In 2001 three earthquakes occurred in Shidian in Yunnan Province, which were the MS=5.2 on April 10, the MS=5.9 on April 12 and the MS=5.3 on June 8. Based on the data from the station Baoshan of Yunnan Telemetry Digi...In 2001 three earthquakes occurred in Shidian in Yunnan Province, which were the MS=5.2 on April 10, the MS=5.9 on April 12 and the MS=5.3 on June 8. Based on the data from the station Baoshan of Yunnan Telemetry Digital Seismograph Network, the variational characteristics of shear-wave splitting on these series of strong earthquakes has been studied by using the systematic analysis method (SAM) of shear-wave splitting. The result shows the time delays of shear-wave splitting basically increase with earthquake activity intensifying. However the time delays abruptly decrease immediately before strong aftershocks. It accords with the stress relaxation before earthquakes, which was found recently in study on shear-wave splitting. The result suggests it is significant for reducing the harm degree of earthquakes to develop the stress-forecasting on earthquake in strong active tectonic zones and economic developed regions or big cities under the danger of strong earthquakes.展开更多
In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2+1)-dimensional variable coeffficient canonical generalized KP (VCCGKP) equation...In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2+1)-dimensional variable coeffficient canonical generalized KP (VCCGKP) equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.展开更多
Painleve property and infinite symmetries of the (2+1)-dimensional higher-order Broer-Kaup (HBK) system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.u...Painleve property and infinite symmetries of the (2+1)-dimensional higher-order Broer-Kaup (HBK) system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to (2+1)-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the (2+1)-dimensional HBK system.展开更多
By applying a direct symmetry method, we get the symmetry of the asymmetric Nizhnik-Novikov-Veselov equation (ANNV). Taking the special case, we have a finite-dimensional symmetry. By using the equivalent vector of ...By applying a direct symmetry method, we get the symmetry of the asymmetric Nizhnik-Novikov-Veselov equation (ANNV). Taking the special case, we have a finite-dimensional symmetry. By using the equivalent vector of the symmetry, we construct an eight-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, we reduce the ANNV equation and obtain some solutions to the reduced equations. Furthermore, we find some new explicit solutions of the ANNV equation. At last, we give the conservation laws of the ANNV equation.展开更多
In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, ...In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the (2+1)-dimensional Lax pair is reduced to (1+1)-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.展开更多
In this paper, the generalized ranch function method is extended to (2+1)-dimensianal canonical generalized KP (CGKP) equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit ...In this paper, the generalized ranch function method is extended to (2+1)-dimensianal canonical generalized KP (CGKP) equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the (2+1)-dimensional OGKP equation with variable coetffcients.展开更多
New exact solutions expressed by the Jacobi elliptic functions are obtained to the (2+1)-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave sol...New exact solutions expressed by the Jacobi elliptic functions are obtained to the (2+1)-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave solutions and triangular periodic wave solutions are obtained as well.展开更多
Using the classical Lie method of infinitesimals, we first obtain the symmetry of the (2+1)-dimensional Burgers-Korteweg-de-Vries (3D-BKdV) equation. Then we reduce the 3D-BKdV equation using the symmetry and giv...Using the classical Lie method of infinitesimals, we first obtain the symmetry of the (2+1)-dimensional Burgers-Korteweg-de-Vries (3D-BKdV) equation. Then we reduce the 3D-BKdV equation using the symmetry and give some exact solutions of the 3D-BKdV equation. When using the direct method, we restrict a condition and get a relationship between the new solutions and the old ones. Given a solution of the 3D-BKdV equation, we can get a new one from the relationship. The relationship between the symmetry obtained by using the classical Lie method and that obtained by using the direct method is also mentioned. At last, we give the conservation laws of the 3D-BKdV equation.展开更多
Based on the binary Bell polynomials,the bilinear representation,bilinear B?cklund transformation and the Lax pair for the dissipative(2+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation are obtained.Moreover,th...Based on the binary Bell polynomials,the bilinear representation,bilinear B?cklund transformation and the Lax pair for the dissipative(2+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation are obtained.Moreover,the infinite conservation laws are also derived.展开更多
Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional brea...Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.展开更多
New exact solutions expressed by the Jacobi elliptic functions are obtained to the long-short wave interaction equations by using the modified F-expansion method. In the limit case, solitary wave solutions and triangu...New exact solutions expressed by the Jacobi elliptic functions are obtained to the long-short wave interaction equations by using the modified F-expansion method. In the limit case, solitary wave solutions and triangular periodic wave solutions are obtained as well.展开更多
基金National Natural Science Foundation of China (40274011, 40074020), MOST (2001BA601B02) and Joint Seismological Science Foundation of China (102068).
