Nonlinear integrable couplings of a generalized super Ablowitz‐Kaup‐Newell‐Segur hierarchy and its super bi‐Hamiltonian structures

For the orthosymplectic Lie superalgebra OSPfalse(2,2false), we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi‐Hamiltonian structures.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

Han

2020

“…As we all know, nonlinear science is widely used in nature. [2][3][4] Since the 1960s, the dust acoustic wave, 5 dust ion-acoustic wave, 6 Langmuir soliton, 7 and dust kinetic Alfven waves 8 have been proved both theoretically and experimentally. This paper mainly describes the dust ion-acoustic wave.…”

Section: Introductionmentioning

confidence: 99%

(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

Tian

Ren

et al. 2019

In this paper, the theoretical model of dust ion-acoustic waves in collisionless, unmagnetized dust plasma is analyzed. According to the (3 + 1)-dimensional nonlinear dynamic propagation equations with the electron distribution in the presence of trapping particles and the kinetic equation of charge of dust particle, based on multiscale analysis and perturbation method, a new (3 + 1)-dimensional modified Burgers equation is constructed. Because of the space property of dust plasma, (3 + 1)-dimensional modified Burgers equation is more suitable than low-dimensional equation to describe the dust ion-acoustic waves. Furthermore, applying the semi-inverse method and the fractional variational principle, time-fractional modified Burgers equation is given, which owns potential value for comprehending the propagation feature of dust ion-acoustic waves in three-dimensional space. Following, the conservation laws of the time-fractional modified Burgers equation are discussed by using the Noether method. Finally, the shock wave solutions of the new model is obtained with the help of the G ′ ∕G-expansion method; meanwhile, the radial basis function method is also used to get the numerical solutions of the (3 + 1)-dimensional time-fractional modified Burgers equation. KEYWORDS(3 + 1)-dimensional time-fractional modified Burgers equation, conservation laws, dust ion-acoustic waves, radial basis function method, shock wave solution MSC CLASSIFICATION 35Q51; 35Q55After solving the equation Ac k =B k , we get the algebraic system at each time step, and we can obtain the solution by using Equation (50).

“…We all knew that the variables satisfied commuting relationship in classical integrable hierarchies. In recent years, super (supersymmetry) integrable hierarchies has aroused the interest of many researchers, including nonlinearization of Lax pairs,() Darboux transformation,() construction of new super integrable hierarchies,() and so forth. () Both commuting and anticommuting variables appeared in super (supersymmetry) integrable hierarchies.…”

Section: Introductionmentioning

confidence: 99%

Generalized nonisospectral super integrable hierarchies

Zhou

Han

et al. 2019

For the Lie superalgebra B(0,1), we choose a set of basis matrices. Then we consider a linear combination of the basis matrices, which is exactly the spectral matrix of the spatial part for the super Ablowitz‐Kaup‐Newell‐Segur (AKNS) hierarchy. The compatible condition of the spatial and temporal spectral problems leads to the well‐known zero curvature equation. Here, when the spectral parameter is independent (dependent) of temporal parameter, we obtain isospectral (nonisospectral) super AKNS hierarchy. Furthermore, we derive the generalized nonisospectral super AKNS hierarchy (GNI‐SAKNS). As another example, similar method is successfully applied to the super Dirac hierarchy, and we obtain the generalized nonisospectral super Dirac hierarchy (GNI‐SD).