General Form Of Solution Research Articles

The nonlinear Riemann–Hilbert problems for a new general Pavlov equation

Manakov and Santini had studied the formal solutions and utilized the Riemann–Hilbert (RH) dressing method to investigate longtime behavior of the solutions and asymptotical implicit solutions of Pavlov equation (Manakov and Santini, 2007) and heavenly equation (Manakov and Santini, 2006). In the paper, we introduce a new Lax pair to construct an integrable system, which can be reduced to the standard Pavlov equation. Therefore, we refer to it as a generalized Pavlov equation. By utilizing the inverse scattering transform (IST) technique, we successfully derive the general formal solutions for the general Pavlov equation. Additionally, through the construction of a new RH problem, we investigate the longtime behavior of the solutions to the general Pavlov equation. The results presented in this paper expand upon the results previously presented by Manakov and Santini in their work (Manakov and Santini, 2007).

Existence and uniqueness of the motion of a particle subject to a unilateral constraint and friction

We prove that there exists a unique solution to the initial value problem describing the motion of a particle subject to a unilateral constraint and Coulomb friction, if the external force acting on the particle is an analytic function of time and of the particle’s position and velocity. Previous work claimed that this problem has a local series solution that corresponds to an analytic function, after any impacts have been resolved. However, a counterexample to that claim was recently discovered, involving a particle starting to slide, in which the series solution is divergent, and thus does not correspond to an analytic function. This paper corrects previous arguments by considering a general formal series solution for a particle that is starting to slide.

Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems

The paper discusses three-dimensional models of the propagation of stochastic internal waves in hydrodynamic systems: ’half-space - half-space’, ’half-space - layer with rigid lid’, and ’layer with solid bottom - layer with rigid lid’. In constructing the models, the layers are considered to be ideal fluids separated by a contact surface. The main objective of the modeling is to obtain a dynamic equation for the stochastic amplitude of surface waves. A comparative analysis of the obtained results has been conducted. In order to control the contribution of nonlinear terms, a dimensionless non-numerical parameter has been introduced. The models are distinguished by boundary conditions that determine the general form of solutions. As a result, a dynamic equation for the stochastic amplitude of internal waves has been derived. After ensemble averaging of the amplitudes, the dynamic equation is formulated in integral form using Fourier-Stieltjes integrals. The dynamic equation reveals two-wave and three-wave interactions, as well as the contribution of dispersion to wave dynamics. An investigation of the boundary case of the transition of internal waves in the ’half-space - half-space’ system to surface waves in the absence of an upper liquid layer confirms the validity of the results.

Study tsunamis through approximate solution of damped geophysical Korteweg-de Vries equation

<abstract><p>The article studied tsunami waves with consideration of important wave properties such as velocity, width, and collision through finding an approximate solution to the damped geophysical Korteweg-de Vries (dGKdV). The addition of the damping term in the GKdV is a result of studying the nonlinear waves in bounded nonplanar geometry. The properties of the wave in bounded nonplanar geometry are different than the unbounded planar geometry, as many experiments approved. Thus, this work reported for the first time the analytical solution for the dGKdV equation using the Ansatz method. The used method assumed a suitable hypothesis and the initial condition of the GKdV. The GKdV is an integrable equation and the solution can be found by several known methods either analytically or numerically. On the other hand, the dGKdV is a nonintegrable equation and does not have an initial exact solution, and this is the challenge. In this work, the novel Ansatz method proved its ability to reach the approximate solution of dGKdV and presented the effect of the damping term as well as the Coriolis effect term in the amplitude of the wave. The advantage of the Ansatz method was that the obtained solution was in a general solution form depending on the exact solution of GKdV. This means the variety of nonlinear wave structures like solitons, lumps, or cnoidal can be easily investigated by the obtained solution. We realized that the amplitude of a tsunami wave decreases if the Coriolis term or damping term increases, while it increases if wave speed increases.</p></abstract>

Characterizations of the solutions to linear interval fractional difference equations

In this work, linear interval fractional difference equations (LIFDEs) are introduced. We investigate the general form of solutions to LIFDEs under the generalized Hukuhara difference. Moreover, we use the theory of vector difference equations to solve first order linear interval difference equations. An example is presented to examine our theoretical results.

