Formal Series Symmetry Research Articles

Duality of positive and negative integrable hierarchies via relativistically invariant fields

It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

Constructing (2+1)-dimensional supersymmetric integrable systems from the Hirota formalism in the superspace**Project supported by the National Natural Science Foundation of China (Grant Nos. 11605102, 11475052, 11675055, and 11626140).

The supersymmetric extensions of two integrable systems, a special negative Kadomtsev–Petviashvili (NKP) system and a (2+1)-dimensional modified Korteweg–de Vries (MKdV) system, are constructed from the Hirota formalism in the superspace. The integrability of both systems in the sense of possessing infinitely many generalized symmetries are confirmed by extending the formal series symmetry approach to the supersymmetric framework. It is found that both systems admit a generalization of type algebra and a Kac-Moody–Virasoro type subalgebra. Interestingly, the first one of the positive flow of the supersymmetric NKP system is another supersymmetric extension of the (2+1)-dimensional MKdV system. Based on our work, a hypothesis is put forward on a series of (2+1)-dimensional supersymmetric integrable systems. It is hoped that our work may develop a straightforward way to obtain supersymmetric integrable systems in high dimensions.

Generalized symmetries of an \U0001d4dd = 1 supersymmetric Boiti–Leon–Manna–Pempinelli system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11275123, 11175092, 11475052, and 11435005), the Shanghai Knowledge Service Platform for Trustworthy Internet of Things, China (Grant No. ZF1213), and the Talent Fund and K CWong Magna Fund in Ningbo University, China.

The formal series symmetry approach (FSSA), a quite powerful and straightforward method to establish infinitely many generalized symmetries of classical integrable systems, has been successfully extended in the supersymmetric framework to explore series of infinitely many generalized symmetries for supersymmetric systems. Taking the \U0001d4a9 = 1 supersymmetric Boiti–Leon–Manna–Pempinelli system as a concrete example, it is shown that the application of the extended FSSA to this supersymmetric system leads to a set of infinitely many generalized symmetries with an arbitrary function f (t). Some interesting special cases of symmetry algebras are presented, including a limit case f (t) = 1 related to the commutativity of higher order generalized symmetries.

Infinitely many generalized symmetries and Painlevé analysis of a (2 + 1)-dimensional Burgers system

Infinitely many generalized symmetries of a coupled (2 + 1)-dimensional Burgers system are obtained by means of the formal series symmetry approach. It is found that the generalized symmetries constitute a closed infinite-dimensional Lie algebra. Three interesting special cases are presented, including a closed infinite-dimensional Lie algebra and a Kac–Moody–Virasoro-type Lie symmetry algebra. From the first one of the positive flow, a new integrable coupled system of the modified Korteweg–de Vries equation and the potential Boiti–Leon–Manna–Pempinelli equation is constructed. In addition, it is demonstrated that the coupled Burgers system can pass the Painlevé test.

A new (2+1)-dimensional generalization of the modified Korteweg-deVries equation and its infinitely many symmetries

Starting from the nonlinear version of bilinear negative Kadomtsev-Petviashvili system, a new (2+1)-dimensional generalization of the modified Korteweg-deVries equation is obtained. For this new system, a set of infinitely many generalized symmetries is found by applying a formal series symmetry approach. And these symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra.

Symmetries and Algebras of a (2+1)-Dimensional MKdV-Type System

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.

Infinitely many symmetries and symmetry reduction of (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

Integrability condition of （2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation are obtained by Painléve-test. Based on this condition and Painléve-test, the bilinear form of GCBS equation is found. Towards this bilinear form infinitely many formal series symmetries are found by the formal series symmetry method, the obtained symmetries are used to get the symmetry reductions of GCBS equation.

Kac–Moody–Virasoro symmetry algebra of a (2+1)-dimensional bilinear system

Based on some known facts of integrable models, this paper proposes a new (2+1)-dimensional bilinear model equation. By virtue of the formal series symmetry approach, the new model is proved to be integrable because of the existence of the higher order symmetries. The Lie point symmetries of the model constitute an infinite dimensional Kac–Moody–Virasoro symmetry algebra. Making use of the infinite Lie point symmetries, the possible symmetry reductions of the model are also studied.

Kac–Moody–Virasoro symmetry algebra and symmetry reductions of the bilinear sinh-Gordon equation in (2 + 1)-dimensions

By means of the formal series symmetry approach proposed in [1], infinite many symmetries and the corresponding Kac–Moody–Virasoro Lie symmetry algebra of a new bilinear (2 + 1)-dimensional sinh-Gordon equation are given. Then, the obtained symmetries are used to get the symmetry reductions of the model. From one of the special reduction we know that the bilinear form of the first member of the negative Kadomtsev–Petviashvili hierarchy is not only a (2 + 1)-dimensional sinh-Gordon extension but also a novel (2 + 1)-dimensional classical Boussinesq extension.

