L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

[CrossRef]

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).

[CrossRef]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).

[CrossRef]

C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).

[CrossRef]

M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).

[CrossRef]
[PubMed]

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).

[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).

[CrossRef]
[PubMed]

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).

[CrossRef]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).

[CrossRef]
[PubMed]

D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).

[CrossRef]
[PubMed]

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).

[CrossRef]
[PubMed]

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).

[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).

[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).

[CrossRef]
[PubMed]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).

[CrossRef]
[PubMed]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).

[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).

[CrossRef]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).

[CrossRef]

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).

[CrossRef]
[PubMed]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).

[CrossRef]
[PubMed]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).

[CrossRef]
[PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).

[CrossRef]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).

[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).

[CrossRef]

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).

[CrossRef]

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).

[CrossRef]

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).

[CrossRef]
[PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).

[CrossRef]
[PubMed]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).

[CrossRef]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).

[CrossRef]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

[CrossRef]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).

[CrossRef]

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).

[CrossRef]

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).

[CrossRef]

M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).

[CrossRef]
[PubMed]

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).

[CrossRef]

C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).

[CrossRef]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).

[CrossRef]

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).

[CrossRef]

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).

[CrossRef]

D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).

[CrossRef]
[PubMed]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).

[CrossRef]
[PubMed]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).

[CrossRef]
[PubMed]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).

[CrossRef]
[PubMed]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).

[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).

[CrossRef]
[PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).

[CrossRef]
[PubMed]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).

[CrossRef]
[PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).

[CrossRef]

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).

[CrossRef]

D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).

[CrossRef]
[PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).

[CrossRef]
[PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).

[CrossRef]
[PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).

[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).

[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).

[CrossRef]
[PubMed]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).

[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).

[CrossRef]
[PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).

[CrossRef]
[PubMed]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

[CrossRef]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).

[CrossRef]
[PubMed]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).

[CrossRef]

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).

[CrossRef]

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).

[CrossRef]

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).

[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).

[CrossRef]

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).

[CrossRef]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).

[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).

[CrossRef]

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).

[CrossRef]

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).

[CrossRef]

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).

[CrossRef]

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099–2111 (1997).

[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).

[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).

[CrossRef]

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).

[CrossRef]

C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).

[CrossRef]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).

[CrossRef]
[PubMed]

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).

[CrossRef]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).

[CrossRef]
[PubMed]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).

[CrossRef]
[PubMed]

M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).

[CrossRef]
[PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).

[CrossRef]
[PubMed]

D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).

[CrossRef]
[PubMed]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).

[CrossRef]
[PubMed]

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).

[CrossRef]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).

[CrossRef]
[PubMed]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).

[CrossRef]

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).

[CrossRef]
[PubMed]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

[CrossRef]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).

[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).

[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).

[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).

[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).

[CrossRef]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).

[CrossRef]
[PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).

[CrossRef]
[PubMed]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).

[CrossRef]
[PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).

[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).

[CrossRef]
[PubMed]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).

[CrossRef]

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).

[CrossRef]

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.