Resonance, fusion and fission dynamics of bifurcation solitons and hybrid rogue wave structures of Sawada–Kotera equation

The goal of this article is to investigate brand-new exact solutions to the Sawada-Kotera (SK) equation. To get at a general form solution to the SK equation, the Hirota bilinear method is used. In order to study the novel soliton solutions, several dispersion coefficients are adopted. For the first time in the literature, we analyze novel multiple-bifurcated soliton and novel rogue wave solutions of the SK equation. We study the interaction of dark soliton with rogue waves and its resonance. Another type of hybrid novel rogue wave solution is investigated, which is the fusion of bright X-shaped and rogue wave. All results are validated and displayed through graphs by using MATLAB-2020.

Traveling wave and general form solutions for the coupled Higgs system

In this study, the coupled Higgs system, which is a special case of the coupled Higgs field equation, which is effective in energy transport in the sub‐particles of the atom, is discussed. With the help of the modified generalized exponential rational function method, which is an important instrument in obtaining traveling wave solutions, both the propagating wave solutions and general form solutions of coupled Higgs system are presented. These solutions are examined under some restrictive conditions as they are presented. It is argued that these solutions differ from the literature. The advantages and disadvantages of the method discussed in the conclusion and discussion section are discussed. In addition, the changes in the behavior of the traveling wave solution are discussed by giving physical meaning to some constants in the traveling wave solutions produced by the method. The effects on the traveling wave solution are analyzed for different values of wave number, wave velocity, and wave frequency, which have physically important meanings. In addition, the behaviors caused by these effects are supported with the help of simulation.

On the Existence of Phonon Coherent States in Nanomaterials

In our work we describe the energy transfer by thermal excitations with accounting a coherence in nanosized systems during heat removal. A general formalism of thermal conductivity by second quantization method is proposed with an account of both the usual phonon model of heat transfer and the formation of coherent Schrödinger states of the oscillator system. An exact general form of solution for a time-dependent problem with arbitrary initial conditions is analytically developed. It is shown that at certain ratios of constants characterizing the interaction of phonons with the electronic subsystem a heat flow does not decay with time in the crystal.

Different dynamics of a periodic mass-in-mass nonlinear chain during a single mode excitation

The multiple scale dynamics of a periodic chain composed of nonlinear mass-in-mass cells is studied. Based on a continuous approach of the one-dimensional chain, dispersion equation is obtained which provides the general form of solutions of the linearized system. Taking into account a single harmonic of the chain around a 1:1 resonance with a targeted mode, fast dynamics of the system leads to the detection of the slow invariant manifold and its stability borders. Slow dynamics permits to predict singularities and equilibrium points leading to frequency responses. The developments predict periodic and non-periodic responses and permit to tune parameters of the chain for the aim of localization of vibrating energy and design of periodic or non-periodic responses.

Research on rockfall impact process based on viscoelastic contact theory

Rockfall collision is the most complex and unpredictable movement pattern in rockfall movement. By examining the rockfall collision process, it will be possible to accurately forecast the route of the rockfall and direct the selection of protective measures. There is no unified quantitative theoretical solution, and the primary research approaches for determining rockfall collision characteristics are experiments and numerical simulation. In this paper, the rockfall is simplified as a sphere, and the rockfall impact mechanics model is constructed based on the viscoelastic contact theory. The time history solution method of the rockfall collision parameters is theoretically obtained by the differential equation solution method in combination with the initial conditions of the rockfall collision. Further research results show that the rockfall impact acceleration, impact velocity and impact displacement time history solutions derived in this paper are close to the logistic method. Finally, the results demonstrate that the results of this study are accurate when compared to the widely used rockfall maximum impact force algorithm. In conclusion, in this study, the rockfall collision process is theoretically quantified, and the general solution form of the time-history law of rockfall impact load is obtained, which has engineering applications.

