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We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation

We consider the wave maps problem with domain

The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

We study the following singularly perturbed problem

A generalization of the regularized long-wave equation is considered, and the existences of smooth soliton, peakon, and periodic solutions are established via the complete discrimination system for polynomial method and the bifurcation method. Concrete examples of these solutions are constructed to verify our conclusions directly. In particular, we construct a special kind of smooth soliton solution, namely a Gaussian soliton solution, and give two sufficient conditions for the existence of such a solution by the extended trial equation method. To the best of our knowledge, this is the first time that a Gaussian soliton solution has been constructed for an equation with no logarithmic nonlinearity.

Variable coefficients (3+1)-generalized shallow water wave equation (GSWE) is investigated via modified Hirota bilinear method. This method is presented for the first time. Compared with other methods, it solves solution without setting solution and calculates transformations without making logarithmic transformations. The rational transformation is first utilized to transform GSWE. According to homogeneous balance principle, the relation between F and G in rational transformation can be calculated by utilizing. Solutions that included rogue wave solutions, interaction solutions, breather solutions and so on, are obtained and depicted graphically. Figures are given out to the dynamic characteristics of the solution. Furthermore, the results obtained demonstrate that this approach is more direct, generalized, effective and holds for many nonlinear partial differential equations.

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \\sqrt{-1}\\, u_{t}=u_{xx}-M_{\\xi}u+\\varepsilon|u|^2u, subject to Dirichlet boundary conditions u(t,0)=u(t,\\pi)=0, where M_{\\xi} is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier M_{\\xi}, any solution with the initial datum in the \\delta-neighborhood of a KAM torus still stays in the 2\\delta-neighborhood of the KAM torus for a polynomial long time such as |t|\\leq \\delta^{-\\mathcal{M}} for any given \\mathcal M with 0\\leq \\mathcal{M}\\leq C(\\varepsilon), where C(\\varepsilon) is a constant depending on \\varepsilon and C(\\varepsilon)\\rightarrow\\infty as \\varepsilon\\rightarrow0.

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

With respect to oceanic fluid dynamics, certain models have appeared, e.g., an extended time-dependent (3+1)-dimensional shallow water wave equation in an ocean or a river, which we investigate in this paper. Using symbolic computation, we find out, on one hand, a set of bilinear auto-Bäcklund transformations, which could connect certain solutions of that equation with other solutions of that equation itself, and on the other hand, a set of similarity reductions, which could go from that equation to a known ordinary differential equation. The results in this paper depend on all the oceanic variable coefficients in that equation.

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function.