This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

## References

J. Bourgain, “Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations,”

*Annals of Mathematics*, vol. 148, no. 2, pp. 363–439, 1998. MR1668547 0928.35161 10.2307/121001 J. Bourgain, “Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations,”*Annals of Mathematics*, vol. 148, no. 2, pp. 363–439, 1998. MR1668547 0928.35161 10.2307/121001 J. Bourgain, “Periodic solutions of nonlinear wave equations,” in

*Harmonic Analysis and Partial Differential Equations*, Chicago Lectures in Mathematics, pp. 69–97, University of Chicago Press, Chicago, Ill, USA, 1999. MR1743856 0976.35041 J. Bourgain, “Periodic solutions of nonlinear wave equations,” in*Harmonic Analysis and Partial Differential Equations*, Chicago Lectures in Mathematics, pp. 69–97, University of Chicago Press, Chicago, Ill, USA, 1999. MR1743856 0976.35041 J. Bourgain,

*Green's Function Estimates for Lattice Schrödinger Operators and Applications*, vol. 158 of*Annals of Mathematics Studies*, Princeton University Press, Princeton, NJ, USA, 2005. MR2100420 J. Bourgain,*Green's Function Estimates for Lattice Schrödinger Operators and Applications*, vol. 158 of*Annals of Mathematics Studies*, Princeton University Press, Princeton, NJ, USA, 2005. MR2100420 L. Chierchia and J. You, “KAM tori for 1D nonlinear wave equations with periodic boundary conditions,”

*Communications in Mathematical Physics*, vol. 211, no. 2, pp. 497–525, 2000. MR1754527 0956.37054 10.1007/s002200050824 L. Chierchia and J. You, “KAM tori for 1D nonlinear wave equations with periodic boundary conditions,”*Communications in Mathematical Physics*, vol. 211, no. 2, pp. 497–525, 2000. MR1754527 0956.37054 10.1007/s002200050824 W. Craig and C. E. Wayne, “Newton's method and periodic solutions of nonlinear wave equations,”

*Communications on Pure and Applied Mathematics*, vol. 46, no. 11, pp. 1409–1498, 1993. MR1239318 0794.35104 10.1002/cpa.3160461102 W. Craig and C. E. Wayne, “Newton's method and periodic solutions of nonlinear wave equations,”*Communications on Pure and Applied Mathematics*, vol. 46, no. 11, pp. 1409–1498, 1993. MR1239318 0794.35104 10.1002/cpa.3160461102 Y. Gao, Y. Li, and J. Zhang, “Invariant tori of nonlinear Schrödinger equation,”

*Journal of Differential Equations*, vol. 246, no. 8, pp. 3296–3331, 2009. MR2507958 1172.35070 10.1016/j.jde.2009.01.031 Y. Gao, Y. Li, and J. Zhang, “Invariant tori of nonlinear Schrödinger equation,”*Journal of Differential Equations*, vol. 246, no. 8, pp. 3296–3331, 2009. MR2507958 1172.35070 10.1016/j.jde.2009.01.031 J. Geng and J. You, “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces,”

*Communications in Mathematical Physics*, vol. 262, no. 2, pp. 343–372, 2006. MR2200264 1103.37047 10.1007/s00220-005-1497-0 J. Geng and J. You, “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces,”*Communications in Mathematical Physics*, vol. 262, no. 2, pp. 343–372, 2006. MR2200264 1103.37047 10.1007/s00220-005-1497-0 J. Geng and X. Ren, “Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation,”

*Journal of Differential Equations*, vol. 249, no. 11, pp. 2796–2821, 2010. MR2718667 1206.35024 10.1016/j.jde.2010.04.003 J. Geng and X. Ren, “Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation,”*Journal of Differential Equations*, vol. 249, no. 11, pp. 2796–2821, 2010. MR2718667 1206.35024 10.1016/j.jde.2010.04.003 J. Geng, “Almost periodic solutions for a class of higher dimensional Schrödinger equations,”

*Frontiers of Mathematics in China*, vol. 4, no. 3, pp. 463–482, 2009. MR2525751 1181.35210 10.1007/s11464-009-0029-1 J. Geng, “Almost periodic solutions for a class of higher dimensional Schrödinger equations,”*Frontiers of Mathematics in China*, vol. 4, no. 3, pp. 463–482, 2009. MR2525751 1181.35210 10.1007/s11464-009-0029-1 J. Pöschel, “Quasi-periodic solutions for a nonlinear wave equation,”

