Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.
We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices,and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings.Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.
In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.
The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations is analyzed to shed light oi1 the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differentii equations and their corresponding exact solutions with generalized separated variables.
The exact explicit traveling solutions to the two completely integrable sixthorder nonlinear equations KdV6 are given by using the method of dynamical systems and Cosgrove's work.It is proved that these traveling wave solutions correspond to some orbits in the 4-dimensional phase space of two 4-dimensional dynamical systems.These orbits lie in the intersection of two level sets defined by two first integrals.
A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.
Let Λ be an isolated non-trivial transitive set of a C1 generic diffeomorphism f ∈ Diff（M）. We show that the space of invariant measures supported on A coincides with the space of accumulation measures of time averages on one orbit. Moreover, the set of points having this property is residual in Λ （which implies that the set of irregular＋ points is also residual in Λ）. As an application, we show that the non-uniform hyperbolicity of irregular＋ points in A with totally 0 measure （resp., the non-uniform hyperbolicity of a generic subset in Λ） determines the uniform hyperbolicity of Λ.
Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is not a real solution. This paper answers the question as to whether or not nonlinear wave equations exist for which a ＂real＂ loopsolution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions.
Let M be a smooth compact manifold and A be a compact invariant set. In this article, we prove that, for every robustly transitive set A, flA satisfies a Cl-genericstable shadowable property （resp., Cl-generic-stable transitive specification property or Cl-generic-stable barycenter property） if and only if A is a hyperbolic basic set. In particular, flA satisfies a Cl-stable shadowable property （resp., Cl-stable transitive specification property or Cl-stable barycenter property） if and only if A is a hyperbolic basic set. Similar results are valid for volume-preserving case.
