With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various...With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.展开更多
Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rat...Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.展开更多
In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions ...In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.展开更多
The new multiple （G＇/G）-expansion method is proposed in this paper to seek the exact double traveling wave solutions of nonlinear partial differential equations. With the aid of symbolic computation, this new metho...The new multiple （G＇/G）-expansion method is proposed in this paper to seek the exact double traveling wave solutions of nonlinear partial differential equations. With the aid of symbolic computation, this new method is applied to construct double traveling wave solutions of the coupled nonlinear Klein-Gordon equations and the coupled SchrSdinger-Boussinesq equation. As a result, abundant double traveling wave solutions including double hyperbolic tangent function solutions, double tangent function solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via this new method. The new multiple ＇ （G＇/G-expanslon method not only gets new exact solutions of equations directly and effectively, but also expands the scope of the solution. This new method has a very wide range of application for the study of nonlinear partial differential equations.展开更多
Four Cu2＋ complexes of salicylidene-amino acid Schiff base with 1,10-phenanthroline （Phen） or 2,2＇- bipyridine （Bipy） were successfully intercalated in interlayer galleries of Mg/AI-NO3-1ayered double hydroxide ...Four Cu2＋ complexes of salicylidene-amino acid Schiff base with 1,10-phenanthroline （Phen） or 2,2＇- bipyridine （Bipy） were successfully intercalated in interlayer galleries of Mg/AI-NO3-1ayered double hydroxide （LDH） by the swelling-restored method. The hybrids were characterized by elemental analysis, X-ray diffraction, FT-IR spectra, UV-vis DRS, TG-DTA and SEM observation. Good protection of the complexes by LDH in neutral and weak acidic solutions was revealed by UV spectra, cyclic voltammograms and luminescence spectra.展开更多
The nonisospectral effectλ_t=α(t)λsatisfied by spectral parameterλopens up a new scheme for constructing localized waves to some nonlinear partial differential equations.In this paper,we perform this effect on a c...The nonisospectral effectλ_t=α(t)λsatisfied by spectral parameterλopens up a new scheme for constructing localized waves to some nonlinear partial differential equations.In this paper,we perform this effect on a complex nonisospectral nonpotential sine-Gordon equation by the bilinearization reduction method.From an integrable nonisospectral Ablowitz–Kaup–Newell–Segur equation,we construct some exact solutions in double Wronskian form to the reduced complex nonisospectral nonpotential sine-Gordon equation.These solutions,including soliton solutions,Jordan-block solutions and interaction solutions,exhibit localized structure,whose dynamics are analyzed with graphical illustration.The research ideas and methods in this paper can be generalized to other negative order nonisospectral integrable systems.展开更多
With the help of skew-symmetric differential forms, the hidden properties of the mathematical physics equations that describe discrete quantum transitions and emergence the physical structures are investigated. It is ...With the help of skew-symmetric differential forms, the hidden properties of the mathematical physics equations that describe discrete quantum transitions and emergence the physical structures are investigated. It is shown that the mathematical physics equations possess a unique property. They can describe discrete quantum transitions, emergence of physical structures and occurrence observed formations. However, such a property possesses only equations on which no additional conditions, namely, the conditions of integrability, are imposed. The intergrability conditions are realized from the equations themselves. Just under realization of integrability conditions double solutions to the mathematical physics equations, which describe discrete transitions and so on, are obtained. The peculiarity consists in the fact that the integrability conditions do not directly follow from the mathematical physics equations;they are realized under the description of evolutionary process. The hidden properties of differential equations were discovered when studying the integrability of differential equations of mathematical physics that depends on the consistence between the derivatives in differential equations along different directions and on the consistence of equations in the set of equations. The results of this work were obtained with the help of skew-symmetric differential forms that possess a nontraditional mathematical apparatus such as nonidentical relations, degenerate transformations and the transition from nonintegrable manifolds to integrable structures. Such results show that mathematical physics equations can describe quantum processes.展开更多
In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find mult...In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find multiple soliton solutions of nonlinear partial differential equations, this approach is constructive and pure algebraic. The results found here are tested on computer and therefore their validity is ensured.展开更多
Effect of hot and cold dust charge on the propagation of dust-acoustic waves （DAWs） in unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains has been investigated. The reductive perturb...Effect of hot and cold dust charge on the propagation of dust-acoustic waves （DAWs） in unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains has been investigated. The reductive perturbation method is employed to reduce the basic set of fluid equations to the Kortewege-de Vries （KdV） equation. At the critical hot dusty plasma density Nho, the KdV equation is not appropriate for describing the system. Hence, a set of stretched coordinates is considered to derive the modified KdV equation. It is found that the presence of hot and cold dust charge grains not only significantly modifies the basic properties of solitary structure, but also changes the polarity of the solitary profiles. In the vicinity of the critical hot dusty plasma density Nho, neither KdV nor mKdV equation is appropriate for describing the DAWs. Therefore, a further modified KdV （fmKdV） equation is derived, which admits both soliton and double layer solutions.展开更多
文摘With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.
