Addressing the challenges posed by the nonlinear and non-stationary vibrations in rotating machinery,where weak fault characteristic signals hinder accurate fault state representation,we propose a novel feature extrac...Addressing the challenges posed by the nonlinear and non-stationary vibrations in rotating machinery,where weak fault characteristic signals hinder accurate fault state representation,we propose a novel feature extraction method that combines the Flexible Analytic Wavelet Transform(FAWT)with Nonlinear Quantum Permutation Entropy.FAWT,leveraging fractional orders and arbitrary scaling and translation factors,exhibits superior translational invariance and adjustable fundamental oscillatory characteristics.This flexibility enables FAWT to provide well-suited wavelet shapes,effectively matching subtle fault components and avoiding performance degradation associated with fixed frequency partitioning and low-oscillation bases in detecting weak faults.In our approach,gearbox vibration signals undergo FAWT to obtain sub-bands.Quantum theory is then introduced into permutation entropy to propose Nonlinear Quantum Permutation Entropy,a feature that more accurately characterizes the operational state of vibration simulation signals.The nonlinear quantum permutation entropy extracted from sub-bands is utilized to characterize the operating state of rotating machinery.A comprehensive analysis of vibration signals from rolling bearings and gearboxes validates the feasibility of the proposed method.Comparative assessments with parameters derived from traditional permutation entropy,sample entropy,wavelet transform(WT),and empirical mode decomposition(EMD)underscore the superior effectiveness of this approach in fault detection and classification for rotating machinery.展开更多
A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better...A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.展开更多
In this paper,a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems.This method can be used to study the limit cycle behavior of the autonomous systems and to ana...In this paper,a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems.This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system.展开更多
A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and...A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1< kh <1 5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.展开更多
With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for th...With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.展开更多
In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series...In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.展开更多
The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation...The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.展开更多
In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existe...In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.展开更多
In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT pre...In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.展开更多
Applying the method of canonical,transformation,we studied the nonlinear wave equationφ_(xx)-φ_(tt)=dF(φ)/dφ.A Bäcklund transformation(BT)of this equation is obtained.Combining BT of this equation with the we...Applying the method of canonical,transformation,we studied the nonlinear wave equationφ_(xx)-φ_(tt)=dF(φ)/dφ.A Bäcklund transformation(BT)of this equation is obtained.Combining BT of this equation with the well-known BT of sine-Gordon equation(SGE),a formula of nonlinear superposition which contains some arbitrary functions of x and t has been established.This formula leads to a kind of general solution of SGE through its some known solutions.The general soliton solutions and their properties are simply discussed.展开更多
This article describes transformation of nonlinear surface gravity waves under shallow-water conditions with the aid of the suggested semigraphical method. There are given profiles of surface gravity waves on the cres...This article describes transformation of nonlinear surface gravity waves under shallow-water conditions with the aid of the suggested semigraphical method. There are given profiles of surface gravity waves on the crests steepening stages, their leading edges steepening. There are discussed the spectral component influence on the transformation of surface wave profile.展开更多
This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are result...This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.展开更多
In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the it...In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.展开更多
The peculiarities of energy dissipation transferred by solitary waves on defects such as freesurface, grain boundary, region with high concentration of vacancies are studied. One of theways of description of the long ...The peculiarities of energy dissipation transferred by solitary waves on defects such as freesurface, grain boundary, region with high concentration of vacancies are studied. One of theways of description of the long range effect taking place at ion implantation in metallic materialsis suggested.展开更多
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary fun...In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.展开更多
With the aid of computation,we consider the variable-coefficient coupied nonlinear Schr(o|¨)dinger equationswith the effects of group-velocity dispersion,self-phase modulation and cross-phase modulation,which hav...With the aid of computation,we consider the variable-coefficient coupied nonlinear Schr(o|¨)dinger equationswith the effects of group-velocity dispersion,self-phase modulation and cross-phase modulation,which have potentialapplications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers.Based on theobtained nonisospectral linear eigenvalue problems(i.e.Lax pair),we construct the Darboux transformation for such amodel to derive the optical soliton solutions.In addition,through the one-and two-soliton-like solutions,we graphicallydiscuss the features of picosecond solitons in inhomogeneous optical fibers.展开更多
Investigated in this paper is the generalized nonlinear Schrdinger equation with radial symmetry.Withthe help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear metho...Investigated in this paper is the generalized nonlinear Schrdinger equation with radial symmetry.Withthe help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear method.Bcklund transformation in the bilinear form is presented, through which a new solution is constructed.Graphically, wehave found that the solitons are symmetric about x=0, while the soliton pulse width and amplitude will change alongwith the distance and time during the propagation.展开更多
A bilinear Bcklund transformation is presented for the three coupled higher-order nonlinear Schrdingerequations with the inclusion of the group velocity dispersion,third-order dispersion and Kerr-law nonlinearity,...A bilinear Bcklund transformation is presented for the three coupled higher-order nonlinear Schrdingerequations with the inclusion of the group velocity dispersion,third-order dispersion and Kerr-law nonlinearity,whichcan describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses inwavelength-division multiplexed system.Starting from the Bcklund transformation,the analytical soliton solution isobtained from a trivial solution.Simultaneously,the N-soliton-like solution in double Wronskian form is constructed,and the corresponding proof is also given via the Wronskian technique.The results obtained from this paper might bevaluable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communicationsystems.展开更多
In this paper, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.