Ground motion around a semi-cylindrical canyon in saturated soil: P wave incidence

Based on Biot wave theory in saturated two-phase medium, the analytical solution of plane P-wave scattering by semi-circular canyon in half space in saturated two-phase medium is given by using complex function method and Hankel function integral transformation method. Firstly, a semi-circular depression model in saturated half space is constructed, the wave field potential function in the medium is solved, and its general solution form is obtained. Secondly, the unknown coefficient value in the potential function is obtained by substituting the corresponding stress boundary conditions. With the help of Hankel function integral transformation method, the wave potential function in complex coordinates is transformed to meet the horizontal boundary conditions of half space in rectangular coordinates. Finally, the surface displacement amplitude at the semi-circular depression and the surrounding horizontal boundary is obtained through the total wave field potential function, and the variation law of surface displacement amplitude under the influence of different parameters is analyzed.

Spectral functions of the half-filled one-dimensional Hubbard chain within the exchange-correlation potential formalism

The spectral functions of the one-band half-filled 1D Hubbard chain are calculated using the exchange-correlation potential formalism developed recently. The exchange-correlation potential is adopted from the exact potential derived from the Hubbard dimer. Within an approximation in which the full Green function is replaced by a non-interacting one, the spectral functions can be calculated analytically. Despite the simplicity of the approximation, the resulting spectra are in favorable agreement with the more accurate results obtained from the dynamic density-matrix renormalization group method. In particular, the calculated band gap as a function of $U$ is in close agreement with the exact gap obtained from the Bethe ansatz. In addition, the formal general solution to the equation of motion of the Green function is presented and the difference between the traditional self-energy approach and the exchange-correlation potential formalism is also discussed and elaborated. A simplified Holstein Hamiltonian is considered to further illustrate the general form of the exchange-correlation potential.

A class of isochronous and non-isochronous nonlinear oscillators

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude dependent frequency of oscillations from the integrals of the simple harmonic oscillator equation. To achieve this, we consider the case where the integrals are in the same form both for the linear and the nonlinear oscillators in either of the cases. We also discuss the method of deriving the associated integrals and the general solution in harmonic form for both the types. We demonstrate the applicability of this method up to 2N coupled first order nonlinear ODEs in both the cases. Further, we illustrate the theory with an example in each case.

CONTROLLABILITY OF FREDHOLM’S INTEGRO-DIFFERENTIAL EQUATIONS WITH BY A DEGENERATE KERNEL IN HILBERT SPACES

The work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces. The need to study these equations is related to numerous ones applications of integro- differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere. There are different approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coefficient conditions of solvability and analytical presentation of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods. In this connection, Fredholm’s integro-differential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.

SOME NEW CHARACTERIZATIONS OF PARTIAL ISOMETRIES IN RINGS WITH INVOLUTION

In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.

Rindler geodesics: Lie and Hamiltonian symmetries, conservation laws and Hamiltonian equations

The present paper includes the Lie symmetry analysis of the Rindler’s geodesics. Geometric vector fields of symmetries are found via the standard Lie algorithm. The general form of solutions constructed by the symmetries are given. Hamiltonian equations and Hamiltonian symmetries are calculated by the use of obtained symmetry operators. Finally conservation laws are computed via two different methods. A method which is based on the modifies version of Noether’s theroem by a formal Lagrangian introduced by Ibragimov. Also, another conservation laws are calculated by the direct method.