*Commentarii Mathematici Helvetici*, vol. 71, no. 2, pp. 269–296, 1996. MR1396676 0866.35013 10.1007/BF02566420 J. Pöschel, “Quasi-periodic solutions for a nonlinear wave equation,”*Commentarii Mathematici Helvetici*, vol. 71, no. 2, pp. 269–296, 1996. MR1396676 0866.35013 10.1007/BF02566420 X. Yuan, “A KAM theorem with applications to partial differential equations of higher dimensions,”

*Communications in Mathematical Physics*, vol. 275, no. 1, pp. 97–137, 2007. MR2335770 1132.70009 10.1007/s00220-007-0287-2 X. Yuan, “A KAM theorem with applications to partial differential equations of higher dimensions,”*Communications in Mathematical Physics*, vol. 275, no. 1, pp. 97–137, 2007. MR2335770 1132.70009 10.1007/s00220-007-0287-2 C. E. Wayne, “Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory,”

*Communications in Mathematical Physics*, vol. 127, no. 3, pp. 479–528, 1990. MR1040892 0708.35087 10.1007/BF02104499 euclid.cmp/1104180217 C. E. Wayne, “Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory,”*Communications in Mathematical Physics*, vol. 127, no. 3, pp. 479–528, 1990. MR1040892 0708.35087 10.1007/BF02104499 euclid.cmp/1104180217 V. Benci and P. Rabinowitz, “Critical point theorems for indefinite functionals,”

*Inventiones Mathematicae*, vol. 52, no. 3, pp. 241–273, 1979. MR537061 0465.49006 10.1007/BF01389883 V. Benci and P. Rabinowitz, “Critical point theorems for indefinite functionals,”*Inventiones Mathematicae*, vol. 52, no. 3, pp. 241–273, 1979. MR537061 0465.49006 10.1007/BF01389883 H. Brézis and L. Nirenberg, “Forced vibrations for a nonlinear wave equation,”

*Communications on Pure and Applied Mathematics*, vol. 31, no. 1, pp. 1–30, 1978. MR470377 0378.35040 H. Brézis and L. Nirenberg, “Forced vibrations for a nonlinear wave equation,”*Communications on Pure and Applied Mathematics*, vol. 31, no. 1, pp. 1–30, 1978. MR470377 0378.35040 H. Brézis, J.-M. Coron, and L. Nirenberg, “Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,”

*Communications on Pure and Applied Mathematics*, vol. 33, no. 5, pp. 667–684, 1980. MR586417 0484.35057 10.1002/cpa.3160330507 H. Brézis, J.-M. Coron, and L. Nirenberg, “Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,”*Communications on Pure and Applied Mathematics*, vol. 33, no. 5, pp. 667–684, 1980. MR586417 0484.35057 10.1002/cpa.3160330507 E. R. Fadell and P. H. Rabinowitz, “Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,”

*Inventiones Mathematicae*, vol. 45, no. 2, pp. 139–174, 1978. MR0478189 0403.57001 10.1007/BF01390270 E. R. Fadell and P. H. Rabinowitz, “Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,”*Inventiones Mathematicae*, vol. 45, no. 2, pp. 139–174, 1978. MR0478189 0403.57001 10.1007/BF01390270 P. H. Rabinowitz, “Periodic solutions of nonlinear hyperbolic partial differential equations,”

*Communications on Pure and Applied Mathematics*, vol. 20, no. 1, pp. 145–205, 1967. MR0206507 0152.10003 10.1002/cpa.3160200105 P. H. Rabinowitz, “Periodic solutions of nonlinear hyperbolic partial differential equations,”*Communications on Pure and Applied Mathematics*, vol. 20, no. 1, pp. 145–205, 1967. MR0206507 0152.10003 10.1002/cpa.3160200105 P. H. Rabinowitz, “Free vibrations for a semilinear wave equation,”

*Communications on Pure and Applied Mathematics*, vol. 31, no. 1, pp. 31–68, 1978. MR470378 0341.35051 10.1002/cpa.3160310103 P. H. Rabinowitz, “Free vibrations for a semilinear wave equation,”*Communications on Pure and Applied Mathematics*, vol. 31, no. 1, pp. 31–68, 1978. MR470378 0341.35051 10.1002/cpa.3160310103 P. H. Rabinowitz,

