We analyze the existence of dark solitons in nonlinear complex Ginzburg-Landau equations. We prove existence results concerned with the initial value problem for these equations in Zhidkov spaces using a new approach with splitting methods. MSC 2010. 47J35.

In this paper, we study the initial value problem for infinite dimensional fractional non-autonomous reaction-diffusion equations. Applying general timesplitting methods, we prove the existence of solutions globally defined in time using convex sets as invariant regions. We expose examples, where biological and pattern formation systems, under suitable assumptions, achieve global existence.We also analyze the asymptotic behavior of solutions.

We consider the so-called complex Ginzburg-Landau equations with a polynomial nonlinearity in the real line. We prove existence results concerned with the initial value problem for these equations in Zhidkov spaces with a new approach using splitting methods.

In this article, we will analyze the existence of Peregrine type solutions for the fractional diffusion reaction equation by applying Splitting-type methods. These functions that have two main characteristics, they are direct sum of functions of periodic type and functions that tend to zero at infinity. Global existence results are obtained for each particular characteristic, for then finally combining both results.

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