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Abstract: We introduce a class of reductions of the two-component KP hierarchy that includes the Hirota–Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterizing the reduction. We derive the reduction conditions in terms of the Lax operator and higher linear operators of the hierarchy, as well as in terms of the basic two-component KP system of equations. PubDate: 2022-04-01

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Abstract: The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such. PubDate: 2022-04-01

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Abstract: Based on a determining equation set and master function, we consider a Cauchy matrix scheme for three semidiscrete lattice Korteweg–de Vries-type equations. The Lax integrability of these equations is discussed. Various types of solutions, including soliton solutions, Jordan-block solutions, and mixed solutions are derived by solving the determining equation set. Specifically, we find \(1\) -soliton, \(2\) -soliton, and the simplest Jordan-block solutions for the semidiscrete lattice potential Korteweg–de Vries equation. PubDate: 2022-04-01

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Abstract: We for the first time study the integrable nonlocal nonlinear Gerdjikov–Ivanov (GI) equation with variable coefficients. The variable-coefficient nonlocal GI equation is constructed using a Lax pair. On this basis, the Darboux transformation is studied. Exact solutions of the variable-coefficient nonlocal GI equation are then obtained by constructing the \(2n\) -fold Darboux transformation of the equation. The results show that the solution of the GI equation with variable coefficients is more general than that of its constant-coefficient form. By taking special values for the coefficient function, we can obtain specific exact solutions, such as a kink solution, a periodic solution, a breather solution, a two-soliton interaction solution, etc. The exact solutions are represented visually with the help of images. PubDate: 2022-04-01

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Abstract: We discuss the gravitational collapse of a spherically symmetric perfect fluid distribution of uniformly contracting stars. In a uniformly contracting star, the relative volume element (relative distance in the case of spherical symmetry) between any two neighboring fluid particles is preserved irrespective of the radial coordinate. The physical meaning is that during collapse each small volume element of the fluid distribution preserves its spatial position. This new class of gravitational collapse is analogous to the reverse phenomenon of motion of galaxies during the expansion of the Universe. We discuss the shearing solution of a perfect fluid distribution executing the uniform expansion, which is a scalar and obeys the equation of state \(p=p(\rho)\) . The field equation is solved in complete generality, such that that the Oppenheimer–Snyder solution with homogeneous density and the Thompson–Whitrow shear-free solution arise as particular cases. PubDate: 2022-04-01

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Abstract: An extended MKdV hierarchy associated with a \(3\times3\) matrix spectral problem is derived by resorting to the Lenard recursion series and zero-curvature equation. The three-sheeted Riemann surface \(\mathcal K_{m-1}\) for the extended MKdV hierarchy is defined by the zeros of the characteristic polynomial of the Lax matrix together with two points at infinity. On \(\mathcal K_{m-1}\) , we introduce the Baker–Akhiezer function and a meromorphic function, and then obtain their explicit representations in terms of the Riemann theta function with the aid of algebraic geometry tools. The asymptotic expansions of the meromorphic function give rise to quasiperiodic solutions for the entire extended MKdV hierarchy. PubDate: 2022-04-01

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Abstract: We consider a realization of representations of the Lie algebra \(\mathfrak{o}_5\) in the space of functions on the group \(Spin_5\simeq Sp_4\) . In the representations, we take a Gelfand–Tsetlin-type basis associated with the restriction \(\mathfrak{o}_5\downarrow\mathfrak{o}_3\) . Such a basis is useful in problems appearing in quantum mechanics. We explicitly construct functions on the group that correspond to basis vectors. As in the cases of \(\mathfrak{gl}_3\) and \(\mathfrak{sp}_4\) Lie algebras, these functions can be expressed in terms of \(A\) -hypergeometric functions (this does not hold for higher-rank algebras of these series). Using this realization, we obtain formulas for the action of generators. PubDate: 2022-04-01

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Abstract: The inverse spectral problem method is used to integrate the nonlinear Schrödinger equation with some additional terms in the class of infinite-gap periodic functions. We reveal the evolution of spectral data for a periodic Dirac operator whose coefficients solve the Cauchy problem for a nonlinear Schrödinger equation with some additional terms. Several examples are given to illustrate the algorithm described in the paper. PubDate: 2022-04-01

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Abstract: We construct the deformed ladder operators in the presence of a minimal length to study the one- and two-mode squeezed harmonic oscillator. The generalized Hamiltonian of the system is expressed in terms of a deformed \(su(1,1)\) algebra. The realizations of this algebra allow us to convert the purely quantum mechanical problem of the model into a differential equation. By means of the Nikiforov–Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions, in the momentum space, are expressed in terms of hypergeometric functions. Our study shows that the domain of existence of the energy levels is extended and this extension is due to the deformation parameter. PubDate: 2022-04-01

