Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, severa...Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.展开更多
In this paper,the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)system is studied by using residual symmetry and the consistent Riccati expansion(CRE)method,respectively.The residual symmetry of the DSSH system is localized to...In this paper,the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)system is studied by using residual symmetry and the consistent Riccati expansion(CRE)method,respectively.The residual symmetry of the DSSH system is localized to Lie point symmetry in a properly prolonged system,based on which we get a new Bäcklund transformation for this system.New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system.Moreover,the DSSH system proves to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.展开更多
The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal r...The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.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
The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for...The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.展开更多
A consistent tanh expansion(CTE) method is used to study the modified Boussinesq equation. It i proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a ...A consistent tanh expansion(CTE) method is used to study the modified Boussinesq equation. It i proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by th Painlev′e analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependen variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initia value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among soliton and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wav interaction behaviors are studied both in analytical and graphical ways.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
A nonlocal coupled Kadomtsev–Petviashivili(ncKP) system with shifted parity(P_(s)^(x)) and delayed time reversal(T_(d)) symmetries is generated from the local coupled Kadomtsev–Petviashivili(cKP) system. By introduc...A nonlocal coupled Kadomtsev–Petviashivili(ncKP) system with shifted parity(P_(s)^(x)) and delayed time reversal(T_(d)) symmetries is generated from the local coupled Kadomtsev–Petviashivili(cKP) system. By introducing new dependent variables which have determined parities under the action of P_(s)^(x)T_(d)^(d), the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the c KP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.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
In this paper, a （2＋1）-dimensional MKdV-type system is considered. By applying the formal series symmetry approach, a set of infinitely many generalized symmetries is obtained. These symmetries constitute a closed ...In this paper, a （2＋1）-dimensional MKdV-type system is considered. By applying the formal series symmetry approach, a set of infinitely many generalized symmetries is obtained. These symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra. Thus the complete integrability of this system is confirmed.展开更多
The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis...The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.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
The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial different...The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.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
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
The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painl...The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE) solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.展开更多
We study a forced variable-coefficient extended Korteweg-de Vries(KdV)equation in fluid dynamics with respect to internal solitary wave.Bäcklund transformations of the forced variable-coefficient extended KdV equ...We study a forced variable-coefficient extended Korteweg-de Vries(KdV)equation in fluid dynamics with respect to internal solitary wave.Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevéexpansion.When the variable coefficients are time-periodic,the wave function evolves periodically over time.Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations.One-parameter group transformations and one-parameter subgroup invariant solutions are presented.Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method.The consistent Riccati expansion(CRE)solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE.Interaction phenomenon between cnoidal waves and solitary waves can be observed.Besides,the interaction waveform changes with the parameters.When the variable parameters are functions of time,the interaction waveform will be not regular and smooth.展开更多
The difficulty of the prediction of military aircraft purchase price lies in the small sample data, and the sample data have the complicated non-linear characteristics. By analyzing the influence of parameters of airc...The difficulty of the prediction of military aircraft purchase price lies in the small sample data, and the sample data have the complicated non-linear characteristics. By analyzing the influence of parameters of aircraft purchase price, SVR is proposed to predict the aircraft purchasing price model, and uses the model to predict the aircraft purchase price. The calculation results show that the prediction of the purchase price to establish military aircraft model has higher prediction accuracy.展开更多
The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed...The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.展开更多
The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between ...The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters.The soliton molecules including the bright and bright soliton,the dark and bright soliton,and the dark and dark soliton are exhibited in figures 1–3,respectively.Meanwhile,the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevémethod.The consistent Riccati expansion(CRE)method is applied to the extended mKdV equation.It demonstrates that the extended mKdV equation is a CRE solvable system.A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.11305031 and 11305106Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No.Yq2013205
文摘Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.
基金supported by the National Natural Science Foundation of China under Grant Nos.12175148 and 11975156。
文摘In this paper,the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)system is studied by using residual symmetry and the consistent Riccati expansion(CRE)method,respectively.The residual symmetry of the DSSH system is localized to Lie point symmetry in a properly prolonged system,based on which we get a new Bäcklund transformation for this system.New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system.Moreover,the DSSH system proves to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11305031,11365017,and 11305106)the Natural Science Foundation of Guangdong Province,China(Grant No.S2013010011546)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.LQ13A050001)the Science and Technology Project Foundation of Zhongshan,China(Grant Nos.2013A3FC0264 and 2013A3FC0334)the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province,China(Grant No.Yq2013205)
文摘The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.
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Supported by the Global Change Research Program China under Grant No.2015CB953904the National Natural Science Foundations of China under Grant Nos.11435005,11175092,and 11205092+1 种基金Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11305106 and 11505154
文摘A consistent tanh expansion(CTE) method is used to study the modified Boussinesq equation. It i proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by th Painlev′e analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependen variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initia value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among soliton and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wav interaction behaviors are studied both in analytical and graphical ways.
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supported by the National Natural Science Foundation of China under Grant Nos.12175148,11975156.
文摘A nonlocal coupled Kadomtsev–Petviashivili(ncKP) system with shifted parity(P_(s)^(x)) and delayed time reversal(T_(d)) symmetries is generated from the local coupled Kadomtsev–Petviashivili(cKP) system. By introducing new dependent variables which have determined parities under the action of P_(s)^(x)T_(d)^(d), the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the c KP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.
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.10735030, 10675065, and 90503006, and PCSIRT (IRT0734)the National Basic Research Programme of China under Grant No.2007CB814800
文摘In this paper, a （2＋1）-dimensional MKdV-type system is considered. By applying the formal series symmetry approach, a set of infinitely many generalized symmetries is obtained. These symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra. Thus the complete integrability of this system is confirmed.
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 nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.
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 meters restrain as the relation:Foundation of China Grant No.11775146.
文摘The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.
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.
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Project supported by the National Natural Science Foundation of China(Grant Nos.11775047,11775146,and 11865013)the Science and Technology Project Foundation of Zhongshan City,China(Grant No.2017B1016).
文摘The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE) solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11775047,11775146,and 11865013).
文摘We study a forced variable-coefficient extended Korteweg-de Vries(KdV)equation in fluid dynamics with respect to internal solitary wave.Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevéexpansion.When the variable coefficients are time-periodic,the wave function evolves periodically over time.Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations.One-parameter group transformations and one-parameter subgroup invariant solutions are presented.Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method.The consistent Riccati expansion(CRE)solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE.Interaction phenomenon between cnoidal waves and solitary waves can be observed.Besides,the interaction waveform changes with the parameters.When the variable parameters are functions of time,the interaction waveform will be not regular and smooth.
文摘The difficulty of the prediction of military aircraft purchase price lies in the small sample data, and the sample data have the complicated non-linear characteristics. By analyzing the influence of parameters of aircraft purchase price, SVR is proposed to predict the aircraft purchasing price model, and uses the model to predict the aircraft purchase price. The calculation results show that the prediction of the purchase price to establish military aircraft model has higher prediction accuracy.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11775146,11435005,and 11472177Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.
基金supported by the National Natural Science Foundation of China Grant Nos.11775146,11305106,11975156,11775047 and 11835011。
文摘The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters.The soliton molecules including the bright and bright soliton,the dark and bright soliton,and the dark and dark soliton are exhibited in figures 1–3,respectively.Meanwhile,the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevémethod.The consistent Riccati expansion(CRE)method is applied to the extended mKdV equation.It demonstrates that the extended mKdV equation is a CRE solvable system.A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.
