Advertisement

A Breather Construction for a Semilinear Curl-Curl Wave Equation with Radially Symmetric Coefficients

  • Michael PlumEmail author
  • Wolfgang Reichel
Original Paper

Abstract

We consider the semilinear curl-curl wave equation
$$s\left( x \right)\partial _t^2U + \nabla \times \nabla \times + q\left( x \right)U \pm V\left( x \right){\left| U \right|^{p - {\text{ 1}}}}U = 0for\left( {x,t} \right) \in {\mathbb{R}^3} \times \mathbb{R}.$$
For any p < 1 we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the + and the - case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U(x; t + a(x)), where U is a particular breather and a: ℝ3 → ℝ an arbitrary radially symmetric C2-function.

2000 Mathematics Subject Classification

Primary: 35L71 Secondary: 34C25 

Key words and phraes

Axially symmetric gravitational fields boundary value problem existence and uniqueness of solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30 (1973), 1262–1264.MathSciNetCrossRefGoogle Scholar
  2. [2]
    G. T. Adamashvili and D. J. Kaup, Optical breathers in nonlinear anisotropic and dispersive media. Phys. Rev. E 73 (2006), 066613.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Stanley Alama and Yan Yan Li, Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differential Equations 96 (1992), 89–115.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Antonio Azzollini, Vieri Benci, Teresa D’Aprile, and Donato Fortunato, Existence of static solutions of the semilinear maxwell equations. Ricerche di Matematica 55 (2006), 123–137.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Th. Bartsch and J. Mederski, Nonlinear time-harmonic maxwell equations in an anisotropic bounded medium. arXiv:1509.01994[math.AP].Google Scholar
  6. [6]
    Thomas Bartsch, Tomáš Dohnal, Michael Plum, and Wolfgang Reichel, Ground states of a nonlinear curl-curl problem in cylindrically symmetric media. NoDEA Nonlinear Differential Equations Appl. 23 (2016), 23–52.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Vieri Benci and Donato Fortunato, Towards a unified field theory for classical electrodynamics. Arch. Ration. Mech. Anal. 173 (2004), 379–414.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), 313–345.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Bjöorn Birnir, Henry P. McKean, and Alan Weinstein, The rigidity of sine-Gordon breathers. Comm. Pure Appl. Math. 47 (1994), 1043–1051.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Carsten Blank, Martina Chirilus-Bruckner, Vincent Lescarret, and Guido Schneider, Breather solutions in periodic media. Comm. Math. Phys. 302 (2011), 815–841.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Martina Chirilus-Bruckner and Clarence Eugene Wayne, Inverse spectral theory for uniformly open gaps in a weighted Sturm-Liouville problem. J. Math. Anal. Appl. 427 (2015), 1168–1189.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Teresa D’Aprile and Gaetano Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations. Adv. Differential Equations 16 (2011), 435–466.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Jochen Denzler, Nonpersistence of breather families for the perturbed sine Gordon equation. Comm. Math. Phys. 158 (1993), 397–430.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Hirsch and W. Reichel, Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients. arXiv:1606.04415[math.AP].Google Scholar
  15. [15]
    Guillaume James, Bernardo Sánchez-Rey, and Jesúus Cuevas, Breathers in inhomogeneous nonlinear lattices: an analysis via center manifold reduction. Rev. Math. Phys. 21 (2009), 1–59.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7 (1994), 1623–1643.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Jaros law Mederski, Ground states of time-harmonic semilinear Maxwell equations in ℝ3 with vanishing permittivity. Arch. Ration. Mech. Anal. 218 (2015), 825–861.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73 (2005), 259–287.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Dmitry E. Pelinovsky, Gideon Simpson, and Michael I. Weinstein, Polychromatic solitary waves in a periodic and nonlinear Maxwell system. SIAM J. Appl. Dyn. Syst. 11 (2012), 478–506.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Walter A. Strauss, Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing 2016

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology (KIT)Institute for AnalysisKarlsruheGermany

Personalised recommendations