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September 2019, 18(5): 2217-2242. doi: 10.3934/cpaa.2019100

On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas-SP, Brazil

Received  June 2017 Revised  September 2017 Published  April 2019

Fund Project: The author is supported by CNPq grants 402849/2016-7 and 303098/2016-3

Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.

Citation: Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
References:
[1]

J. Angulo and F. Linares, Periodic pulses of coupled nonlinear Schrödinger equations in optics, Indiana Univ. Math. J., 56 (2007), 847-877. doi: 10.1512/iumj.2007.56.2884.

[2]

H. Berestycki and P-L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[3]

M. ColinT. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.

[4]

M. ColinT. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380. doi: 10.1619/fesi.52.371.

[5]

M. ColinL Di Menza and J. C-Saut, Solitons in quadratic media, Nonlinearity, 29 (2016), 1000-1035. doi: 10.1088/0951-7715/29/3/1000.

[6]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774. doi: 10.1002/cpa.3160410602.

[7]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.

[8]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry Ⅱ, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.

[9]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. doi: 10.1016/j.anihpc.2012.10.007.

[10]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.

[11]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences, 185, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7.

[12]

T. Kapitula and K. Promislow, Stability indices for constrained self-adjoint operators, Proc. Amer. Math. Soc., 140 (2012), 865-880. doi: 10.1090/S0002-9939-2011-10943-2.

[13]

T. KapitulaP. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195 (2004), 263-282. doi: 10.1016/j.physd.2004.03.018.

[14]

T. KapitulaP. G. Kevrekidis and B. Sandstede, Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 201 (2005), 199-201. doi: 10.1016/j.physd.2004.11.015.

[15]

Y. S. Kivshar, et al., Multi-component optical solitary waves, Phys. A, 288 (2000), 407-412. doi: 10.1016/S0378-4371(00)00420-9.

[16]

Y. S. KivsharA. A. Sukhorukov and M. Saltiel, Two-color multistep cascading and parametric soliton-induced waveguides, Phys. Rev. E, 60 (1999), 5056-5059. doi: 10.1103/PhysRevE.60.R5056.

[17]

U. Kota, Final state problem for a system of nonlinear Schrödinger equations with three wave interaction, J. Evol. Equ., 16 (2016), 173-191. doi: 10.1007/s00028-015-0297-z.

[18]

K. Koynov and S. Saltiel, Nonlinear phase shift via multistep $\chi^2$ cascading, Opt. Commun., 152 (1998), 96-100. doi: 10.1016/S0030-4018(98)00114-X.

[19]

M. K. Kwong, Uniqueness of positive solutions of $-\Delta u+u = u^p$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4939-2181-2.

[21]

O. Lopes, Stability of solitary waves for some coupled systems, Nonlinearity, 19 (2006), 95-113. doi: 10.1088/0951-7715/19/1/006.

[22]

O. Lopes, Uniqueness of a positive symmetric solution to an ODE system, Electron. J. Differential Equations, 2009 (2009), 1-8.

[23]

O. Lopes, Stability of solitary waves for a three-wave interaction model, Electron. J. Differential Equations, 2014 (2014), 1-9.

[24]

F. Natali and A. Pastor, Orbital instability of standing waves for the quadratic-cubic Klein-Gordon-Schrödinger system, Z. Angew. Math. Phys., 66 (2015), 1341-1354. doi: 10.1007/s00033-014-0467-9.

[25]

A. Pastor, Orbital stability of periodic travelling waves for coupled nonlinear nonlinear Schrödinger equations, Electron. J. Differential Equations, 2010 (2010), 1-19.

[26]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155. doi: 10.1016/j.jmaa.2012.10.003.

[27]

D. E. Pelinovsky, Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 461 (2005), 783-812. doi: 10.1098/rspa.2004.1345.

[28]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 65, Providence, 1986. doi: 10.1090/cbms/065.

[29] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008. doi: 10.1142/p493.
[30]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[31]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.

[32]

A. C. Yew, Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations, Indiana Univ. Math. J., 49 (2000), 1079-1124. doi: 10.1512/iumj.2000.49.1826.

