Three Dimensional Vortices in the Nonlinear Wave Equation

Marino Badiale; Vieri Benci; Sergio Rolando

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 1, page 105-134
  • ISSN: 0392-4041

Abstract

top
We prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation - Δ u + μ | y | 2 u + λ u = g ( u ) , u H 1 ( N ) , N u 2 | y | 2 𝑑 x < , where x = ( y , z ) k × N - k , N > k 2 , μ > 0 and λ 0 . The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to ( ) with bounded Palais-Smale sequences.

How to cite

top

Badiale, Marino, Benci, Vieri, and Rolando, Sergio. "Three Dimensional Vortices in the Nonlinear Wave Equation." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 105-134. <http://eudml.org/doc/290553>.

@article{Badiale2009,
abstract = {We prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation \begin\{equation\} \tag\{\dag\} -\Delta u + \frac\{\mu\}\{|y|^\{2\}\} u + \lambda u = g(u), \quad u \in H^\{1\}(\mathbb\{R\}^\{N\}), \quad \int\_\{\mathbb\{R\}^\{N\}\} \frac\{u^\{2\}\}\{|y|^\{2\}\} \, dx < \infty,\end\{equation\} where $x=(y,z) \in \mathbb\{R\}^\{k\} \times \mathbb\{R\}^\{N-k\}$, $N > k \ge 2$, $\mu > 0$ and $\lambda \ge 0$. The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to $(\dag)$ with bounded Palais-Smale sequences.},
author = {Badiale, Marino, Benci, Vieri, Rolando, Sergio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {105-134},
publisher = {Unione Matematica Italiana},
title = {Three Dimensional Vortices in the Nonlinear Wave Equation},
url = {http://eudml.org/doc/290553},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Badiale, Marino
AU - Benci, Vieri
AU - Rolando, Sergio
TI - Three Dimensional Vortices in the Nonlinear Wave Equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 105
EP - 134
AB - We prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation \begin{equation} \tag{\dag} -\Delta u + \frac{\mu}{|y|^{2}} u + \lambda u = g(u), \quad u \in H^{1}(\mathbb{R}^{N}), \quad \int_{\mathbb{R}^{N}} \frac{u^{2}}{|y|^{2}} \, dx < \infty,\end{equation} where $x=(y,z) \in \mathbb{R}^{k} \times \mathbb{R}^{N-k}$, $N > k \ge 2$, $\mu > 0$ and $\lambda \ge 0$. The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to $(\dag)$ with bounded Palais-Smale sequences.
LA - eng
UR - http://eudml.org/doc/290553
ER -

