Energy Scattering for the Generalized Davey—Stewartson Equations

Abstract Considring the generalized Davey-Stewartson equation i-△u+λ│u│~pu+μE(│u│~q)│u│~(q-2)u=0,where λ>0,μ≥0,E=F~(-1)(ξ_1~2│ξ│~2)F,we obtain the existence of scattering operator in ∑(R~n):u{u∈H~1(R~n):│x│u∈L~2(R~n)}.     

Davey-Stewartson方程在Banach空间中的指数吸引子

我们考虑了DS方程在Banach空间X^α_p中的指数吸引子,并且得到其分形维度估计.     

Long-time decay of the solutions of the Davey—Stewartson II equations

Summary Using the method of inverse scattering, the sup-norms of the solutions of the Davey—Stewartson II equations are shown to decay in the order of 1/¦t¦ as ¦t¦ goes to infinity. In the focusing case this result is obtained for small initial data, whereas in the defocusing case it is obtained for general initial data.     

二维空间中广义Davey-Stewartson系统整体解存在的最佳条件

根据基态的特征 ,首先在二维空间中导出了广义Davey Stewartson系统解爆破和整体存在的最佳条件 ;其次得到了整体解存在的一个最佳充分条件 ;最后证明了当初值为多小时 ,该系统的整体解存在 .     

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

Heteroclinic Breather-Wave Solutions for Davey-Stewartson Equation

Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.     

Anisotropic collapse in three-dimensional dipolar Bose-Einstein condensates

We study the (3+1)-dimensional Gross-Pitaevskii / Nonlinear Schrödinger equation describing a dipolar Bose-Einstein condensate. Bound states are computed using accurate numerical techniques. When the dipolar strength is negative, the total number of atoms vs. frequency relationship for these bound states is multi-valued and possesses a cusp point, which corresponds to a “candlestick” ground state. Direct simulations of this ground state exhibit strongly-anisotropic collapse of its nucleus, with different contraction rates along the dipole axis and perpendicular to it. We propose an anisotropic self-similar theory to explain this dynamics. The physical implications are discussed.     

New Formal Solutions of Davey-Stewartson Equation via Combined tanh Function Method with Symmetry Method

We combine the tanh function method with the symmetry group method to construct new type of solutions of Davey-Stewartson equation and implemente it in a computer algebraic system. As a result, some new types of solutions are obtained. This method is also applied to other differential equations if the nonlinear evolution equations admit nontrivial one-parameter group of transformation.     

New Types of Travelling Wave Solutions From （2＋l）-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixtb-aegree nonnneal term, we study the （2＋l ）-dimensional Davey-Stewartson equation and new types of travelling wave solutions are obtained, which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

Almost conservation law and global rough solutions to a nonlinear Davey-Stewartson equation

We prove an “almost conservation law” to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in Hs(R²), and .