Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations

The Lax integrability of the coupled KdV equations derived from two-layer fluids [S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids, J. Phys. A: Math. Gen. 39 (2006) 513-527] is investigated by means of prolongation technique. As a result, the Lax pairs of some Painlevé integrable coupled KdV equations and several new coupled KdV equations are obtained. Finally, the Miura transformations and some coupled modified KdV equations associated with the Lax integrable coupled KdV equations are derived by an easy way.     

On a class of solutions of KdV

Practically every book on the Inverse Scattering Transform method for solving the Cauchy problem for KdV and other integrable systems refers to this method as nonlinear Fourier transform. If this is indeed so, the method should lead to a nonlinear analogue of the Fourier expansion formula . In this paper a special class of solutions of KdV whose role is similar to that of e^i(kx-ω(k)t) is discussed. The theory of these solutions, referred to here as harmonic breathers, is developed and it is shown that these solutions may be used to construct more general solutions of KdV similarly to how the functions e^i(kx-ω(t)) are used to perform the same task in the theory of Fourier transform. A nonlinear superposition formula for general solutions of KdV similar to the Fourier expansion formula is conjectured.     

Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations

Zero curvature formulations, pseudo-potentials, modified versions, “Miura transformations”, conservation laws, and nonlocal symmetries of the Korteweg–de Vries, Camassa–Holm and Hunter–Saxton equations are investigated from a unified point of view: these three equations belong to a two-parameter family of equations describing pseudo-spherical surfaces, and therefore their basic integrability properties can be studied by geometrical means.      

Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Sufficient and necessary conditions for the embeddings between Besov spaces and modulation spaces are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces is shown.     

Birkhoff coordinates for KdV on phase spaces of distributions

The purpose of this paper is to extend the construction of Birkhoff coordinates for the KdV equation from the phase space of square integrable 1-periodic functions with mean value zero to the phase space of mean value zero distributions from the Sobolev space endowed with the symplectic structure More precisely, we construct a globally defined real-analytic symplectomorphism where is a weighted Hilbert space of sequences supplied with the canonical Poisson structure so that the KdV Hamiltonian for potentials in is a function of the actions alone.     

The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability

Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin-Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices.     

Antiplectic Structure, Diffeomorphism and Generalized KdV Family

In this paper we show that the generalized KdV, generalized Camassa–Holm equations and the corresponding Möbius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by on the space of differential operators and on the space of immersion curves, respectively. These are Euler–Poincaré type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler–Poincaré flow with respect to metric, and this induces generalized Camassa–Holm equation. In the final section we discuss the Antiplectic pair in dimensions.Dedicated to Professor George Wilson on his 65th birthday with great respect and admiration.     

Convex hypersurfaces and L estimates for Schrödinger equations

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the L^p-L^q estimate of the solution operator in the free case. This estimate, combined with the results of fractionally integrated groups, allows us to further obtain the L^p estimate of solutions for the initial data belonging to a dense subset of L^p in the case of integrable potentials.     

Cauchy problem for the Ostrovsky equation in spaces of low regularity

In this article we consider the initial value problem for the Ostrovsky equation:

     

Spreading speeds and traveling waves for periodic evolution systems

The theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case. Then these abstract results are applied to a periodic system modeling man-environment-man epidemics, a periodic time-delayed and diffusive equation, and a periodic reaction-diffusion equation on a cylinder.     

Dynamics of two-compartment Gray-Scott equations

In this work the existence of a global attractor is proved for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension n≤3. The grouping estimation method combined with a new decomposition approach is introduced to overcome the difficulties in proving the absorbing property and the asymptotic compactness of this four-component reaction-diffusion systems with cubic autocatalytic nonlinearity and linear coupling. The finite dimensionality of the global attractor is also proved.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.