# American Institute of Mathematical Sciences

## 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

Received  August 2020 Revised  January 2021 Published  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021100
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York–London, 1975.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.  Google Scholar [9] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.   Google Scholar [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.  Google Scholar [11] M. Colin, T. 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.  Google Scholar [12] M. Colin, T. 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.  Google Scholar [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] O. Lopes, Stability of solitary waves for a three-wave interaction model, Electron. J. Differential Equations, 2014 (2014), 1-9.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York–London, 1975.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003. doi: 10.1090/cln/010.  Google Scholar [9] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.   Google Scholar [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.  Google Scholar [11] M. Colin, T. 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.  Google Scholar [12] M. Colin, T. 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.  Google Scholar [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075060.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] O. Lopes, Stability of solitary waves for a three-wave interaction model, Electron. J. Differential Equations, 2014 (2014), 1-9.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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