A spectral method is used to derive a series of equations for axisymmetric Couette-Taylor flow. A three-modes system, which is similar to the Lorenz systems, is obtained by a suitable three-modes truncation of the Navier-Stokes equations for the incompressible flow between two concentric rotating cylinders. The stability of the three-modes systems is discussed. Moreover, the existence of its attractor and the estimation of Hausdorff dimension are given.

We establish the existence of positive bound state solutions for the singular quasilinear Schrödinger equation i\frac{∂ψ}{∂t}=-div(ρ(|∇ψ|^2)∇ψ)+ω(|ψ|^2)ψ-λρ(|ψ|^2)ψ, x∈Ω, t > 0, where ω(τ^2)τ→+∞ as τ → 0 and,λ > 0 is a parameter and Ω is a ball in R^N. This problem is studied in connection with the following quasilinear eigenvalue problem with Dirichlet boundary condition -div(ρ(|∇Ψ|^2)∇Ψ)=λ_1ρ(|Ψ|^2)Ψ, x∈Ω. Indeed, we establish the existence of solutions for the above Schrödinger equation when λ belongs to a certain neighborhood of the first eigenvalue λ_1 of this eigenvalue problem. Themain feature of this paper is that the nonlinearity ω(|ψ|^2)ψ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter λ combined with the nonlinear nonhomogeneous term div(ρ(|∇ψ|^2)∇ψ) which leads us to treat this problem in an appropriateOrlicz space. The proofs are based on various techniques related to variational methods and implicit function theorem.

In the first part of the present paper we deal with the first boundary value problem for general second-order differential equation in plane angle. The criterion of non-trivial solvability is obtained for such problem in space C^2 of functions having polynomial growth at infinity. In the second part so-called "almost Cauchy" problem in a polygon for high order differential equation without respect of type is investigated. The necessary condition of uniqueness violation of solution is appeared to be sufficient in case of problem with one boundary condition.

We prove the existence of solitons (finite energy solitary wave) for a Boussinesq system that arise in the study of the evolution of small amplitude long water waves including surface tension. This Boussinesq system reduces to the generalized Benney-Luke equation and to the generalized Kadomtsev-Petviashivili equation in appropriate limits. The existence of solitons follows by a variational approach involving the Mountain Pass Theorem without the Palais-Smale condition. For surface tension sufficiently strong, we show that a suitable renormalized family of solitons of this model converges to a nontrivial soliton for the generalized KP-I equation.

In this work, we study the similarity analysis of the generalized nonlinear Burgers' equation having diffusivity and viscosity as a general functions of the particle velocity. We perform the symmetry analysis and exhibit exact solutions for various forms of the diffusivity and viscosity.

We prove the global existence of weak solutions of the one-dimensional compressible Navier-stokes equations with density-dependent viscosity. In particular, we assume that the initial density belongs to L^1 and L^∞, module constant states at x=-∞ and x=+∞, which may be different. The initial vacuum is permitted in this paper and the results may apply to the one-dimensional Saint-Venant model for shallow water.