*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, vol. 65 of*CBMS Regional Conference Series in Mathematics*, American Mathematical Society, Providence, RI, USA, 1986. MR845785 P. H. Rabinowitz,*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, vol. 65 of*CBMS Regional Conference Series in Mathematics*, American Mathematical Society, Providence, RI, USA, 1986. MR845785 D. Bambusi and M. Berti, “A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs,”

*SIAM Journal on Mathematical Analysis*, vol. 37, no. 1, pp. 83–102, 2005. MR2176924 1105.37045 10.1137/S0036141003436107 D. Bambusi and M. Berti, “A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs,”*SIAM Journal on Mathematical Analysis*, vol. 37, no. 1, pp. 83–102, 2005. MR2176924 1105.37045 10.1137/S0036141003436107 D. Bambusi and S. Paleari, “Families of periodic solutions of resonant PDEs,”

*Journal of Nonlinear Science*, vol. 11, no. 1, pp. 69–87, 2001. MR1819863 0994.37040 10.1007/s003320010010 D. Bambusi and S. Paleari, “Families of periodic solutions of resonant PDEs,”*Journal of Nonlinear Science*, vol. 11, no. 1, pp. 69–87, 2001. MR1819863 0994.37040 10.1007/s003320010010 M. Berti and P. Bolle, “Periodic solutions of nonlinear wave equations with general nonlinearities,”

*Communications in Mathematical Physics*, vol. 243, no. 2, pp. 315–328, 2003. MR2021909 1072.35015 10.1007/s00220-003-0972-8 M. Berti and P. Bolle, “Periodic solutions of nonlinear wave equations with general nonlinearities,”*Communications in Mathematical Physics*, vol. 243, no. 2, pp. 315–328, 2003. MR2021909 1072.35015 10.1007/s00220-003-0972-8 M. Berti and P. Bolle, “Multiplicity of periodic solutions of nonlinear wave equations,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 56, no. 7, pp. 1011–1046, 2004. MR2038735 1064.35119 M. Berti and P. Bolle, “Multiplicity of periodic solutions of nonlinear wave equations,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 56, no. 7, pp. 1011–1046, 2004. MR2038735 1064.35119 M. Berti and P. Bolle, “Cantor families of periodic solutions for completely resonant nonlinear wave equations,”

*Duke Mathematical Journal*, vol. 134, no. 2, pp. 359–419, 2006. MR2248834 1103.35077 10.1215/S0012-7094-06-13424-5 euclid.dmj/1155045505 M. Berti and P. Bolle, “Cantor families of periodic solutions for completely resonant nonlinear wave equations,”*Duke Mathematical Journal*, vol. 134, no. 2, pp. 359–419, 2006. MR2248834 1103.35077 10.1215/S0012-7094-06-13424-5 euclid.dmj/1155045505 B. V. Lidskii and E. I. Shul'man, “Periodic solutions of the equation ${u}_{tt}-{u}_{xx}+{u}^{3}=0$,”

*Functional Analysis and Its Applications*, vol. 22, no. 4, pp. 332–33, 1988. MR977006 B. V. Lidskii and E. I. Shul'man, “Periodic solutions of the equation ${u}_{tt}-{u}_{xx}+{u}^{3}=0$,”*Functional Analysis and Its Applications*, vol. 22, no. 4, pp. 332–33, 1988. MR977006 G. Gentile, V. Mastropietro, and M. Procesi, “Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions,”

*Communications in Mathematical Physics*, vol. 256, no. 2, pp. 437–490, 2005. MR2160800 1094.35021 10.1007/s00220-004-1255-8 G. Gentile, V. Mastropietro, and M. Procesi, “Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions,”*Communications in Mathematical Physics*, vol. 256, no. 2, pp. 437–490, 2005. MR2160800 1094.35021 10.1007/s00220-004-1255-8 G. Gentile and M. Procesi, “Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrödinger equation,”

