Resumo: A desnaturação térmica do DNA, ou seja, a separação das duas cadeias, é um fenômeno causado pela amplitude das vibrações das bases, portanto, é necessário saber como essa separação é gerada para implementar o modelo alternativo do comportamento de fusão como uma função da sequência de nucleotídeos e terapias para combater o câncer. Propomos usar o modelo de Peyrard-Bishop (PB) de DNA não linear estendido para incluir um potencial enarmônico que represente a interação de empilhamento aromático entre n-e (n-1)-enésimos pares de bases consecutivos para tratar o problema. Nós usamos métodos de diferença finita para determinar o valor médio do deslocamento para o potencial "corcunda de Morse" do modelo Peyrard-Bishop do DNA. Mostramos como o "pseudo-Schrödinger" estendido combinado com método de diferença finita pode ser usado para obter os deslocamentos do valor médio para a desnaturação térmica do DNA com o potencial Morse "da corcunda". Palavras-chave:Método das diferenças finitas, equação pseudo-Schrödinger, "corcunda" do potencial de Morse ___________________________________________________________________________ Abstract: The thermal denaturation of DNA, i.e. the separation of the two strands is a phenomenon caused by the amplitude of the vibrations of the bases, therefore it is necessary to know how such separation is generated in order to implement alternative model of the melting behavior as a function of nucleotide sequence and therapies to combat the cancer. We propose to use the extended nonlinear Peyrard-Bishop(PB) model of DNA to include an anharmonic potential representing the aromatic stacking interaction between n-and (n-1)-th consecutive base pairs to treat the problem. We use Finite-difference methods for determine the mean value of the displacement for the "hump Morse" potential of the Peyrard-Bishop model of DNA. We show how the extended "pseudoSchrödinger" combined with finite difference method can be used to obtain the mean value displacements for the thermal denaturation of DNA with "hump" Morse potential.

Objectives. Analyze the DNA dynamics in PeyrardBishop-Dauxois model (PBD) with different control parameters using its energy center of the mobile "breather". Materials and methods. We used the Peyrard-Bishop-Dauxois mathematical model and the MATLAB software for studying the DNA dynamic using Morse potential, Symmetric Morse and the "hump" potential for simulating the interactions which arise the pile up. Results. It has been observed that the analytical and computational methods allow to detect the influence of the potentials of the PBD model in the behavior of the energy center in the presence of a couple of base A(adenine) or T(thymine) using the control of parameter α=-0.30 and velocity of mobile breather: v 0 =0.1. In the case of Morse potential, the center of energy respect to the mobile breather undergoes a change in its trajectory and produce a DNA breathing. Conclusions. Analytical and computational approaches can be used for obtaining differences respect to the DNA dynamics using different control parameters: velocity of BM and inhomogeneity. The potential "hump" may decrease the reflective effect with the indicated parameters to the effect on the energy center to the mobile breather.

In this article, we deal with the nonlinear Schrödinger equation with nonlocal regional diffusion false(−normalΔfalse)ραu+Vfalse(xfalse)u=ffalse(x,ufalse)0.30emin0.30emℝN,0.30emu∈Hαfalse(ℝNfalse), where 0 < α < 1, n ≥ 2, and f:ℝN×ℝ→ℝ is a continuous function. The operator false(−normalΔfalse)ρα is a variational version of the nonlocal regional Laplacian defined as ∫ℝN(−Δ)ραu(x)φ(x)dx=∫ℝN∫B(0,ρ(x))[u(x+z)−u(x)][φ(x+z)−φ(x)]|z|N+2αdzdx, where ρ∈Cfalse(ℝN,ℝ+false) be a positive function. Considering that ρ, V, and f(· , t) are periodic or asymptotically periodic at infinity, we prove the existence of ground state solution of () by using Nehari manifold and comparison method. Furthermore, in the periodic case, by combining deformation‐type arguments and Lusternik–Schnirelmann theory, we prove that problem () admits infinitely many geometrically distinct solutions.

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