pages, 259 KiB

Open AccessArticle

The Friedrichs Extension of Elliptic Operators with Conditions on Submanifolds of Arbitrary Dimension

by Anton Savin

Mathematics 2024, 12(3), 418; https://doi.org/10.3390/math12030418 - 27 Jan 2024

Cited by 1 | Viewed by 698

We describe the Friedrichs extension of elliptic symmetric pseudodifferential operators on a closed smooth manifold with the domain consisting of functions vanishing on a given submanifold. In summary, the Friedrichs extension is an elliptic Sobolev problem defined in terms of boundary and coboundary [...] Read more.

We describe the Friedrichs extension of elliptic symmetric pseudodifferential operators on a closed smooth manifold with the domain consisting of functions vanishing on a given submanifold. In summary, the Friedrichs extension is an elliptic Sobolev problem defined in terms of boundary and coboundary operators, and the number of boundary and coboundary conditions in the problem depends on the order of the operator and the codimension of the submanifold. In this paper, the discreteness of the spectrum is proved, and singularities of eigenfunctions are described. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

8 pages, 229 KiB

Open AccessArticle

On Global Solutions of Hyperbolic Equations with Positive Coefficients at Nonlocal Potentials

by Andrey B. Muravnik

Mathematics 2024, 12(3), 392; https://doi.org/10.3390/math12030392 - 25 Jan 2024

Cited by 2 | Viewed by 741

Abstract

We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to [...] Read more.

We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to change its sign. Earlier, only the case where the said sign is constant was investigated. We find a condition relating the coefficient at the nonlocal term of the investigated equation and the length of the translation, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

15 pages, 294 KiB

Open AccessArticle

Miura-Type Transformations for Integrable Lattices in 3D

by Ismagil T. Habibullin, Aigul R. Khakimova and Alfya U. Sakieva

Mathematics 2023, 11(16), 3522; https://doi.org/10.3390/math11163522 - 15 Aug 2023

Cited by 1 | Viewed by 742

Abstract

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form [...] Read more.

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form

u_{n + 1, x}^{j} = f (u_{n, x}^{j}, u_{n}^{j + 1}, u_{n}^{j}, u_{n + 1}^{j}, u_{n + 1}^{j - 1})

up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders,

N = 1

and

N = 2

, complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case

N = 1

, an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

23 pages, 418 KiB

Open AccessArticle

Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients

by Natalia P. Bondarenko

Mathematics 2023, 11(16), 3455; https://doi.org/10.3390/math11163455 - 9 Aug 2023

Cited by 3 | Viewed by 968

Abstract

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order

n \geq 2

. We show that every matrix function of this class is associated with some [...] Read more.

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order

n \geq 2

. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases

n = 2

and

n = 4

in more detail. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

12 pages, 331 KiB

Open AccessArticle

On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra

by Sergey Buterin

Mathematics 2023, 11(12), 2688; https://doi.org/10.3390/math11122688 - 13 Jun 2023

Cited by 4 | Viewed by 1996

Abstract

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the [...] Read more.

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

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9 pages, 862 KiB

Open AccessArticle

On the Dynamics of Solitary Waves to a (3+1)-Dimensional Stochastic Boiti–Leon–Manna–Pempinelli Model in Incompressible Fluid

by Wael W. Mohammed, Farah M. Al-Askar, Clemente Cesarano and M. El-Morshedy

Mathematics 2023, 11(10), 2390; https://doi.org/10.3390/math11102390 - 22 May 2023

Cited by 2 | Viewed by 1435

Abstract

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to [...] Read more.

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liquid in fluid mechanics, the acquired solutions may be applied to explain a lot of fascinating physical phenomena. To address how Brownian motion effects the exact solutions of the SBLMPE, we present some 2D and 3D diagrams. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

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21 pages, 398 KiB

Open AccessFeature PaperArticle

Application of Fatou’s Lemma for Strong Homogenization of Attractors to Reaction–Diffusion Systems with Rapidly Oscillating Coefficients in Orthotropic Media with Periodic Obstacles

by Kuanysh A. Bekmaganbetov, Gregory A. Chechkin and Vladimir V. Chepyzhov

Mathematics 2023, 11(6), 1448; https://doi.org/10.3390/math11061448 - 16 Mar 2023

Cited by 2 | Viewed by 1299

Abstract

We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the [...] Read more.

We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the strong topology of the corresponding trajectory dynamical systems. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of considered systems to the trajectory attractors of the corresponding homogenized reaction–diffusion systems which contain an additional potential. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

23 pages, 401 KiB

Open AccessArticle

Geometric Approximation of Point Interactions in Two-Dimensional Domains for Non-Self-Adjoint Operators

by Denis Ivanovich Borisov

Mathematics 2023, 11(4), 947; https://doi.org/10.3390/math11040947 - 13 Feb 2023

Cited by 1 | Viewed by 1185

Abstract

We define the notion of a point interaction for general non-self-adjoint elliptic operators in planar domains. We show that such operators can be approximated in a geometric way by cutting out a small cavity around the point, at which the interaction is concentrated. [...] Read more.

We define the notion of a point interaction for general non-self-adjoint elliptic operators in planar domains. We show that such operators can be approximated in a geometric way by cutting out a small cavity around the point, at which the interaction is concentrated. On the boundary of the cavity, we impose a special Robin-type boundary condition with a nonlocal term. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting one with a point interaction containing an arbitrary prescribed complex-valued coupling constant. The mentioned convergence holds in a few operator norms, and for each of these norms we establish an estimate for the convergence rate. As a corollary of the norm resolvent convergence, we prove the convergence of the spectrum. Full article

(This article belongs to the Special Issue Spectral Theory and Its Applications in Problems of Mathematical Physics)

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