Nodal Multi-peak Standing Waves of Fourth-Order Schrödinger Equations with Mixed Dispersion

We consider the existence and concentration properties of standing waves for a fourth-order Schrödinger equation with mixed dispersion, which was introduced to regularize and stabilize solutions to the classical time-dependent Schrödinger equation. This leads to study multi-peak solutions to the following singularly perturbed fourth-order nonlinear Schrödinger equation $$\begin{aligned} {\varepsilon ^{\text {4}}}{\Delta ^{\text {2}}}u - \beta {\varepsilon ^2}\Delta u + V(x)u = |u{|^{p - 2}}u{\text { in }}{\mathbb {R}^N},{\text { }}u \in {H^2}({\mathbb {R}^N}). \end{aligned}$$ ε 4 Δ 2 u - β ε 2 Δ u + V ( x ) u = | u | p - 2 u in R N , u ∈ H 2 ( R N ) . We first establish a local $${W^{4,p}}$$ W 4 , p -estimate for a class of fourth-order semilinear elliptic equations, which is a key to get the uniform and global $${L^\infty }$$ L ∞ -estimate of solutions to the considered singularly perturbed equation above. Next, under certain assumptions on $$\beta $$ β and the potential V(x), we construct a family of sign-changing multi-peak solutions with a unique maximum (or minimum) point on each component. We prove that these solutions concentrate around any prescribed finite set of local minima (possibly degenerate) of the potential V(x). Compared with the classical singularly perturbed Schrödinger equation, the presence of a fourth-order term in the problem above forces the development of new techniques to obtain qualitative properties of multi-peak solutions.

Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.