基金supported financially by FundamentalResearch Program of Shanxi Province(No.202103021223056).
文摘Addressing the challenges posed by the nonlinear and non-stationary vibrations in rotating machinery,where weak fault characteristic signals hinder accurate fault state representation,we propose a novel feature extraction method that combines the Flexible Analytic Wavelet Transform(FAWT)with Nonlinear Quantum Permutation Entropy.FAWT,leveraging fractional orders and arbitrary scaling and translation factors,exhibits superior translational invariance and adjustable fundamental oscillatory characteristics.This flexibility enables FAWT to provide well-suited wavelet shapes,effectively matching subtle fault components and avoiding performance degradation associated with fixed frequency partitioning and low-oscillation bases in detecting weak faults.In our approach,gearbox vibration signals undergo FAWT to obtain sub-bands.Quantum theory is then introduced into permutation entropy to propose Nonlinear Quantum Permutation Entropy,a feature that more accurately characterizes the operational state of vibration simulation signals.The nonlinear quantum permutation entropy extracted from sub-bands is utilized to characterize the operating state of rotating machinery.A comprehensive analysis of vibration signals from rolling bearings and gearboxes validates the feasibility of the proposed method.Comparative assessments with parameters derived from traditional permutation entropy,sample entropy,wavelet transform(WT),and empirical mode decomposition(EMD)underscore the superior effectiveness of this approach in fault detection and classification for rotating machinery.
文摘A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.
基金The project partly supported by the Foundation of Zhongshan University Advanced Research Center
文摘In this paper,a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems.This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system.
文摘A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1< kh <1 5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.
基金Natural Science Foundation of Gansu Province of China
文摘With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.
文摘In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.
基金supported by the Natural Science Foundation of Liaoning Province,China(Grant No.201602678).
文摘The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.
基金supported by the National Natural Science Foundation of China(12071491,12001113)。
文摘In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.
基金Supported by National Natural Science Foundation of China (61135001, 61075029, 61074179, 61074155) and the Postdoctoral Science Foundation of China (20110491692)
文摘In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.
文摘Applying the method of canonical,transformation,we studied the nonlinear wave equationφ_(xx)-φ_(tt)=dF(φ)/dφ.A Bäcklund transformation(BT)of this equation is obtained.Combining BT of this equation with the well-known BT of sine-Gordon equation(SGE),a formula of nonlinear superposition which contains some arbitrary functions of x and t has been established.This formula leads to a kind of general solution of SGE through its some known solutions.The general soliton solutions and their properties are simply discussed.
文摘This article describes transformation of nonlinear surface gravity waves under shallow-water conditions with the aid of the suggested semigraphical method. There are given profiles of surface gravity waves on the crests steepening stages, their leading edges steepening. There are discussed the spectral component influence on the transformation of surface wave profile.
文摘This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.
基金supported by the Shanghai Leading Academic Discipline Project under Grant No.XTKX2012by the Natural Science Foundation of Shanghai under Grant No.12ZR1446800,Science and Technology Commission of Shanghai municipalityby the National Natural Science Foundation of China under Grant Nos.11201302 and11171220.
文摘In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.
文摘The peculiarities of energy dissipation transferred by solitary waves on defects such as freesurface, grain boundary, region with high concentration of vacancies are studied. One of theways of description of the long range effect taking place at ion implantation in metallic materialsis suggested.
文摘In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023 the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.BUAA-SKLSDE-09KF-04+2 种基金Beijing University of Aeronautics and Astronautics,by the National Basic Research Program of China (973 Program) under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos.20060006024 and 200800130006Chinese Ministry of Education
文摘With the aid of computation,we consider the variable-coefficient coupied nonlinear Schr(o|¨)dinger equationswith the effects of group-velocity dispersion,self-phase modulation and cross-phase modulation,which have potentialapplications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers.Based on theobtained nonisospectral linear eigenvalue problems(i.e.Lax pair),we construct the Darboux transformation for such amodel to derive the optical soliton solutions.In addition,through the one-and two-soliton-like solutions,we graphicallydiscuss the features of picosecond solitons in inhomogeneous optical fibers.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund (No.BUAASKLSDE-09KF-04)+2 种基金Supported Project (No.SKLSDE-2010ZX-07) of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China (973 Program) under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘Investigated in this paper is the generalized nonlinear Schrdinger equation with radial symmetry.Withthe help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear method.Bcklund transformation in the bilinear form is presented, through which a new solution is constructed.Graphically, wehave found that the solitons are symmetric about x=0, while the soliton pulse width and amplitude will change alongwith the distance and time during the propagation.
基金the Key Project of the Ministry of Education under Grant No.106033the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024National Natural Science Foundation of China under Grant No.60372095
文摘A bilinear Bcklund transformation is presented for the three coupled higher-order nonlinear Schrdingerequations with the inclusion of the group velocity dispersion,third-order dispersion and Kerr-law nonlinearity,whichcan describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses inwavelength-division multiplexed system.Starting from the Bcklund transformation,the analytical soliton solution isobtained from a trivial solution.Simultaneously,the N-soliton-like solution in double Wronskian form is constructed,and the corresponding proof is also given via the Wronskian technique.The results obtained from this paper might bevaluable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communicationsystems.
文摘In this paper, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.