The Riemann-Hilbert approach for the focusing Hirota equation with single and double poles

We study the Cauchy problem of the focusing Hirota equation with fastly decay initial value at infinity via Riemann-Hilbert (RH) approach. The Jost function and scattering matrix related to the initial value are obtained by spectral analysis, and their analytical, asymptotic and symmetric properties are analyzed. The solution of the Hirota equation is derived by solving the corresponding RH problem, which contains two cases of scattering data with single zeros and double zeros. The general form of solutions with no reflection potential is obtained. The two types of the discrete spectrum are selected for the purpose of comparing the influence of different eigenvalues on solution propagation behavior.

Single-photon emission computed tomography in the scattering medium with the property of “scattering straight back”

The problem of radiation scattering in integral reconstruction algorithms for single-photon emission computed tomography (SPECT) has not been completely solved. This is primarily due to the fact that the radiation transport equation (RTE) does not have an exact analytical solution in general form for a scattering indicatrix corresponding to a real medium. A special form of the scattering indicatrix is proposed here that corresponds to a scattering medium with a property named “scattering straight back.” With this scattering indicatrix, it is possible to obtain an exact analytical solution of the RTE with exact boundary conditions for an arbitrary distribution of radiation sources. From this exact solution, a new expression can be obtained for the measured data, named as generalized attenuated exponential Radon transform. A new definition of projections is proposed to obtain an exact solution of the inverse tomographic problem. This solution has the same level of rigor as the exponential Radon transform for a purely absorbing medium. The proposed algorithm is compared with the traditional SPECT algorithm (the inverse exponential Radon transform) using Monte Carlo simulation based on the nuclear physics software package Geant4. It is demonstrated that the new method is superior to the traditional method in terms of both the standard deviation criterion and better visual distinction of the details in the reconstructed tomogram. The results obtained can be used to improve the existing algorithms for image reconstruction in SPECT, as well as to aid in developing new designs of emission tomographs.

VOLUMETRIC HYDRAULIC DRIVE MODELLING IN TRANSMISSIONS OF TRACTORS AND SELF-PROPELLED MACHINES

The development of a mathematical description of the individual nodes of the object of study makes the mathematical models non-linear, using both deterministic and probabilistic characteristics. Moreover, it is not possible to obtain a solution in general form for such mathematical models without significant simplifications. Modern computer technology allows you to move on to the modular method of compiling mathematical models when a bank of models of individual units and assemblies is developed, the combination of which makes it possible in a short time to collect the required generalized mathematical model or supplement it with newly created modules. The main elements of the aggregates together determine the nature of the transient processes and dynamic loads in the transmission at various operating modes, and will be considered as some modules from which you can build the mathematical model necessary for the study. One of these modules is a volumetric hydraulic drive (GCP), which are increasingly being used in the transmissions of modern tractors and self-propelled agricultural machines. In the presented article, a mathematical description of axial piston hydraulic machines is considered, in which the power and kinematic gear ratios in the GCP can be adjusted both at the pump and at the hydraulic motor. The dynamic characteristics of hydraulic machines, fluid leakage and its elastic properties, as well as variable values of hydraulic drive efficiency are taken into account. The simulation results were compared with experimental studies. The discrepancy between them was in the range (5-8)%. As the objects of study were used: a model of the T-150E caterpillar tractor with independent full-threaded UCP of the left and right sides; harvesting machine KS-6B with OGP on the rear drive axle; harvesting machine KS-6V with independent UCP on the rear and front drive axles; layout of a combine harvester with independent UCP of the left and right sides of the rear drive wheels.

Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case

In this paper, we prove the unique solvability of the Cauchy problem for a linear inhomogeneous equation in a Banach space that is solved with respect to the Gerasimov–Caputo fractional derivative. We assume that the operator acting on the unknown function in the equation generates a set of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. The general form of solutions to the Cauchy problem is obtained. The general results are applied to the study of the unique solvability of a certain class of initial-boundary-value problems for partial differential equations solvable with respect to the Gerasimov–Caputo fractional derivative by time, containing in the simplest case initial-boundary-value problems for fractional diffusion and diffusion-wave equations.