Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Abstract

In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the coexistence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

Complex dynamics in a quasi-periodic plasma perturbations model

“…Our research results reveal a classical Hopf bifurcation in the studied system, and play an important role for the better understanding of the complex dynamics of QPP model subject to time delay. Finally, we mention that when the delay is continuous and modeled by a convolution, the problem on the periodic phenomenon can be restricted on the critical manifold, the limit cycle can be detected by the zeros of Melnikov function, see [17], [18], [19].…”

“…Recall that a canard is a solution of a singularly perturbed dynamical system (16) following the stable invariant manifold M s , passing near a bifurcation point located on the fold of the slow invariant manifold M 0 , and then following the unstable invariant manifold M u . Consider the differential-algebraic system (18…”

An exact expression of positive periodic solution for a first-order singular equation

“…As is well known, singular equations have a wide range of applications in many fields, and the existence of positive ω-periodic solutions to singular equations plays a significant role in solving many practical problems. There is a good amount of work on periodic solutions for singular equations (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references cited therein). In 2003, Agarwal and O'Regan [4] provided some results on positive ω-periodic solutions of Eq.…”

Weak and strong singularities problems to Liénard equation

“…Lazer and Solimini's work has attracted the attention of many scholars in singular equations. More recently, the Poincaré-Birkhoff twist theorem [2][3][4], Schauder's fixed point theorem [5][6][7][8], the Leray-Schauder alternative principle [9][10][11], coincidence degree theory [12][13][14][15],the Krasnoselskii fixed point theorem in cones [16,17] and Leray-Schauder degree theory [18,19] have been employed to discuss the existence of a positive periodic solution of singular equations.…”