This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak so...This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak solutions can be get based on Galerkin approximation. With the linear operator B constructed by BOGOVSKII, we show that the density ■is square integrable up to the boundary.展开更多
Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fer...Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.展开更多
By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distin...By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.展开更多
We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first resul...We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.展开更多
A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equatio...A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equation, a singularity structure analysis for this system is carried out.Through a detailed analysis in two cases by using Kruskal's simplified method, the sIto system is found to pass the Painleve test, and thus is Painlevé integrable.展开更多
This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied (TsT) transformations on S^3 and AdS3, respectively. It finds that the U(...This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied (TsT) transformations on S^3 and AdS3, respectively. It finds that the U(1) currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and ADS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.展开更多
This paper is concerned with the generalized variable-coefficient nonlinear evolution equation(vc-NLEE).The complete integrability classification is presented,and the integrable conditions for the generalized variab...This paper is concerned with the generalized variable-coefficient nonlinear evolution equation(vc-NLEE).The complete integrability classification is presented,and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlevé analysis.Then,the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method,and the Lax pairs(LP) of the vc-NLEEs are constructed in terms of the integrable conditions.展开更多
Very recently, Yu, Le and Zhou introduced the so called △B1^* and △B2^* conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the...Very recently, Yu, Le and Zhou introduced the so called △B1^* and △B2^* conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.展开更多
The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli (mKdV-BLMP) system is investigated using the Painlevé analysis approach. It is shown that this coupled system ...The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli (mKdV-BLMP) system is investigated using the Painlevé analysis approach. It is shown that this coupled system possesses the Painlevé property in both the principal and secondary branches. Then, the consistent Riccati expansion (CRE) method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally, starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.展开更多
The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e prope...The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.展开更多
We provide necessary conditions in order that the Hamiltonian systems with Hamiltonian ,?and one of the following potentials ?are integrable in the Liouville sense.
The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system an...The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.展开更多
In this paper, integrability conditions and an integrating algorithm of fully rheonomous affine constraints (FRACs) for the partially integrable case are studied. First, some preliminaries on the FRACs are illustrated...In this paper, integrability conditions and an integrating algorithm of fully rheonomous affine constraints (FRACs) for the partially integrable case are studied. First, some preliminaries on the FRACs are illustrated. Next, necessary and sufficient conditions on the partially integrable case for the FRACs are derived. Then, an integrating algorithm to calculate independent first integrals of the FRACs for the partially integrable case is derived. Moreover, the existence of an inverse function utilized in the algorithm is proven. After that, an example is presented for evaluation of the effectiveness of the proposed method. As a result, it turns out that the proposed integrating algorithm can easily calculate independent first integrals for given partially integrable FRACs, and thus this new algorithm is expected to be applied to various research fields.展开更多
This paper presents a complete integrability condition for fully rheonomous affine constraints in terms of the rheonomous bracket. We first define fully rheonomous affine constraints and develop geometric representati...This paper presents a complete integrability condition for fully rheonomous affine constraints in terms of the rheonomous bracket. We first define fully rheonomous affine constraints and develop geometric representation for them. Next, the rheonomous bracket is explained and some properties of it are derived. We then investigate a necessary and sufficient condition on complete integrability for the fully rheonomous affine constraints based on the rheonomous bracket as an extension of Frobenius’ theorem. The effectiveness and the availability of the new results are also evaluated via an example.展开更多
There exists an Ehresmann connection on the fibred constrained sub-manifold defined by Pfaffian differential constraints. It is proved that curvature of the connection is closely related to the d-delta commutation rel...There exists an Ehresmann connection on the fibred constrained sub-manifold defined by Pfaffian differential constraints. It is proved that curvature of the connection is closely related to the d-delta commutation relation in the classical nonholonomic mechanics. It is also proved that conditions of complete integrability for Pfaffian systems in Frobenius sense are equivalent to the three requirements upon the conditional variations in the classical calculus of variations: (1) the variations belong to the constrained manifold, (2) variational operators commute with differential operators, (3) variations satisfy the Chetaev's conditions. Thus this theory verifies the conjecture or experience of researchers of mechanics on the integrability conditions in terms of variation calculus.展开更多
This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on t...This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness,the authors prove that such operators are bounded on the mixed radial-angular spaces.Meanwhile,corresponding vector-valued versions are also obtained.展开更多
In this paper, we first discuss the relationship between the McShane integral and Pettis integral for vector-valued functions. Then by using the embedding theorems for the fuzzy number space E^1, we give a new equival...In this paper, we first discuss the relationship between the McShane integral and Pettis integral for vector-valued functions. Then by using the embedding theorems for the fuzzy number space E^1, we give a new equivalent condition for (K) integrabihty of a fuzzy set-valued mapping F : [a, b] → E^1.展开更多
We present the multi-component Hunter-Saxton andμ-Camassa-Holm systems.It is shown that the multicomponent Camassa-Holm,Hunter-Saxton andμ-Camassa-Holm systems are geometrically integrable,namely they describe pseud...We present the multi-component Hunter-Saxton andμ-Camassa-Holm systems.It is shown that the multicomponent Camassa-Holm,Hunter-Saxton andμ-Camassa-Holm systems are geometrically integrable,namely they describe pseudo-spherical surfaces.As a consequence,their infinite number of conservation laws can be directly constructed.For the three-component Camassa-Holm and Hunter-Saxton systems,their nonlocal symmetries depending on the pseudo-potentials are obtained.展开更多
By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerica...By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.展开更多
This paper is concerned with partial Bochner integrals with emphasis on a geometrical study of partial Bochner integrability. Convex sets involving the range of L-1-bounded functions are constructed. These constructio...This paper is concerned with partial Bochner integrals with emphasis on a geometrical study of partial Bochner integrability. Convex sets involving the range of L-1-bounded functions are constructed. These constructions provide a complete characterization of partial Bochner integrable functions.展开更多
基金supported by the National Natural Science Foundation of China(11271305,11531010)
文摘This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak solutions can be get based on Galerkin approximation. With the linear operator B constructed by BOGOVSKII, we show that the density ■is square integrable up to the boundary.