(2+1)-dimensional (M+N)-component AKNS system: Painlevé integrability, infinitely many symmetries, similarity reductions and exact solutions

The (2+1)-dimensional (M+N)-component AKNS system that is derived from the inner parameter dependent symmetry constraint of the KP equation is studied in detail. First, the Painlevé integrability of the model is proved by using the standard WTC and Kruskal approach. Using the formal series symmetry approach, the generalized KMV symmetry algebra and the related symmetry group are found. The two-dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized KMV symmetry algebra and the direct method. Abundant localized coherent structures are revealed by the variable separation approach. Some special types of the localized excitations like the multiple solitoffs, dromions, lumps, ring solitons, breathers and instantons are plotted also.

Symmetries and dromion solution of a (2+1)-dimensional nonlinear Schrödinger equation

The Painleve property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schrodinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painleve property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.

Study of a (2 + 1)-dimensional Broer-Kaup equation

The Painlev? property and infinitely many symmetries of a (2 + 1)-dimensional Broer-Kaup equation which is obtained from the constraints of the KP equation are studied in this paper. The Painlev? property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method.

Higher-Dimensional Integrable Models with a Common Recursion Operator**The project supported by National Natural Science Foundation of China and Natural Science Foundation of Zhejiang Province of China

Starting from any one of hereditary symmetries, we can construct a type of integrable models with arbitrary dimensions. The models with different dimensions obtained from a same hereditary symmetry possess a common recursion operator. The symmetry structures of the models are studied in their potential forms. Using the formal series symmetry approach, we can get various sets of formal series symmetries with some arbitrary functions. Generally, these sets of series symmetries are not truncated for arbitrary functions. The series symmetries will all be truncated if the arbitrary functions are fixed as polynomials. Some sets of nontruncated symmetries constitute generalized Virasoro algebras. The more details about the symmetries and algebras are discussed for a concrete -dimensional KdV equation.

Abundant Symmetry Structures of the Breaking Soliton Equation11The project supported by National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province of China

The formal series symmetry approach of obtaining symmetries of a higher-dimensional partial differential equation is treated as an alternative way. For a breaking soliton equation which possesses a -dimensional-like recursion operator, six sets of generalized symmetries are explicitly given. It is known that the truncated formal series symmetries of the KP and Toda equations constitute the generalized algebra. From this paper we find that the generalized algebra can also be realized by means of the nontruncated formal series symmetries.

SYMMETRIES OF THE(2 +1) DIMENSIONAL DISCRETE Toda EQUATION

We extend the formal series symmetry theory proposed in Refs. [1-4] to the (2 + 1) dimensional discrete Toda equation. Two sets of infinitely many symmetries of this equation are found. Every one set of symmetries constitute a generalized W∞ algebra which contains the usual W∞ algebra as its subalgebra.

Non-truncated formal series symmetries and generalized Virasoro algebra: AKNS-type breaking soliton equation

Using the formal series symmetry approach, we obtain three sets of generalized symmetries of the AKNS-type breaking soliton equation. These sets of formal series symmetries are not truncated except that the arbitrary functions of the formal series symmetries are fixed as polynomials of the corresponding independent variables. Differing from the truncated cases such as KP and Toda field equations, these non-truncated symmetries constitute another type of generalization of the Virasoro algebra.

Formal Series Symmetry Approach for a Generalized Type of High Dimensional Models11The project supported by National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.

A formal series symmetry formula is given for a quite generalized partial differential-difference vector equation. The details about the symmetries of the KP equation are restudied by means of the potential KP equations.

Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation

The formal series symmetry approach is applied to the conditionally integrable nonlinear evolution equation. A set of infinitely many symmetries for a 3+1-dimensional model, Jimbo–Miwa equation with a conditional equation (Kadomtsev–Petviashvilli equation), are obtained by an explicitly constructable formula. Similar to the usual KP equation, these symmetries constitute a generalized W∞ symmetry algebra.

Formal Series Symmetries and Truncated Symmetries of a Bilinear Schrödinger Equation

For a type of nonlinear Schrödinger equations, we get a set of formal series symmetries. For a special integrable bilinear Schrödinger equation in -dimensional spacetime, some truncated symmetries which constitute an infinite dimensional Lie algebra are obtained.

Generalized symmetries and algebras of the two-dimensional differential-difference Toda equation

Applying the formal series symmetry theory to a differential-difference system, two-dimensional Toda equation, we found that as with the continuous Toda field theory, there exist two sets of infinitely many symmetries for the discrete Toda system. Each set of symmetries constitutes a generalized Winfinity algebra.