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March  2022, 27(3): 1511-1547. doi: 10.3934/dcdsb.2021100

Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction

Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami Osawa, Hachioji, Tokyo 192-0397, Japan

* Corresponding author: Yuki Osada

Received  August 2020 Revised  January 2021 Published  March 2022 Early access  March 2021

In this paper we study several $ L^2 $-constrained variational problems associated with a three component system of nonlinear Schrödinger equations with three wave interaction. We consider the existence and the orbital stability of minimizers for these variational problems. We also investigate an asymptotic expansion of the minimal energy and the asymptotic behavior of a minimizer for the variational problem when the attractive effect of three wave interaction is sufficiently large.

Citation: Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1511-1547. doi: 10.3934/dcdsb.2021100
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York–London, 1975. 
[2]

A. H. Ardila, Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction, Nonlinear Anal., 167 (2018), 1-20.  doi: 10.1016/j.na.2017.10.013.

[3]

S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Continuous Dynam. Systems - A, 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.

[4]

S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differ. Equ., 263 (2017), 3197-3229.  doi: 10.1016/j.jde.2017.04.034.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[7]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.

[9]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330. 

[10]

M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031.

[11]

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.

[12]

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.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.

[14]

T. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508. doi: 10.1063/1.5028208.

[15]

T. Gou and L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144 (2016), 10-22.  doi: 10.1016/j.na.2016.05.016.

[16]

N. Ikoma and Y. Miyamoto, Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 48, 20 pp. doi: 10.1007/s00526-020-1703-0.

[17]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.

[18]

E. H. Lieb and M. Loss, Analysis, volume 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2 edition, 2001. doi: 10.1090/gsm/014.

[19]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[21]

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

[22]

L. Lu, $L^2$ normalized solutions for nonlinear Schrödinger systems in $\mathbb{R}^3$, Nonlinear Anal., 191 (2020), 111621, 19 pp. doi: 10.1016/j.na.2019.111621.

[23]

A. Pomponio, Ground states for a system of nonlinear Schrödinger equations with three wave interaction, J. Math. Phys., 51 (2010), 093513, 20pp. doi: 10.1063/1.3486069.

[24]

M. Shibata, A new rearrangement inequality and its application for $L^2$–constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York–London, 1975. 
[2]

A. H. Ardila, Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction, Nonlinear Anal., 167 (2018), 1-20.  doi: 10.1016/j.na.2017.10.013.

[3]

S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Continuous Dynam. Systems - A, 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.

[4]

S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differ. Equ., 263 (2017), 3197-3229.  doi: 10.1016/j.jde.2017.04.034.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[7]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.

[9]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330. 

[10]

M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031.

[11]

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.

[12]

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.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.

[14]

T. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508. doi: 10.1063/1.5028208.

[15]

T. Gou and L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144 (2016), 10-22.  doi: 10.1016/j.na.2016.05.016.

[16]

N. Ikoma and Y. Miyamoto, Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 48, 20 pp. doi: 10.1007/s00526-020-1703-0.

[17]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.

[18]

E. H. Lieb and M. Loss, Analysis, volume 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2 edition, 2001. doi: 10.1090/gsm/014.

[19]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[21]

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

[22]

L. Lu, $L^2$ normalized solutions for nonlinear Schrödinger systems in $\mathbb{R}^3$, Nonlinear Anal., 191 (2020), 111621, 19 pp. doi: 10.1016/j.na.2019.111621.

[23]

A. Pomponio, Ground states for a system of nonlinear Schrödinger equations with three wave interaction, J. Math. Phys., 51 (2010), 093513, 20pp. doi: 10.1063/1.3486069.

[24]

M. Shibata, A new rearrangement inequality and its application for $L^2$–constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1.

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