Spinor electrodynamics in terms of gauge-invariant quantities

We give a formulation of classical spinor electrodynamics in terms of gauge-invariant quantities. The set of invariants consists of bilinear combinations of spinor fields (currents), a real-valued covector field, and a complex scalar field of modulus one. The presented result is a first step towards formulating quantum electrodynamics in terms of gauge-invariant fields.     

A Connecting Link between Lorentz and Dirac Theories of Electrons

A theory, that is an initial step towards bridging the gap between Lorentz and Dirac theories of electrons, is presented. An electromagnetic Lagrangian density is postulated, such that, the theory can be cast into a form, similar to Dirac theory. Electromagnetic interpretation of conserved currents, and of usual bilinear covariants in the Dirac theory, is obtained. A counterpart of Abraham-Lorentz equation, with oscillatory solutions is derived. Self-energy of electrons is expressed in terms of their self-potentials and self-currents.     

On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

Abstract

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τ -functions.     

Completeness of observables and particle trajectories in quantum mechanics

The complete set of observables (bilinear Hermitian forms) is determined for the Schrödinger equation and their connection with the curvature and torsion of the curves, where conservation laws are fulfilled, is established. It is shown that these curves for a free particle, in the general case, are spiral lines with the radius and step length defined by the observables at the initial point (both parameters are proportional to the de Broglie wavelength). A spiral line turns to a straight line under some conditions. The trajectory variations are considered in the problem with a potential step and a rectangular barrier. It is shown that spiral lines can be transformed into straight lines and vice versa. All observables, which are changed along the potential barrier, can be restored under some constraints on the potential. The Hermitian transformations at the potential step are connected with the Lorentz transformations. A qualitative explanation of the double-slit experiment for extremely low intensity of the particles' source in the absence of the interference conditions is suggested.     

Bell Polynomials to the Kadomtsev-Petviashivili Equation with Self-Consistent Sources

Bell Polynomials play an important role in the characterization of bilinear equation. Bell Polynomials are extended to construct the bilinear form, bilinear Bäcklund transformation and Lax pairs for the Kadomtsev-Petviashvili equation with self-consistent sources.     

Structure of resonances and formation of stationary points in symmetrical chains of bilinear oscillators

Dynamics of strongly nonlinear systems can in many cases be modelled by bilinear oscillators, which are the oscillators whose springs have different stiffnesses in compression and tension. This underpins the analysis of a wide range of phenomena, from oscillations of fragmented structures, connections and mooring lines to deformation of geological media. Single bilinear oscillators were studied previously and the presence of multiple resonances both super- and sub-harmonic was found. Less attention was paid to systems of multiple bilinear oscillators that describe many natural and engineering processes such as for example the behaviour of fragmented solids. Here we fill this gap concentrating on the simplest case – 1D symmetrical chains of bilinear oscillators. We show that the presence and structure of resonances in a symmetric chain of bilinear oscillators with fixed ends depends upon the number of oscillating masses. Two elementary chains act as the basic ones: a single mass bilinear chain (a mass connected to the fixed points by two bilinear springs) that behaves as a linear oscillator with a single resonance and a two mass chain that is a coupled bilinear oscillator (two masses connected by three bilinear springs). The latter has multiple resonances. We demonstrate that longer chains either do not have resonances or get decomposed, in the resonance, into either the single mass or two mass elementary chains with stationary masses in between. The resonance frequencies are inherited from the basic chains of decomposition. We show that if the number of masses is odd the chain can be decomposed into the single mass bilinear chains separated by stationary masses. It then inherits the resonances of the single mass bilinear chain. The chains with the number of masses minus 2 divisible by 3 can be decomposed into the two mass bilinear chains separated by stationary masses and inherit the resonances of the two mass chains. The chains whose lengths satisfy both criteria (such as chains with 5, 11, 17 … masses) allow both types of resonances.     

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.     

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.     

Exact Solutions for a Nonisospectral and Variable-Coefficient Kadomtsev-Petviashvili Equation

The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the bilinear Backlund transformation from its Lax pairs and find solutions with the help of the obtained bilinear Bgcklund transformation.     

Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko

Based on the Hirota bilinear operators and their generalized bilinear derivatives, we formulate two new (2+1)-dimensional nonlinear partial differential equations, which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schiff equation and the generalized Bogoyavlensky-Konopelchenko equation as particular examples, and the other has the same bilinear form with different $D_p$-operators. A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches. A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.     

Exact Solutions to a (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painlevé expansion at the constant level term, the Hirota bilinear form is obtained for a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ε-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskianform is presented and proved by direct substitution into the bilinear equation.     

The Bilinear Integrability,N-soliton and Riemann-theta function solutions of B-type KdV Equation

In this paper, the bilinear integrability for B-type KdV equation have been explored. According to the relation to tau function, we find the bilinear transformation and construct the bilinear form with an auxiliary variable of the B-type KdV equation. Based on the truncation form, the Bäcklund transformation has been constructed. Furthermore, the N-soliton solutions and Riemann-theta function 1-periodic solutions of the B-type KdV equation are obtained.     

Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation

In this paper, by virtue of symbolic computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrödinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Bäcklund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.     

New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation

Based on the binary Bell polynomials, the bilinear form for the Boiti-Leon-Manna-Pempinelli equation is obtained. The new exact solutions are presented with an arbitrary function in y, and soliton interaction properties are discussed by the graphical analysis. Further, the bilinear Bäcklund transformation is derived by the binary Bell polynomials, and the corresponding Lax pair is obtained by linearizing the bilinear equation.     

A Cartesian method for fitting the bathymetry and tracking the dynamic position of the shoreline in a three-dimensional, hydrodynamic model

This paper presents a Cartesian method for the simultaneous fitting of the bathymetry and shorelines in a three-dimensional, hydrodynamic model for free-surface flows. The model, named LESS3D (Lake & Estuarine Simulation System in Three Dimensions), solves flux-based finite difference equations in the Cartesian-coordinate system (x,y,z). It uses a bilinear bottom to fit the bottom topography and keeps track the dynamic position of the shoreline. The resulting computational cells are hybrid: interior cells are regular Cartesian grid cells with six rectangular faces, and boundary/bottom cells (at least one face is the water–solid interface) are unstructured cells whose faces are generally not rectangular. With the bilinear interpolation, the shape of a boundary/bottom cell can be determined at each time step. This allows the Cartesian coordinate model to accurately track the dynamic position of the shorelines. The method was tested with a laboratory experiment of a Tsunami runup case on a circular island. It was also tested for an estuary in Florida, USA. Both model applications demonstrated that the Cartesian method is quite robust. Because the present method does not require any coordinate transformation, it can be an attractive alternative to curvilinear grid model.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

Supersymmetric two-boson equation: Bilinearization and solutions

A bilinear formulation for the supersymmetric two-boson equation is derived. As applications, some solutions are calculated for it. We also construct a bilinear Bäcklund transformation.     

Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

Abstract

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirota's bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.