*Communications in Mathematical Physics*, vol. 262, no. 3, pp. 533–553, 2006. MR2202301 1106.35095 10.1007/s00220-005-1409-3 G. Gentile and M. Procesi, “Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrödinger equation,”*Communications in Mathematical Physics*, vol. 262, no. 3, pp. 533–553, 2006. MR2202301 1106.35095 10.1007/s00220-005-1409-3 X. Yuan, “Quasi-periodic solutions of completely resonant nonlinear wave equations,”

*Journal of Differential Equations*, vol. 230, no. 1, pp. 213–274, 2006. MR2270553 1146.35307 10.1016/j.jde.2005.12.012 X. Yuan, “Quasi-periodic solutions of completely resonant nonlinear wave equations,”*Journal of Differential Equations*, vol. 230, no. 1, pp. 213–274, 2006. MR2270553 1146.35307 10.1016/j.jde.2005.12.012 M. Berti and M. Procesi, “Quasi-periodic solutions of completely resonant forced wave equations,”

*Communications in Partial Differential Equations*, vol. 31, no. 6, pp. 959–985, 2006. MR2233048 1100.35011 10.1080/03605300500358129 M. Berti and M. Procesi, “Quasi-periodic solutions of completely resonant forced wave equations,”*Communications in Partial Differential Equations*, vol. 31, no. 6, pp. 959–985, 2006. MR2233048 1100.35011 10.1080/03605300500358129 M. Procesi, “Quasi-periodic solutions for completely resonant non-linear wave equations in ID and 2D,”

*Discrete and Continuous Dynamical Systems*, vol. 13, no. 3, pp. 541–552, 2005. MR2152330 10.3934/dcds.2005.13.541 M. Procesi, “Quasi-periodic solutions for completely resonant non-linear wave equations in ID and 2D,”*Discrete and Continuous Dynamical Systems*, vol. 13, no. 3, pp. 541–552, 2005. MR2152330 10.3934/dcds.2005.13.541 P. Baldi, “Quasi-periodic solutions of the equation ${{\lightv}}_{tt}-{{\lightv}}_{xx}+{{\lightv}}^{3}=f({\lightv})$,”

*Discrete and Continuous Dynamical Systems*, vol. 15, no. 3, pp. 883–903, 2006. MR2220754 1124.35039 10.3934/dcds.2006.15.883 P. Baldi, “Quasi-periodic solutions of the equation ${{\lightv}}_{tt}-{{\lightv}}_{xx}+{{\lightv}}^{3}=f({\lightv})$,”*Discrete and Continuous Dynamical Systems*, vol. 15, no. 3, pp. 883–903, 2006. MR2220754 1124.35039 10.3934/dcds.2006.15.883 Y. Ma and W. Lou, “Quasi-periodic solutions of completely resonant wave equations with quasi-periodically forced vibrations,”

*Acta Applicandae Mathematicae*, vol. 112, no. 3, pp. 309–322, 2010. MR2737171 1207.35031 10.1007/s10440-010-9574-6 Y. Ma and W. Lou, “Quasi-periodic solutions of completely resonant wave equations with quasi-periodically forced vibrations,”*Acta Applicandae Mathematicae*, vol. 112, no. 3, pp. 309–322, 2010. MR2737171 1207.35031 10.1007/s10440-010-9574-6 D. Bambusi, “Lyapunov center theorem for some nonlinear PDE's: a simple proof,”

*Annali della Scuola Normale Superiore di Pisa. Classe di Scienze*, vol. 29, no. 4, pp. 823–837, 2000. MR1822409 D. Bambusi, “Lyapunov center theorem for some nonlinear PDE's: a simple proof,”*Annali della Scuola Normale Superiore di Pisa. Classe di Scienze*, vol. 29, no. 4, pp. 823–837, 2000. MR1822409 A. Ambrosetti and M. Badiale, “Homoclinics: Poincaré-Melnikov type results via a variational approach,”

*Annales de l'Institut Henri Poincare (C) Non Linear Analysis*, vol. 15, no. 2, pp. 233–252, 1998. MR1614571 1004.37043 10.1016/S0294-1449(97)89300-6 A. Ambrosetti and M. Badiale, “Homoclinics: Poincaré-Melnikov type results via a variational approach,”*Annales de l'Institut Henri Poincare (C) Non Linear Analysis*, vol. 15, no. 2, pp. 233–252, 1998. MR1614571 1004.37043 10.1016/S0294-1449(97)89300-6