The four-dimensional Martínez Alonso–Shabat equation: Reductions and nonlocal symmetries

We consider the four-dimensional integrable Martínez Alonso–Shabat equation and list three integrable three-dimensional reductions thereof. We also present a four-dimensional integrable modified Martínez Alonso–Shabat equation together with its Lax pair.We further construct an infinite hierarchy of commuting nonlocal symmetries (and not just the shadows, as it is usually the case in the literature) for the Martínez Alonso–Shabat equation.     

Corrigendum to “The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators” [J. Geom. Phys. 85 (2014) 75–80]

We correct an error in Theorem 1 from our article stated in the title. The ensuing results are added as well.     

The renormalized stress-energy tensor and its associated operators for photons in curved four-dimensional space–time

We establish expressions for the renormalized stress-energy (or energy-momentum) tensor and its associated operators relative to photons as a Klein–Gordon field of non-zero rest-mass particles (with gravitational interaction) in curved four-dimensional space–time.     

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

The Darboux–Egoroff system of PDEs with any number $n\ge 3$ of independent variables plays an essential role in the problems of describing $n$-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators.The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev.In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an $(n-1)$-component vector modified KdV hierarchy.The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds.We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.     

The Monge–Ampère equation: Various forms and numerical solution

We present three novel forms of the Monge–Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge–Ampère problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge–Ampère equation is a sum of such distributions.     

The Gibbons–Tsarev equation: symmetries,invariant solutions,and applications

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation.     

Differential operators on the algebra of densities and factorization of the generalized Sturm–Liouville operator

Letters in Mathematical Physics - We consider factorization problem for differential operators on the commutative algebra of densities (defined either algebraically or in terms of an auxiliary...     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

The Schrödinger–Langevin equation with and without thermal fluctuations

The Schrödinger–Langevin equation (SLE) is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the excited states of the quantum subsystem. We then study analytically and numerically the SLE ability to bring a quantum subsystem to the thermal equilibrium of statistical mechanics. To do so, concepts about statistical mixed states and quantum noises are discussed and a detailed analysis is carried with two kinds of noise and potential. We show that within our assumptions the use of the SLE as an effective open quantum system formalism is possible and discuss some of its limitations.     

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington–Finkelstein-type coordinates is shown. To this end, Kerr geometry is described by a Carter tetrad and the Dirac spinors and matrices are given in a chiral Newman–Penrose dyad representation. Applying Chandrasekhar’s mode ansatz, the Dirac equation is separated into systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are explicitly derived. Their decay is analyzed by means of error estimates. Moreover, the eigenfunctions and eigenvalues of the angular system are discussed. Finally, as an application, the scattering of Dirac waves by the gravitational field of a Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in Kerr geometry in horizon-penetrating coordinates and for the construction of a functional analytic integral representation of the Dirac propagator.     

The modified nonlinear Schrödinger equation: facts and artefacts

We argue that the integrable modified nonlinear Schr?dinger equation with the nonlinearity dispersion term is the true starting point to analytically describe subpicosecond pulse dynamics in monomode fibers. Contrary to the known assertions, solitons of this equation are free of self-steepening and the breather formation is possible. Received 29 September 2001 / Received in final form 25 January 2002 Published online 2 October 2002 RID="a" ID="a"doktorov@dragon.bas-net.by     

Interdiffusion in strongly ionic insulating compounds: The Nernst–Planck equation

In strongly ionic insulating materials, the Nernst–Planck Equation relates the interdiffusion coefficient of the cations (having the same charge) with the corresponding tracer diffusivities and the thermodynamic factor. In this paper, we explore the Nernst–Planck Equation for ionic ternary (quasi-binary) and ionic quaternary (quasi-ternary) systems using the diffusion kinetics formalisms of Darken [Trans. Am. Inst. Min. (Metall.) Eng. 175 184 (1948)], Manning [Phys. Rev. B 4 1111 (1971)] and Moleko, Allnatt and Allnatt [Phil. Mag. A 59 141 (1989)]. It is shown that for the binary system, the Darken and Manning formalisms both give the usual form of the Nernst–Planck Equation. However, the almost exact Moleko, Allnatt and Allnatt formalism (for randomly mixed systems) provides an additional correction factor analogous to the vacancy-wind factor in the well-known Darken–Manning Equation used in binary alloy systems. Nernst–Planck-type equations are also derived for strongly ionic insulating ternary systems and are found to behave similar to the binary case.     

The Riemann–Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann,Robin and Dirichlet boundary conditions

We revisit the Helmholtz equation in a quarter-plane in the framework of the Riemann–Hilbert approach to linear boundary value problems suggested in late 1990s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas’ scheme.     

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we revisit the connection between the Riemann–Roch theorem and the zero-energy solutions of the two-dimensional Dirac equation in the presence of a delta-function-like magnetic field. Our main result is the resolution of a paradox—the fact that the Riemann–Roch theorem correctly predicts the number of zero-energy solutions of the Dirac equation despite counting what seem to be functions of the wrong type.     

The Lindblad and Redfield forms derived from the Born–Markov master equation without secular approximation and their applications

In this paper, we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems. The coefficients of the equations are re-evaluated according to the scheme in[(2019), Phys. Rev. A 99, 022118]. They are complex numbers rather than the real numbers obtained from traditional simplified methods. The dynamics of two models(one is an open threelevel quantum system model, and the other is the model of phycoerythrin 545(PE545) in a photosynthesis system) are studied. It is shown that the secular approximation and the simplified real coefficients may cause a small distortion of the dynamics in some environments, but a large distortion of the dynamics in others. These effects are discussed and characterized by studying the dynamics of nontrivial instances of multi-level systems in the presence of dissipation.     

The modified multiple (G ′ / G )-expansion method and its application to Sharma–Tasso–Olver equation

The modified multiple ( \(G^{\prime }/G\) )-expansion method is proposed in this paper to construct exact solutions of nonlinear evolution equations. The validity and advantage of the proposed method are illustrated by its application to the Sharma–Tasso–Olver equation. As a result, various exact solutions including hyperbolic functions, trigonometric functions and their mixture with parameters are obtained. When some parameters are taken as special values, the known double solitary-like wave solutions are derived from the double hyperbolic function solution. It is shown that the method introduced in this paper has general significance in searching for exact solutions to the nonlinear evolution equations.     

Semiclassical approximation of the Wheeler–DeWitt equation: arbitrary orders and the question of unitarity

We extend the Born–Oppenheimer type of approximation scheme for the Wheeler–DeWitt equation of canonical quantum gravity to arbitrary orders in the inverse Planck mass squared. We discuss in detail the origin of unitarity violation in this scheme and show that unitarity can be restored by an appropriate modification which requires back reaction from matter onto the gravitational sector. In our analysis, we heavily rely on the gauge aspects of the standard Born–Oppenheimer scheme in molecular physics.