文摘In 2001 three earthquakes occurred in Shidian in Yunnan Province, which were the MS=5.2 on April 10, the MS=5.9 on April 12 and the MS=5.3 on June 8. Based on the data from the station Baoshan of Yunnan Telemetry Digital Seismograph Network, the variational characteristics of shear-wave splitting on these series of strong earthquakes has been studied by using the systematic analysis method (SAM) of shear-wave splitting. The result shows the time delays of shear-wave splitting basically increase with earthquake activity intensifying. However the time delays abruptly decrease immediately before strong aftershocks. It accords with the stress relaxation before earthquakes, which was found recently in study on shear-wave splitting. The result suggests it is significant for reducing the harm degree of earthquakes to develop the stress-forecasting on earthquake in strong active tectonic zones and economic developed regions or big cities under the danger of strong earthquakes.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant Nos. 2004zx16 and Q2005A01
文摘In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2+1)-dimensional variable coeffficient canonical generalized KP (VCCGKP) equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004 zx 16
文摘Painleve property and infinite symmetries of the (2+1)-dimensional higher-order Broer-Kaup (HBK) system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to (2+1)-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the (2+1)-dimensional HBK system.
基金The Project supported by the Natural Science Foundation of Shandong Province of China under Grant No.Q2005A01
文摘By applying a direct symmetry method, we get the symmetry of the asymmetric Nizhnik-Novikov-Veselov equation (ANNV). Taking the special case, we have a finite-dimensional symmetry. By using the equivalent vector of the symmetry, we construct an eight-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, we reduce the ANNV equation and obtain some solutions to the reduced equations. Furthermore, we find some new explicit solutions of the ANNV equation. At last, we give the conservation laws of the ANNV equation.
基金Natural Science Foundation of Shandong Province under Grant Nos.2004zx16 and Q2005A01
文摘In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the (2+1)-dimensional Lax pair is reduced to (1+1)-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.
基金The project supported by the Natural Science Foundation of Shandong Province under Grant Nos. 2004zx16 and Q2005A01
文摘In this paper, the generalized ranch function method is extended to (2+1)-dimensianal canonical generalized KP (CGKP) equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the (2+1)-dimensional OGKP equation with variable coetffcients.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004 zx16
文摘New exact solutions expressed by the Jacobi elliptic functions are obtained to the (2+1)-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave solutions and triangular periodic wave solutions are obtained as well.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004 zx 16
文摘Using the classical Lie method of infinitesimals, we first obtain the symmetry of the (2+1)-dimensional Burgers-Korteweg-de-Vries (3D-BKdV) equation. Then we reduce the 3D-BKdV equation using the symmetry and give some exact solutions of the 3D-BKdV equation. When using the direct method, we restrict a condition and get a relationship between the new solutions and the old ones. Given a solution of the 3D-BKdV equation, we can get a new one from the relationship. The relationship between the symmetry obtained by using the classical Lie method and that obtained by using the direct method is also mentioned. At last, we give the conservation laws of the 3D-BKdV equation.
基金Supported by the National Natural Science Foundation of China and China Academy of Engineering Physics under Grant No 11076015.
文摘Based on the binary Bell polynomials,the bilinear representation,bilinear B?cklund transformation and the Lax pair for the dissipative(2+1)-dimensional Ablowitz-Kaup-Newell-Segur(AKNS)equation are obtained.Moreover,the infinite conservation laws are also derived.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004zx16
文摘Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.
文摘New exact solutions expressed by the Jacobi elliptic functions are obtained to the long-short wave interaction equations by using the modified F-expansion method. In the limit case, solitary wave solutions and triangular periodic wave solutions are obtained as well.