References

top
  1. AMBROSETTI, A. - RABINOWITZ, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Zbl0273.49063MR370183
  2. AMBROSETTI, A. - STRUWE, M., Existence of steady vortex rings in an ideal fluid, Arch. Rational Mech. Anal., 108 (1989), 97-109. Zbl0694.76012MR1011553DOI10.1007/BF01053458
  3. BADIALE, M. - BENCI, V. - ROLANDO, S., Solitary waves: physical aspects and mathematical results, Rend. Sem. Math. Univ. Pol. Torino, 62 (2004), 107-154. Zbl1120.37045MR2131956
  4. BADIALE, M. - BENCI, V. - ROLANDO, S., A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. Zbl1149.35033MR2314102DOI10.4171/JEMS/83
  5. BADIALE, M. - ROLANDO, S., Vortices with prescribed L 2 norm in the nonlinear wave equation, preprint 2008. Zbl1172.35064MR2454877DOI10.1515/ans-2008-0410
  6. BADIALE, M. - TARANTELLO, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Rational Mech. Anal., 163 (2002), 259-293. Zbl1010.35041MR1918928DOI10.1007/s002050200201
  7. BENCI, V. - D'APRILE, T., The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, 184 (2002), 109-138. MR1929149DOI10.1006/jdeq.2001.4138
  8. BENCI, V. - FORTUNATO, D., Existence of 3D-vortices in abelian gauge theories, Mediterr. J. Math., 3 (2006), 409-418. Zbl1167.35351MR2274734DOI10.1007/s00009-006-0087-5
  9. BENCI, V. - FORTUNATO, D., Solitary waves in the nonlinear wave equation and in gauge theories, J. Fixed Point Theory Appl., 1 (2007), 61-86. Zbl1122.35121MR2282344DOI10.1007/s11784-006-0008-z
  10. BENCI, V. - FORTUNATO, D., Vortices in abelian gauge theories, work in progress. Zbl1173.81013
  11. BENCI, V. - VISCIGLIA, N., Solitary waves with non vanishing angular momentum, Adv. Nonlinear Stud., 3 (2003), 151-160. Zbl1030.35051MR1955598DOI10.1515/ans-2003-0104
  12. BERESTYCKI, H. - LIONS, P. L., Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. Zbl0533.35029MR695535DOI10.1007/BF00250555
  13. BERGER, M. S., On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Funct. Analysis, 9 (1972), 249-261. Zbl0224.35061MR299966
  14. BRIHAYE, Y. - HARTMANN, B. - ZAKRZEWSKI, W. J., Spinning solitons of a modified non-linear Schroedinger equation, Phys. Rev. D, 69, 087701. MR2094947DOI10.1103/PhysRevD.69.087701
  15. COLEMAN, S. - GLASER, V. - MARTIN, A., Action minima among solutions to a class of euclidean scalar field equation, Comm. Math. Phys, 58 (1978), 211-221. MR468913
  16. COLEMAN, S., Q-Balls, Nucl. Phys. B, 262 (1985), 263-283. MR819656DOI10.1016/0550-3213(85)90286-X
  17. CRASOVAN, L. C. - MALOMED, B. A. - MIHALACHE, D., Spinning solitons in cubic-quintic nonlinear media, Pramana, 57 (2001), 1041- 1059. 
  18. D'APRILE, T., On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations, Diff. Integral Eq., 16 (2003), 349-384. MR1947957
  19. DERRICK, G. H., Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5 (1964), 1252-1254. MR174304DOI10.1063/1.1704233
  20. GIAQUINTA, M., Introduction to regularity theory for nonlinear elliptic systems, Birkhäuser Verlag, 1993. Zbl0786.35001MR1239172
  21. KIM, C. - KIM, S. - KIM, Y., Global nontopological vortices, Phys. Rev. D, 47 (1985), 5434-5443. 
  22. KUSENKO, A. - SHAPOSHNIKOV, M., Supersymmetric Q-balls as dark matter, Phys. Lett. B, 418 (1998), 46-54. 
  23. KUZIN, I. - POHOŽAEV, S., Entire solutions of semilinear elliptic equations, PNLDE, vol. 33, Birkhäuser, 1997. 
  24. LIONS, P. L., Solutions complexes d'équations elliptiques semilinéaires dans N , C. R. Acad. Sci. Paris, série I, 302 (1986), 673-676. Zbl0606.35027MR847751
  25. PALAIS, R. S., The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. Zbl0417.58007MR547524
  26. RAJARAMAN, R., Solitons and instantons, North-Holland Physics Publishing, 1987. MR719693
  27. ROSEN, G., Particlelike solutions to nonlinear complex scalar field theories with positive-definite energy densities, J. Math. Phys., 9 (1968), 996-998. 
  28. SOLIMINI, S., A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. Henry Poincaré - Analyse non linéaire, 12 (1995), 319-337. Zbl0837.46025MR1340267DOI10.1016/S0294-1449(16)30159-7
  29. SHATAH, J., Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys., 91 (1983), 313-327. Zbl0539.35067MR723756
  30. SHATAH, J., Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. Zbl0617.35072MR792821DOI10.2307/2000308
  31. STRAUSS, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-172. Zbl0356.35028MR454365
  32. STRAUSS, W. A., Nonlinear invariant wave equations, Lecture Notes in Phisics, vol. 23, Springer, 1978. MR498955
  33. VOLKOV, M. S., Existence of spinning solitons in field theory, eprint arXiv:hep-th/ 0401030 (2004). 
  34. VOLKOV, M. S. - WÖHNERT, E., Spinning Q-balls, Phys. Rev. D, 66 (2002) 085003. 
  35. WILLEM, M., Minimax theorems, PNLDE, vol. 24, Birkhäuser, 1996. MR1400007DOI10.1007/978-1-4612-4146-1
  36. WITHAM, G. B., Linear and nonlinear waves, John Wiley & Sons, 1974. MR483954

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.