[33]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

[34]

A. C. YewA. R. Champneys and P. J. McKenna, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52. doi: 10.1007/s003329900063.

show all references

References:
[1]

J. Angulo and F. Linares, Periodic pulses of coupled nonlinear Schrödinger equations in optics, Indiana Univ. Math. J., 56 (2007), 847-877. doi: 10.1512/iumj.2007.56.2884.

[2]

H. Berestycki and P-L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[3]

M. ColinT. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.

[4]

M. ColinT. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380. doi: 10.1619/fesi.52.371.

[5]

M. ColinL Di Menza and J. C-Saut, Solitons in quadratic media, Nonlinearity, 29 (2016), 1000-1035. doi: 10.1088/0951-7715/29/3/1000.

[6]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774. doi: 10.1002/cpa.3160410602.

[7]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.

[8]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry Ⅱ, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.

[9]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690. doi: 10.1016/j.anihpc.2012.10.007.

[10]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 555-567. doi: 10.1007/s00030-009-0017-x.

[11]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences, 185, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7.

[12]

T. Kapitula and K. Promislow, Stability indices for constrained self-adjoint operators, Proc. Amer. Math. Soc., 140 (2012), 865-880. doi: 10.1090/S0002-9939-2011-10943-2.

[13]

T. KapitulaP. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195 (2004), 263-282. doi: 10.1016/j.physd.2004.03.018.

[14]

T. KapitulaP. G. Kevrekidis and B. Sandstede, Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 201 (2005), 199-201. doi: 10.1016/j.physd.2004.11.015.

[15]

Y. S. Kivshar, et al., Multi-component optical solitary waves, Phys. A, 288 (2000), 407-412. doi: 10.1016/S0378-4371(00)00420-9.

[16]

Y. S. KivsharA. A. Sukhorukov and M. Saltiel, Two-color multistep cascading and parametric soliton-induced waveguides, Phys. Rev. E, 60 (1999), 5056-5059. doi: 10.1103/PhysRevE.60.R5056.

[17]

U. Kota, Final state problem for a system of nonlinear Schrödinger equations with three wave interaction, J. Evol. Equ., 16 (2016), 173-191. doi: 10.1007/s00028-015-0297-z.

[18]

K. Koynov and S. Saltiel, Nonlinear phase shift via multistep $\chi^2$ cascading, Opt. Commun., 152 (1998), 96-100. doi: 10.1016/S0030-4018(98)00114-X.

[19]

M. K. Kwong, Uniqueness of positive solutions of $-\Delta u+u = u^p$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[20]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4939-2181-2.

[21]

O. Lopes, Stability of solitary waves for some coupled systems, Nonlinearity, 19 (2006), 95-113. doi: 10.1088/0951-7715/19/1/006.

[22]

O. Lopes, Uniqueness of a positive symmetric solution to an ODE system, Electron. J. Differential Equations, 2009 (2009), 1-8.

[23]

O. Lopes, Stability of solitary waves for a three-wave interaction model, Electron. J. Differential Equations, 2014 (2014), 1-9.

[24]

F. Natali and A. Pastor, Orbital instability of standing waves for the quadratic-cubic Klein-Gordon-Schrödinger system, Z. Angew. Math. Phys., 66 (2015), 1341-1354. doi: 10.1007/s00033-014-0467-9.

[25]

A. Pastor, Orbital stability of periodic travelling waves for coupled nonlinear nonlinear Schrödinger equations, Electron. J. Differential Equations, 2010 (2010), 1-19.

[26]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155. doi: 10.1016/j.jmaa.2012.10.003.

[27]

D. E. Pelinovsky, Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 461 (2005), 783-812. doi: 10.1098/rspa.2004.1345.

[28]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 65, Providence, 1986. doi: 10.1090/cbms/065.

[29] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008. doi: 10.1142/p493.
[30]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[31]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.

[32]

A. C. Yew, Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations, Indiana Univ. Math. J., 49 (2000), 1079-1124. doi: 10.1512/iumj.2000.49.1826.

[33]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

[34]

A. C. YewA. R. Champneys and P. J. McKenna, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52. doi: 10.1007/s003329900063.

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