We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The intr...We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the （2＋l）-dimensional DLW system is consistent Riccati expansion （CRE） solvable. If the special form of （CRE）-consistent tanh-function expansion （CTE） is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.展开更多
The residual symmetry relating to the truncated Painlev6 expansion of the Kadomtsev-Petviashvili （KP） equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using ...The residual symmetry relating to the truncated Painlev6 expansion of the Kadomtsev-Petviashvili （KP） equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.展开更多
In nonlinear physics, it is very difficult to study interactions among different types of nonlinear waves. In this paper,the nonlocal symmetry related to the truncated Painleve′ expansion of the（2＋1）-dimensional B...In nonlinear physics, it is very difficult to study interactions among different types of nonlinear waves. In this paper,the nonlocal symmetry related to the truncated Painleve′ expansion of the（2＋1）-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system. Then the corresponding group invariant solutions are found, from which interaction solutions among different types of nonlinear waves can be found.Furthermore, the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation（BT） is obtained. From this BT, novel interactive solutions among different nonlinear excitations are found.展开更多
Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of t...Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.展开更多
Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transform...Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie's first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th B?cklund transformation in the form of the determinant. Moreover, the P_sT_d(the shifted parity and delayed time reversal) symmetric exact solutions(including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation.展开更多
The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to L...The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained： a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.展开更多
We obtain the non-local residual symmetry related to truncated Painlev~ expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation in...We obtain the non-local residual symmetry related to truncated Painlev~ expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system. By using Lie＇s first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly, we also Iocalize the linear superposition of multiple residual symmetries to find the corresponding finite transformations. It is interesting to find that the n-th B^icklund transformation for Burgers equation can be expressed by determinants in a compact way.展开更多
With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation ...With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the(3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.展开更多
The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé...The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.展开更多
The residual symmetry of the generalized Kaup-Kupershmidt （KKK） equation is obtained from the truncated Painleve expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solution...The residual symmetry of the generalized Kaup-Kupershmidt （KKK） equation is obtained from the truncated Painleve expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Frthermore, the gKK equation is proved to integrable in the sense of owning consistent Riecati expansion and some new Bgcklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11371293 and 11505090)the Natural Science Foundation of Shaanxi Province,China(Grant No.2014JM2-1009)+1 种基金the Research Award Foundation for Outstanding Young Scientists of Shandong Province,China(Grant No.BS2015SF009)the Science and Technology Innovation Foundation of Xi’an,China(Grant No.CYX1531WL41)
文摘We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the （2＋l）-dimensional DLW system is consistent Riccati expansion （CRE） solvable. If the special form of （CRE）-consistent tanh-function expansion （CTE） is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11347183,11275129,11305106,11405110,and 11365017)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘The residual symmetry relating to the truncated Painlev6 expansion of the Kadomtsev-Petviashvili （KP） equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11347183,11275129,11305106,11365017,and 11405110)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘In nonlinear physics, it is very difficult to study interactions among different types of nonlinear waves. In this paper,the nonlocal symmetry related to the truncated Painleve′ expansion of the（2＋1）-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system. Then the corresponding group invariant solutions are found, from which interaction solutions among different types of nonlinear waves can be found.Furthermore, the Burgers equation is also studied by using the generalized tanh expansion method and a new Ba¨cklund transformation（BT） is obtained. From this BT, novel interactive solutions among different nonlinear excitations are found.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11975156 and 12175148)。
文摘Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11705077 and 11775104
文摘Starting from the truncated Painlev′e expansion, the residual symmetry of the Alice-Bob modified Kortewegde Vries(AB-mKdV) equation is derived. The residual symmetry is localized and the AB-mKdV equation is transformed into an enlarged system by introducing one new variable. Based on Lie's first theorem, the finite transformation is obtained from the localized residual symmetry. Further, considering the linear superposition of multiple residual symmetries gives rises to N-th B?cklund transformation in the form of the determinant. Moreover, the P_sT_d(the shifted parity and delayed time reversal) symmetric exact solutions(including invariant solution, breaking solution and breaking interaction solution) of AB-mKdV equation are presented and two classes of interaction solutions are depicted by using the particular functions with numerical simulation.
基金supported by the National Natural Science Foundation of China(Grant Nos.11347183,11405110,11275129,and 11305106)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained： a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.
基金supported by the National Natural Science Foundation of China(Grant Nos.11347183,11275129,11305106,11365017,and 11405110)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘We obtain the non-local residual symmetry related to truncated Painlev~ expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system. By using Lie＇s first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly, we also Iocalize the linear superposition of multiple residual symmetries to find the corresponding finite transformations. It is interesting to find that the n-th B^icklund transformation for Burgers equation can be expressed by determinants in a compact way.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11835011 and 12074343)。
文摘With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the(3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)the K C Wong Magna Fund in Ningbo University。
文摘The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11405110,11275129,11472177the Natural Science Foundation of Zhejiang Province of China under Grant Nos.LY18A050001 and LQ13A050002
文摘The residual symmetry of the generalized Kaup-Kupershmidt （KKK） equation is obtained from the truncated Painleve expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Frthermore, the gKK equation is proved to integrable in the sense of owning consistent Riecati expansion and some new Bgcklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.
