Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the(2+1)-dimensional coupled nonlinear Schr¨odinger(CNLS)equations that can describe interactions of two Rossby waves propagating in stratified fluids.The(2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately.The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method,and the influence factors of modulational instability are analyzed.The results suggest that,when two Rossby waves with slightly different wave numbers propagate in the stratified fluids,the intensity of bright soliton decreases with the increases of dark soliton coefficients.In addition,the size of modulational instable area is related to the amplitude and wave number in y direction.

Long-Time Asymptotics for the Nonlocal MKdV Equation

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation(nonlocal mKdV) qt(x, t)+qxxx(x, t)-6 q(x, t)q(-x,-t)qx(x, t) = 0, which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal m KdV equation. In contrast with the classical mKdV equation,we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the ChenLee-Liu equation i？tu ＋ ？xxu-i｜u2｜？xu = 0 on the half line（-∞, 0] with decaying initial value. Assuming that the solution u（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit（x, t） dependence and is given in terms of the spectral functions{a（λ）, b（λ）}and{A（λ）, B（λ）}, which are obtained from the initial data u0（x） = u（x, 0） and the boundary data g0（t） = u（0, t）, g1（t） = ux（0, t）, respectively. The spectral functions are not independent,but satisfy a so-called global relation.

Seiberg-Witten/Whitham Equations and Instanton Corrections in N = 2 Supersymmetric Yang-Mills Theory

We obtain the instanton correction recursion relations for the low energy effective prepotential in pure Ν = 2SU(n) supersymmetric Yang-Mills gauge theory from Whitham hierarchy and Seiberg-Witten/Whitham equations. These formulae provide us a powerful tool to calculate arbitrary order instanton corrections coefficients from the perturbative contributions of the effective prepotential in Seiberg-Witten gauge theory. We apply this idea to evaluate one-and two-order instanton corrections coefficients explicitly in SU(n) case in detail through the dynamical scale parameter expressed in terms of Riemann's theta-function.