In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a ...In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.展开更多
In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients a...In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.展开更多
The elementary and systematic binary Bell polynomials method is applied to the generalized NizhnikNovikov-Veselov (GNNV) equation.The bilinear representation,bilinear B&cklund transformation,Lax pair and infinitec...The elementary and systematic binary Bell polynomials method is applied to the generalized NizhnikNovikov-Veselov (GNNV) equation.The bilinear representation,bilinear B&cklund transformation,Lax pair and infiniteconservation laws of the GNNV equation are obtained directly,without too much trick like Hirota’s bilinear method.展开更多
Binary Bell Polynomials play an important role in the characterization of bilinear equation. The bilinear form, bilinear B?cklund transformation and Lax pairs for the modified Kadomtsev-Petviashvili equation are deriv...Binary Bell Polynomials play an important role in the characterization of bilinear equation. The bilinear form, bilinear B?cklund transformation and Lax pairs for the modified Kadomtsev-Petviashvili equation are derived from the Binary Bell Polynomials.展开更多
We investigate the extended （2＋ 1）-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax ...We investigate the extended （2＋ 1）-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund 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.展开更多
Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation i...Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.展开更多
Three-dimensional Information Decoupling System Based on PSD were designed based on LabVIEW, in order to achieve precision, timeliness, reliability require-ments of the PSD used in the ATP system of Satellite Earth qu...Three-dimensional Information Decoupling System Based on PSD were designed based on LabVIEW, in order to achieve precision, timeliness, reliability require-ments of the PSD used in the ATP system of Satellite Earth quantum communication. Firstly, the laser light source was driven by a stepper motor to scan on the PSD photosensitive surface, and the voltage value was collected and calculated to get the spot position. Analyzing the cause of nonlinear, a mathematical model was built between the actual value and the measured value by using binary quadratic polynomial method, PSD nonlinear correction function would be got. Then, the object micro displacement and angle offset were measured by combining optical triangulation method, and the error of the measurement results was corrected. Experimental results showed that, after the correction, the measuring deviation could be significantly reduced, the PSD performance calibration requirements was achieved, the efficiency of the system was developed greatly by using LabVIEW.展开更多
In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bel...In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.展开更多
In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the ...In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Backlund transformations are derived.展开更多
With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method ...With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,61021004,10735030Shanghai Leading Academic Discipline Project under Grant No.B412Program for Changjiang Scholars and Innovative Research Team in University(IRT0734)
文摘In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.
基金supported by the National Natural Science Foundation of China(Grant No.10831003)the Natural Science Foundation of Zhejiang Province,China(Grant Nos.Y6100791 and R6090109)
文摘In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10735030,11075055,61021004,90718041,Shanghai Leading Academic Discipline Project (No. B412)Program for Changjiang Scholars and Innovative Research Team in University (IRT0734)
文摘The elementary and systematic binary Bell polynomials method is applied to the generalized NizhnikNovikov-Veselov (GNNV) equation.The bilinear representation,bilinear B&cklund transformation,Lax pair and infiniteconservation laws of the GNNV equation are obtained directly,without too much trick like Hirota’s bilinear method.
基金supported by the National Natural Science Foundation of China(11301183).
文摘Binary Bell Polynomials play an important role in the characterization of bilinear equation. The bilinear form, bilinear B?cklund transformation and Lax pairs for the modified Kadomtsev-Petviashvili equation are derived from the Binary Bell Polynomials.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11075055 and 11275072)the Innovative Research Team Program of the National Natural Science Foundation of China (Grant No. 61021004)+1 种基金the Shanghai Knowledge Service Platform for Trustworthy Internet of Things, China(Grant No. ZF1213)the National High Technology Research and Development Program of China (Grant No. 2011AA010101)
文摘We investigate the extended （2＋ 1）-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund 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.
基金supported by the National Nature Science Foundation of China under Grant No.11871116Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11。
文摘Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.
文摘Three-dimensional Information Decoupling System Based on PSD were designed based on LabVIEW, in order to achieve precision, timeliness, reliability require-ments of the PSD used in the ATP system of Satellite Earth quantum communication. Firstly, the laser light source was driven by a stepper motor to scan on the PSD photosensitive surface, and the voltage value was collected and calculated to get the spot position. Analyzing the cause of nonlinear, a mathematical model was built between the actual value and the measured value by using binary quadratic polynomial method, PSD nonlinear correction function would be got. Then, the object micro displacement and angle offset were measured by combining optical triangulation method, and the error of the measurement results was corrected. Experimental results showed that, after the correction, the measuring deviation could be significantly reduced, the PSD performance calibration requirements was achieved, the efficiency of the system was developed greatly by using LabVIEW.
基金Supported by the National Natural Science Foundation of China under Grant No. 60772023the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02+1 种基金the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications)the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 200800130006, Chinese Ministry of Education
文摘In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10735030 and 11075055Innovative Research Team Program of the National Natural Science Foundation of China under Grant No. 61021004
文摘In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Backlund transformations are derived.
基金Supported by Shandong Provincial Key Laboratory of Marine Ecology and Environment&Disaster Prevention and Mitigation project under Grant No.2012010National Natural Science Foundation of China under Grant No.11271007+1 种基金Special Funds for Theoretical Physics of the National Natural Science Foundation of China under Grant No.11447205Shandong University of Science and Technology Research Fund under Grant No.2012KYTD105
文摘With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.