基金National Natural Science Foundation of China under Grant No.10371070the Special Found for Major Specialities of Shanghai Education CommitteeChina Postdoctoral Science Foundation
文摘Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.
文摘In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11202106、61201444)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20123228120005)+4 种基金the Jiangsu Information and Communication Engineering Preponderant Discipline Platformthe Natural Science Foundation of Jiangsu Province(Grant No.BK20131005)the Jiangsu Qing Lan Projectthe Natural Sciences Fundation of the Universities of Jiangsu Province(Grant No.13KJB170016)the Fundamental Research Funds for the Southeast University(Grant No.CDLS-2016-03)
文摘The new multiple （G＇/G）-expansion method is proposed in this paper to seek the exact double traveling wave solutions of nonlinear partial differential equations. With the aid of symbolic computation, this new method is applied to construct double traveling wave solutions of the coupled nonlinear Klein-Gordon equations and the coupled SchrSdinger-Boussinesq equation. As a result, abundant double traveling wave solutions including double hyperbolic tangent function solutions, double tangent function solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via this new method. The new multiple ＇ （G＇/G-expanslon method not only gets new exact solutions of equations directly and effectively, but also expands the scope of the solution. This new method has a very wide range of application for the study of nonlinear partial differential equations.
基金supported by Beijing Municipal Natural Science Foundation(No.2112022)Key Laboratory of Radiopharmaceuticals of Ministry of Education(College of Chemistry, Beijing Normal University) and Analytical and Testing Center of Beijing Normal University
文摘Four Cu2＋ complexes of salicylidene-amino acid Schiff base with 1,10-phenanthroline （Phen） or 2,2＇- bipyridine （Bipy） were successfully intercalated in interlayer galleries of Mg/AI-NO3-1ayered double hydroxide （LDH） by the swelling-restored method. The hybrids were characterized by elemental analysis, X-ray diffraction, FT-IR spectra, UV-vis DRS, TG-DTA and SEM observation. Good protection of the complexes by LDH in neutral and weak acidic solutions was revealed by UV spectra, cyclic voltammograms and luminescence spectra.
基金supported by the National Natural Science Foundation of China(Grant No.12071432)Zhejiang Provincial Natural Science Foundation(Grant No.LZ24A010007)。
文摘The nonisospectral effectλ_t=α(t)λsatisfied by spectral parameterλopens up a new scheme for constructing localized waves to some nonlinear partial differential equations.In this paper,we perform this effect on a complex nonisospectral nonpotential sine-Gordon equation by the bilinearization reduction method.From an integrable nonisospectral Ablowitz–Kaup–Newell–Segur equation,we construct some exact solutions in double Wronskian form to the reduced complex nonisospectral nonpotential sine-Gordon equation.These solutions,including soliton solutions,Jordan-block solutions and interaction solutions,exhibit localized structure,whose dynamics are analyzed with graphical illustration.The research ideas and methods in this paper can be generalized to other negative order nonisospectral integrable systems.
文摘With the help of skew-symmetric differential forms, the hidden properties of the mathematical physics equations that describe discrete quantum transitions and emergence the physical structures are investigated. It is shown that the mathematical physics equations possess a unique property. They can describe discrete quantum transitions, emergence of physical structures and occurrence observed formations. However, such a property possesses only equations on which no additional conditions, namely, the conditions of integrability, are imposed. The intergrability conditions are realized from the equations themselves. Just under realization of integrability conditions double solutions to the mathematical physics equations, which describe discrete transitions and so on, are obtained. The peculiarity consists in the fact that the integrability conditions do not directly follow from the mathematical physics equations;they are realized under the description of evolutionary process. The hidden properties of differential equations were discovered when studying the integrability of differential equations of mathematical physics that depends on the consistence between the derivatives in differential equations along different directions and on the consistence of equations in the set of equations. The results of this work were obtained with the help of skew-symmetric differential forms that possess a nontraditional mathematical apparatus such as nonidentical relations, degenerate transformations and the transition from nonintegrable manifolds to integrable structures. Such results show that mathematical physics equations can describe quantum processes.
基金The project supported by '973' Project under Grant No.2004CB318000Doctor Start-up Foundation of Liaoning Province under Grant No.1040225Science and Technology Research Project of Liaoning Education Bureau
文摘In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find multiple soliton solutions of nonlinear partial differential equations, this approach is constructive and pure algebraic. The results found here are tested on computer and therefore their validity is ensured.
文摘Effect of hot and cold dust charge on the propagation of dust-acoustic waves （DAWs） in unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains has been investigated. The reductive perturbation method is employed to reduce the basic set of fluid equations to the Kortewege-de Vries （KdV） equation. At the critical hot dusty plasma density Nho, the KdV equation is not appropriate for describing the system. Hence, a set of stretched coordinates is considered to derive the modified KdV equation. It is found that the presence of hot and cold dust charge grains not only significantly modifies the basic properties of solitary structure, but also changes the polarity of the solitary profiles. In the vicinity of the critical hot dusty plasma density Nho, neither KdV nor mKdV equation is appropriate for describing the DAWs. Therefore, a further modified KdV （fmKdV） equation is derived, which admits both soliton and double layer solutions.
