The Cauchy–Riemann equations in the unit ball of l^{2}

We study the local exactness of the \(\overline{\partial }\) operator in the Hilbert space \(l^2\) for a particular class of \((0,1)\) -forms \(\omega \) of the type \(\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i\) , \(z = (z_i)\) in \(l^2\) . We suppose each function \(\omega ^i\) of class \(C^\infty \) in the closed unit ball of \(l^2\) , of the form \(\omega ^i(z) = \sum _k \omega ^i_k\left( z^k\right) \) , where \(\mathbf N = \bigcup I_k\) is a partition of \(\mathbf N\) , \((\) card \(I_k < +\infty )\) and \(z^k\) is the projection of \(z\) on \(\mathbf C^{I_k}\) . We establish sufficient conditions for exactness of \(\omega \) related to the expansion in Fourier series of the functions \(\omega ^i_k\) .     

On {phi}-contractibility of the Lebesgue–Fourier algebra of a locally compact group

For a locally compact group G, we present some characterizations for f{phi}-contractibility of the Lebesgue–Fourier algebra LA(G){mathcal{L}A(G)} endowed with convolution or pointwise product.     

On the nonvanishing of abstract Cauchy–Riemann cohomology groups

In this paper we prove infinite dimensionality of some local and global cohomology groups on abstract Cauchy–Riemann manifolds.     

Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces

We establish an analog of Hörmander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

Complicated asymptotic behavior of solutions for the Cauchy problem of Cahn–Hilliard equation

In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.     

Two-soliton solutions of the complex short pulse equation via Riemann–Hilbert approach

Based on the basic conclusions of the Riemann–Hilbert method for solving the initial value problem of the complex short-pulse equations, the general form of the two-soliton solutions of the complex short pulse equation is given in this paper. Under the two different assumptions of the scattering coefficient, the expression of the two-soliton solutions of the equation is given specifically.     

On exact solutions to the Kolmogorov–Feller equation

The integrodifferential Kolmogorov–Feller equation describing the stochastic dynamics of a system subjected to a regular “force” and a random external disturbance in the form of short pulses with random “amplitudes” and occurrence times is considered. The equation is written in differential form. A method for finding the regular force from a given stationary probability distribution is described. The method is illustrated by examples.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

Riemann–Hilbert approach and $$N$$ -soliton solutions of the two-component Kundu–Eckhaus equation

Theoretical and Mathematical Physics - We study the two-component Kundu–Eckhaus equation with a zero boundary condition at infinity. Based on the spectral analysis of the Lax pair, a...     

Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.