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Abstract: We find a black hole solution in a five-dimensional fully anisotropic holographic model of light quarks. The model is described by the Einstein action with a dilaton field and three Maxwell tensors. The third Maxwell term is related to an external magnetic field. Influence of the external magnetic field on the \(5\) -dimensional black hole solution is considered. PubDate: 2022-03-01

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Abstract: By introducing shift relations satisfied by a matrix \(\boldsymbol{r}\) , we propose a generalized Cauchy matrix scheme and construct a discrete second-order Ablowitz–Kaup–Newell–Segur equation. A modified form of this equation is given. By applying an appropriate skew continuum limit, we obtain the semi-discrete counterparts of these two discrete equations; in the full continuum limit, we derive continuous nonlinear equations. Solutions, including soliton solutions, Jordan-block solutions, and mixed solutions, of the resulting discrete, semi-discrete, and continuous Ablowitz–Kaup–Newell–Segur-type equations are presented. The reductions to discrete, semi-discrete, and continuous nonlinear Schrödinger equations and modified nonlinear Schrödinger equation are also discussed. PubDate: 2022-03-01

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Abstract: In the framework of the Bogoliubov–de Gennes equation, we study the spinless \(p\) -wave superconductor in an infinite strip in the presence of some impurity. We analytically determine the wave functions of stable bound states with energies close to edge points of the energy gap. We prove that for a small impurity potential, the contribution of the nearest subbands to the wave functions in the case of energy values close to edge points is very small, and these energy levels are significantly closer to the gap edge than in the one-dimensional case. We also study the bound states with nearly zero energy values; in contrast to the one-dimensional case, they do not have the “particle–hole” symmetry. In the cases under study, in addition to the bound states, there also exit resonance states related to them. PubDate: 2022-03-01

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Abstract: We prove that the Pohlmeyer–Lund–Regge system is, up to coordinate changes, the unique two-component variational system of chiral type with an irreducible metric that admits a Lax representation with values in the algebra \(\mathfrak{so}(3)\) PubDate: 2022-03-01

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Abstract: We use the inverse scattering theory to integrate the differential–difference sine-Gordon equation with a self-consistent source. PubDate: 2022-03-01

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Abstract: We consider the \(\lambda\) -model on the Cayley tree of order \(k=2\) . Under some conditions, we study translation-invariant Gibbs measures. Moreover, we investigate whether these Gibbs measures are extremal or nonextremal in the set of all Gibbs measures. PubDate: 2022-03-01

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Abstract: We construct a dynamical model of the deformation of a classical diffusion process into superdiffusion implementing the interaction between the diffusion background medium and the external medium. We show how the transformation law of energy characteristics of this deformation is formed gradually. PubDate: 2022-03-01

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Abstract: The initial-boundary value problems for the matrix Lakshmanan–Porsezian–Daniel system are studied by utilizing the Fokas unified transform approach. First, the spectral analysis of the \(4\times4\) Ablowitz–Kaup–Newell–Segur-type matrix Lax pair is performed. Second, solutions of the matrix Lakshmanan–Porsezian–Daniel system are reconstructed from a \(4\times4\) matrix Riemann–Hilbert problem. It is proved in addition that the spectral functions are not independent but are related by the so-called global relation. PubDate: 2022-03-01

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Abstract: Various phase-space distributions, from the celebrated Wigner function, to the Husimi \(Q\) function and the Glauber–Sudarshan \(P\) distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the \(s\) -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter \(s\) , we interpret the \(s\) -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the \(s\) -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the \(s\) -ordered phase-space distribution. In this connection, the parameter \(-s/2\) (rather than the original \(s\) ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter \(s\) , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the \(s\) -ordered phase-space distribution is derived. PubDate: 2022-03-01

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Abstract: For the Vlasov equation with a self-consistent field, a connection is established between the dispersion relation and the Schur algebraic complement of the generator of the corresponding dynamical system. An estimate of the instability index is obtained in terms of the Hankel transform of the background distribution of electrons, the sign of which is determined using the saddle point method. PubDate: 2022-03-01

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Abstract: Two coupled systems involving both bosonic and fermionic fields are proposed as super generalizations of the \(K(-2,-2)\) equation \(u_t=\partial_x^3(u^{-2}/2)-\partial_x(2u^{-2})\) . Linear spectral problems are presented to certify their integrability and lead to infinitely many conservation laws. Based on natural conservation laws, reciprocal transformations are defined that map one super \(K(-2,-2)\) equation to Kupershmidt’s super modified Korteweg–de Vries (mKdV) equation, and the other super \(K(-2,-2)\) equation to the supersymmetric mKdV equation. By means of these connections, bi-Hamiltonian formulations are established for the super \(K(-2,-2)\) equations. PubDate: 2022-03-01