Search results for "Nonlinear system"

On the numerical evaluation of algebro-geometric solutions to integrable equations

2011

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis no…

Nonlinear Eigenvalue Problems of Schrödinger Type Admitting Eigenfunctions with Given Spectral Characteristics

2002

The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form in a real Hilbert space ℋ with a semi-bounded self-adjoint operator A0, while for every y from a dense subspace X of ℋ, B(y ) is a symmetric operator. The left-hand side is assumed to be related to a certain auxiliary functional ψ, and the associated linear problems are supposed to have non-empty discrete spectrum (y ∈ X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (∗) on a sphere SR ≔ {y ∈ X | ∥y∥ℋ = R} whose ψ-value is the n-th Ljusternik-Schnirelman level of ψ| and whose corresponding eigenvalue is t…

Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

2014

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.

Dynamics of closed ecosystems described by operators

2014

Abstract We adopt the so-called occupation number representation , originally used in quantum mechanics and recently adopted in the description of several classical systems, in the analysis of the dynamics of some models of closed ecosystems. In particular, we discuss two linear models, for which the solution can be found analytically, and a nonlinear system, for which we produce numerical results. We also discuss how a dissipative effect could be effectively implemented in the model.

Nonlinear scalar field equations with general nonlinearity

2018

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. T…

Knot placement in the Distributed non-linear lag models (DNLM) framework

2021

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

2008

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Doubly nonlinear periodic problems with unbounded operators

2004

Abstract The solvability of the evolution system v ′( t )+ B ( t ) u ( t )∋ f ( t ), v ( t )∈ A ( t ) u ( t ), 0 t T , with the periodic condition v (0)= v ( T ) is investigated in the case where A (t) are bounded, possibly degenerate, subdifferentials and B (t) are unbounded subdifferentials.

Unifying vectors and matrices of different dimensions through nonlinear embeddings

2020

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, derivi…

Coupled common fixed point theorems in partially ordered G-metric spaces for nonlinear contractions

2014

The aim of this paper is to prove coupled coincidence and coupled common fixed point theorems for a mixed $g$-monotone mapping satisfying nonlinear contractive conditions in the setting of partially ordered $G$-metric spaces. Present theorems are true generalizations of the recent results of Choudhury and Maity [Math. Comput. Modelling 54 (2011), 73-79], and Luong and Thuan [Math. Comput. Modelling 55 (2012) 1601-1609].