文摘Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11201290 and 71103118)
文摘By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.
文摘We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975156 and 11775146)the Natural Science Foundation of Zhejiang Province,China(Grant No.LY18A050001)
文摘A supersymmetric version of the Ito equation is proposed by extending the independent and dependent variables for the classic Ito equation.To investigate the integrability of the N = 1 supersymmetric Ito(sIto) equation, a singularity structure analysis for this system is carried out.Through a detailed analysis in two cases by using Kruskal's simplified method, the sIto system is found to pass the Painleve test, and thus is Painlevé integrable.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 90403019 and 10575080)
文摘This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied (TsT) transformations on S^3 and AdS3, respectively. It finds that the U(1) currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and ADS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.
基金Project supported by the National Natural Science Foundation of China(Grant No.11171041)the High-Level Personnel Foundation of Liaocheng University(Grant No.31805)
文摘This paper is concerned with the generalized variable-coefficient nonlinear evolution equation(vc-NLEE).The complete integrability classification is presented,and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlevé analysis.Then,the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method,and the Lax pairs(LP) of the vc-NLEEs are constructed in terms of the integrable conditions.
基金Supported by National Natural Science Foundation of China (10901044)
文摘Very recently, Yu, Le and Zhou introduced the so called △B1^* and △B2^* conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.
基金Supported by the Natural Science Foundation of Zhejiang Province of China under Grant No LY14A010005
文摘The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli (mKdV-BLMP) system is investigated using the Painlevé analysis approach. It is shown that this coupled system possesses the Painlevé property in both the principal and secondary branches. Then, the consistent Riccati expansion (CRE) method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally, starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.
基金The NSF (11371166,11301210)of ChinaNational 973 Project (2012CB821200) of ChinaJilin Province Youth Science Foundation
文摘The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.
文摘We provide necessary conditions in order that the Hamiltonian systems with Hamiltonian ,?and one of the following potentials ?are integrable in the Liouville sense.
文摘The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.
文摘In this paper, integrability conditions and an integrating algorithm of fully rheonomous affine constraints (FRACs) for the partially integrable case are studied. First, some preliminaries on the FRACs are illustrated. Next, necessary and sufficient conditions on the partially integrable case for the FRACs are derived. Then, an integrating algorithm to calculate independent first integrals of the FRACs for the partially integrable case is derived. Moreover, the existence of an inverse function utilized in the algorithm is proven. After that, an example is presented for evaluation of the effectiveness of the proposed method. As a result, it turns out that the proposed integrating algorithm can easily calculate independent first integrals for given partially integrable FRACs, and thus this new algorithm is expected to be applied to various research fields.
文摘This paper presents a complete integrability condition for fully rheonomous affine constraints in terms of the rheonomous bracket. We first define fully rheonomous affine constraints and develop geometric representation for them. Next, the rheonomous bracket is explained and some properties of it are derived. We then investigate a necessary and sufficient condition on complete integrability for the fully rheonomous affine constraints based on the rheonomous bracket as an extension of Frobenius’ theorem. The effectiveness and the availability of the new results are also evaluated via an example.
文摘There exists an Ehresmann connection on the fibred constrained sub-manifold defined by Pfaffian differential constraints. It is proved that curvature of the connection is closely related to the d-delta commutation relation in the classical nonholonomic mechanics. It is also proved that conditions of complete integrability for Pfaffian systems in Frobenius sense are equivalent to the three requirements upon the conditional variations in the classical calculus of variations: (1) the variations belong to the constrained manifold, (2) variational operators commute with differential operators, (3) variations satisfy the Chetaev's conditions. Thus this theory verifies the conjecture or experience of researchers of mechanics on the integrability conditions in terms of variation calculus.
基金supported by the NSFC(11771358,11701333,11871101)。
文摘This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along"polynomial curves".Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness,the authors prove that such operators are bounded on the mixed radial-angular spaces.Meanwhile,corresponding vector-valued versions are also obtained.
文摘In this paper, we first discuss the relationship between the McShane integral and Pettis integral for vector-valued functions. Then by using the embedding theorems for the fuzzy number space E^1, we give a new equivalent condition for (K) integrabihty of a fuzzy set-valued mapping F : [a, b] → E^1.
基金Supported by the National Natural Science Foundation of China for Distinguished Young Scholars(No 10925104).
文摘We present the multi-component Hunter-Saxton andμ-Camassa-Holm systems.It is shown that the multicomponent Camassa-Holm,Hunter-Saxton andμ-Camassa-Holm systems are geometrically integrable,namely they describe pseudo-spherical surfaces.As a consequence,their infinite number of conservation laws can be directly constructed.For the three-component Camassa-Holm and Hunter-Saxton systems,their nonlocal symmetries depending on the pseudo-potentials are obtained.
文摘By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.
文摘This paper is concerned with partial Bochner integrals with emphasis on a geometrical study of partial Bochner integrability. Convex sets involving the range of L-1-bounded functions are constructed. These constructions provide a complete characterization of partial Bochner integrable functions.