Pulse-like spatial patterns described by higher-order model equations
type: Item
instance of: scholarly article
title: Pulse-like spatial patterns described by higher-order model equations
zbMATH Open document ID: 0920.34048
author: A. I. Rotariu-Bruma
author: W. C. Troy
author: Lambertus A. Peletier
published in: Journal of Differential Equations
publication date: 1999-09-09T00:00:00Z
review text: Consider the scalar differential equation \[ \gamma \frac{d^4 u}{dx^4} = \frac{d^2 u}{dx^2} + f(u) \quad\text{for }\gamma > 0, \tag{\(*\)} \] under the assumptions: \((H_1)\) \(f \in C^2 (R,R)\) has at least two zeroes \(s=a\) and \(s=1\) \((a <1)\). \((H_2)\) \(f > 0\) on \((a,1)\), \(f' (a) > 0\), \(f' (1):= - \sigma < 0\). Let \(F(u):= \int^1_u f(s)ds\) and let \(b\) be the infimum of \(s<1\) such that \(F>0\) on \((s,1)\). \((H_3)\) \(-\infty < b < a\), \(f <0\) on \([b,a)\). \((H_4)\) \(f(s) \leq f'(1)(s-1)\) for \(s \in [b,1)\). The authors look for solutions to \((*)\) satisfying \((u,du/dx , d^2 u/dx^2 ,d^3 u/dx^3) \rightarrow (1,0,0,0)\) as \(x\rightarrow \pm \infty\) that is, \(u\) tends to the stable equilibrium point of the system \(du/dt = f(u)\). These homoclinic solutions are called solitons. In particular, the authors look for solitons which are symmetric with respect to \(x=0\). Under the conditions above, the authors prove the existence of even solitons for \(0 < \gamma \leq 1/4\sigma\) by means of a shooting argument and exploiting the property that \((*)\) has a first integral. Moreover, they estimate the amplitude \(u(0)\) and investigate the asymptotic behavior for \(x\rightarrow \pm \infty\). Under additional assumptions, the uniqueness of a symmetric soliton is proven. Furthermore, it is proved that for all \(\gamma > 0\) there is a family of periodic solutions to \((*)\). The obtained results are applied to a generalized nonlinear Schrödinger equation which arises in the theory of pulse propagation through optical fibers.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 1241037
zbMATH Keywords: symmetric homoclinic solutions
zbMATH Keywords: existence of even solitons
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: periodic solution
zbMATH Keywords: generalized nonlinear Schrödinger equation
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3480
OpenAlex ID: W1998112328
cites work: Homoclinic orbits in the dynamic phase-space analogy of an elastic strut
cites work: Multidimensional nonlinear diffusion arising in population genetics
cites work: Uniqueness of solutions for the extended Fisher-Kolmogorov equation
cites work: Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system
cites work: Global existence of homoclinic and periodic orbits for a class of autonomous Hamiltonian systems
cites work: A global investigation of solitary-wave solutions to a two-parameter model for water waves
cites work: Pattern formation outside of equilibrium
cites work: Q4136233
cites work: The approach of solutions of nonlinear diffusion equations to travelling front solutions
cites work: Q4297497
cites work: Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems
cites work: Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria
cites work: Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation
cites work: Existence of solitary waves for water-wave models
cites work: Q5768532
cites work: Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: Kinks
cites work: Spatial Patterns Described by the Extended Fisher--Kolmogorov Equation: Periodic Solutions
cites work: Chaotic spatial patterns described by the extended Fisher-Kolmogorov equation
cites work: A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation
cites work: Q4888372
cites work: Q5533735
cites work: Homoclinic period blow-up in reversible and conservative systems
DOI: 10.1006/JDEQ.1998.3480
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description: scientific article; zbMATH DE number 1241037