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"Evolution equations"
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Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models
Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution
equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its
domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using
Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, we study the
existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, we use the center
manifold theorem to establish a Hopf bifurcation theorem for age structured models.
eBook
Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are C^3-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.
eBook
Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics
This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [
1
] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.
Journal Article
Nonlinear Evolution Equations
The book introduces the existence, uniqueness, regularity and the long time behavior of solutions with respect to space and time, and the explosion phenomenon for some evolution equations, including the KdV equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Zakharov equations, the Landau-Lifshitz equations, the Boussinesq equation, the Navier-Stokes equations and the Newton-Boussinesq equations etc., as well as the basic concepts and research methods of infinite-dimensional dynamical systems. This book presents fundamental elements and important advances in nonlinear evolution equations. It is intended for senior university students, graduate students, postdoctoral fellows and young teachers to acquire a basic understanding of this field, while providing a reference for experienced researchers and teachers in natural sciences and engineering technology to broaden their knowledge.
eBook
Localized solutions of (5+1)-dimensional evolution equations
In this research, the general linear evolution equations (EEs) in (5+1) dimensions are studied. All the mixed second-order derivatives are included in this aforementioned model. Using the Hirota bilinear operator and symbolic computation, the localized solutions–the abundant lump solutions are constructed. Particularly, it is found that only four groups of linear (5+1)-dimensional EEs are found that they have abundant lump solutions, and no interactions between the lump and other solutions are found via the positive definite quadratic functions. Finally, four examples corresponding to the above-mentioned cases are given to validate the obtained results, and the corresponding graphs are presented to show the dynamic behaviors of the abundant lump solutions of these given examples.
Journal Article
Critical non-linearity for some evolution equations with Fujita-type critical exponent
We consider the Cauchy problem for a class of non-linear evolution equations in the form L(∂t,∂x)u=F(∂tℓu),(t,x)∈[0,∞)×Rn;here, L(∂t,∂x) is a linear partial differential operator with constant coefficients, of order m≥1 with respect to the time variable t, and ℓ is a natural number satisfying 0≤ℓ≤m-1. For several different choices of L, many authors have investigated the existence of global (in time) solutions to this problem when F(s)=|s|p is a power non-linearity, looking for a critical exponentpc>1 such that global small data solutions exist in the supercritical case p>pc, whereas no global weak solutions exist, under suitable sign assumptions on the data, in the subcritical case 1
Journal Article
On growth and instability for semilinear evolution equations: an abstract approach
We propose a new approach to the study of (nonlinear) growth and instability for semilinear abstract evolution equations with compact nonlinearities. We show, in particular, that compact nonlinear perturbations of linear evolution equations can be treated as linear ones as far as the growth of their solutions is concerned. We obtain exponential lower bounds of solutions for initial values from a dense set in resolvent or spectral terms. The abstract results are applied, in particular, to the study of energy growth for semilinear backward damped wave equations.
Journal Article
The improved F-expansion method with Riccati equation and its applications in mathematical physics
The improved F-expansion method combined with Riccati equation is one of the most effective analytical methods in finding the exact traveling wave solutions to non-linear evolution equations in mathematical physics. In this article, this method is implemented to investigate new exact solutions to the Drinfel’d–Sokolov–Wilson (DSW) equation and the Burgers equation. The performance of this method is reliable, direct, and simple to execute compared to other existing methods. The obtained solutions in this work are imperative and significant for the explanation of some practical physical phenomena.
Journal Article
A long-wave model for a falling upper convected Maxwell film inside a tube
A long-wave asymptotic model is developed for a viscoelastic falling film along the inside of a tube; viscoelasticity is incorporated using an upper convected Maxwell model. The dynamics of the resulting model in the inertialess limit is determined by three parameters: Bond number Bo, Weissenberg number We and a film thickness parameter $a$. The free surface is unstable to long waves due to the Plateau–Rayleigh instability; linear stability analysis of the model equation quantifies the degree to which viscoelasticity increases both the rate and wavenumber of maximum growth of instability. Elasticity also affects the classification of instabilities as absolute or convective, with elasticity promoting absolute instability. Numerical solutions of the nonlinear evolution equation demonstrate that elasticity promotes plug formation by reducing the critical film thickness required for plugs to form. Turning points in travelling wave solution families may be used as a proxy for this critical thickness in the model. By continuation of these turning points, it is demonstrated that in contrast to Newtonian films in the inertialess limit, in which plug formation may be suppressed for a film of any thickness so long as the base flow is strong enough relative to surface tension, elasticity introduces a maximum critical thickness past which plug formation occurs regardless of the base flow strength. Attention is also paid to the trade-off of the competing effects introduced by increasing We (which increases growth rate and promotes plug formation) and increasing Bo (which decreases growth rate and inhibits plug formation) simultaneously.
Journal Article