By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （...By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （2＋1）-dimensional breaking soliton system is derived. Based on the derived solitary wave excitation, we obtain some special annihilation solitons and chaotic solitons in this short note.展开更多
A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three c...A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.展开更多
Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broe...Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broer-Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.展开更多
We study the localized coherent structures ofa generally nonintegrable (2+ 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads ...We study the localized coherent structures ofa generally nonintegrable (2+ 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.展开更多
The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryf...The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.展开更多
We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equ...We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.展开更多
Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system...Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction_vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed.It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint_Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint_Venant theorem.The symplectic solution method is not the same as the classical semi_inverse method and breaks through the limit of the traditional semi_inverse solution.The symplectic solution method will have vast application.展开更多
By means of the multilinear variable separation(MLVS) approach, new interaction solutions with low-dimensional arbitrary functions of the(2+1)-dimensional Nizhnik–Novikov–Veselovtype system are constructed. Four-dro...By means of the multilinear variable separation(MLVS) approach, new interaction solutions with low-dimensional arbitrary functions of the(2+1)-dimensional Nizhnik–Novikov–Veselovtype system are constructed. Four-dromion structure, ring-parabolic soliton structure and corresponding fusion phenomena for the physical quantity U =λ(lnf)_(xy) are revealed for the first time. This MLVS approach can also be used to deal with the(2+1)-dimensional Sasa–Satsuma system.展开更多
This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the ...This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.展开更多
This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs.If the nonlinear problem involves uncertainty,we need to characterize the unce...This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs.If the nonlinear problem involves uncertainty,we need to characterize the uncer-tainty by using a few random variables.The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification.The computational cost is very expensive.To overcome the difficulty,a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in terms of stochastic basis functions and deterministic basis functions.No it-eration is performed at each enrichment step.These basis functions are model-oriented and involve offline computation.To efficiently identify the stochastic basis functions,we utilize the greedy algorithm to select some optimal sam-ples.Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples,the solutions of which are used to train the deterministic basis functions.With the deterministic basis functions,we can obtain the corresponding stochastic basis functions by solv-ing linear differential systems.The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase.This is very desirable for scientific computation.Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differential equations.展开更多
基金The project supported by the Natural Science Foundation of Zhejiang Province under Grant No. Y604106, the Foundation of New Century 151 Talent Engineering of Zhejiang Province, and the Natural Science Foundation of Zhejiang Lishui University under Grant No. KZ05010 Acknowledgments The authors would like to thank professor Chun-Long Zheng for his fruitful and helpful suggestions.
文摘By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （2＋1）-dimensional breaking soliton system is derived. Based on the derived solitary wave excitation, we obtain some special annihilation solitons and chaotic solitons in this short note.
基金supported by the National Natural Science Foundation of China (10772014)
文摘A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.
文摘Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broer-Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.
文摘We study the localized coherent structures ofa generally nonintegrable (2+ 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.
基金The project supported by National Natural Science Foundation of China
文摘The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.
基金Supported by the National Natural Science Foundation of China under Grant No 10447007, and the Natural Science Foundation of Shaanxi Province under Grant No 2005A13.
文摘We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.
文摘Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction_vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed.It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint_Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint_Venant theorem.The symplectic solution method is not the same as the classical semi_inverse method and breaks through the limit of the traditional semi_inverse solution.The symplectic solution method will have vast application.
基金supported by the National Natural Science Foundation of China (11771395)。
文摘By means of the multilinear variable separation(MLVS) approach, new interaction solutions with low-dimensional arbitrary functions of the(2+1)-dimensional Nizhnik–Novikov–Veselovtype system are constructed. Four-dromion structure, ring-parabolic soliton structure and corresponding fusion phenomena for the physical quantity U =λ(lnf)_(xy) are revealed for the first time. This MLVS approach can also be used to deal with the(2+1)-dimensional Sasa–Satsuma system.
基金The project supported by National Natural Science Foundation of China under Grant No. 10447007 and the Natural Science Foundation of Shaanxi Province of China under Grant No. 2005A13
文摘This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.
基金supported by the National Natural Science Foundation of China (Grant No.12101217)by the China Postdoctoral Science Foundation (Grant No.2022M713875)by the Natural Science Foundation of Hunan Province (Grant No.2022J40113).
文摘This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs.If the nonlinear problem involves uncertainty,we need to characterize the uncer-tainty by using a few random variables.The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification.The computational cost is very expensive.To overcome the difficulty,a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in terms of stochastic basis functions and deterministic basis functions.No it-eration is performed at each enrichment step.These basis functions are model-oriented and involve offline computation.To efficiently identify the stochastic basis functions,we utilize the greedy algorithm to select some optimal sam-ples.Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples,the solutions of which are used to train the deterministic basis functions.With the deterministic basis functions,we can obtain the corresponding stochastic basis functions by solv-ing linear differential systems.The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase.This is very desirable for scientific computation.Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differential equations.
