An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.The auxiliary parameter has an effect on the convergence of homotopy series solutions.Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.
JIAO XiaoYu1,GAO Yuan1 & LOU SenYue1,2,3 1 Department of Physics,Shanghai Jiao Tong University,Shanghai 200240,China
By the generalized sub-equation expansion method and symbolic computation, this paper investigates the （3＋1）dimensional Gross-Pitaevskii equation with time-and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.
This paper analyzes the characteristics of super typhoons （STYs） over the western North Pacific （WNP） from 1965 to 2005 and describes the seasonal variability of STY activity. The relation between STY activity and the E1 Nifio-Southern Oscillation （ENSO） as well as the possible reason for the influence of the ENSO on STY activity are also investigated. The results showed that about one fifth of the tropical cyclones （TCs） over the WNP could reach the rank of STY. Most STYs appeared from July to November while there was a highest ratio between number of STYs and total number of TCs in November. Most STYs appeared east of the Philippine Sea. In E1 Nino years, affected by sea surface temperature （SST）, monsoon trough and weak vertical wind shear, TC formation locations shifted eastward and there were more STYs than in La Nifia years when the affecting factors changed.
We studied the impact of sea surface temperature anomaly(SSTA) in the Japan Sea and the sea area east of Japan on the winter rainfall and air temperature in Northeast(NE) China using the singular value decomposition(SVD) and empirical orthogonal function(EOF). The monthly-mean rainfall data observed at 160 stations in China, monthly-mean sea surface temperature(SST) of the Hadley Center for Climate Prediction and Research and monthly-mean air temperature from the NCEP reanalysis during 1960–2011 were used. Correlation analysis indicates that the SSTAs in the Japan Sea in September may last for three or four months and are an important index for forecasting the winter rainfall and air temperature in NE China. Positive SSTAs in the central Japan Sea and in the sea area east of Tokyo correspond to positive rainfall anomaly and negative air temperature anomaly in NE China. With the rise of SST in the Japan Sea, a weak cyclone appears over the Japan Sea. The northeasterly wind transports water vapor from the Okhotsk to NE China, resulting in more rainfall and lower air temperature. Negative SSTA years are accompanied by warmer air temperature and less snow in NE China. The 1000 h Pa geopotential height anomaly and wind anomaly fields are simulated by IAP-9L model, which supports the analysis results.
We present three families of soliton solutions to the generalized （3＋l）-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.
The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.
By symbolic computation and a direct method, this paper presents some exact analytical solutions of the one-dimensional generalized inhomogeneous higher-order nonlinear Schrodinger equation with variable coefficients, which include bright solitons, dark solitons, combined solitary wave solutions, dromions, dispersion-managed solitons, etc. The abundant structure of these solutions are shown by some interesting figures with computer simulation.
