We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz Zakharov–Shabat system and Kubelka–Munk equations. show that they all can be reduced to solving one simple antilinear ordinary differential equation u′x=fxux¯ or its nonhomogeneous version u′x=fxux¯+gx, x∈0,x0⊂R. point out some of the advantages proposed reformulation c...

Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in fully periodic case with initial data Sobolev spaces $H^s$, $s>1$, is proved. Frequency dependent time localization utilized to control derivative nonlinearity. The new ingredient improve on previous results a nonlinear Loomis-Whitney-type inequality.

The generalized Zakharov system couples a dispersive field E (scalar or vectorial) and J nondispersive fields {nj}j=1 with a propagating speed of 1/2j . In this paper, we extend our one-dimensional time-splitting spectral method (TSSP) for the generalized Zakharov system into higher dimension. A main new idea is to reformulate the multidimensional wave equations for the nondispersive fields int...

Dimension breaking occurs when the solution of a nonlinear partial differential equation (PDE) depending on n independent variables bifurcates to one depending on n + 1. A central hypothesis in the theory of dimension breaking is that a certain operator should have a non-zero purely imaginary eigenvalue. This hypothesis is difficult to verify in general. We present a geometric theory for verify...

some necessary and sufficient conditions are given for the existence of a g-positive (g-repositive) solution to adjointable operator equations $ax=c,axa^{left( astright) }=c$ and $axb=c$ over hilbert $c^{ast}$-modules, respectively. moreover, the expressions of these general g-positive (g-repositive) solutions are also derived. some of the findings of this paper extend some known results in the...

The main idea of this paper is to study the chaotic behavior Zakharov–Kuznetsov equation with perturbation. By taking traveling wave transformation, we transform perturbed dual-power law and triple-power nonlinearity into planar dynamic systems, then analyze how external terms affect behavior. We emphasize here that there no phenomenon for non-perturbed ZK equation, thus it only caused by terms.

this article is devoted to study of the autoconvolution equations and generalized mittag-leffler functions. these types of equations are given in terms of the laplace transform convolution of a function with itself. we state new classes of the autoconvolution equations of the first kind and show that the generalized mittag-leffler functions are solutions of these types of equations. in view of ...

adomian decomposition method has been applied to solve many functional equations so far. in this article, we have used this method to solve the fuzzy differential equation under generalized differentiability. we interpret a fuzzy differential equation by using the strongly generalized differentiability. also one concrete application for ordinary fuzzy differential equation with fuzzy input data...