# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022039
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## On the Cauchy problem for a nonlocal nonlinear Schrödinger equation

 1 School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China 2 Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan

*Corresponding author: Amin Esfahani

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: H. W. was supported by Key Project of Natural Science Foundation of Educational Committee of Henan Province(No. 20A110007). A. E. was supported by the Social Policy Grant (SPG) funded by Nazarbayev University, Kazakhstan

This paper considers the one-dimensional Schrödinger equation with nonlocal nonlinearity that describes the interactions of nonlinear dispersive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the conditions under which the solutions are uniformly bounded in the energy space.

Citation: Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022039
##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. [2] H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4. [3] J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6. [4] D. Cai, A. Majda, D. McLaughlin and E. Tabak, Dispersive wave turbulence in one dimension, Phys. D., 152/153 (2001), 551-572.  doi: 10.1016/S0167-2789(01)00193-2. [5] Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863. [6] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low-regularity ill-posedness of canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040. [7] A. Erdélyi, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954. [8] A. Esfahani, Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives, Z. Angew. Math. Phys., 66 (2015), 3345-3356.  doi: 10.1007/s00033-015-0586-y. [9] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math, 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591. [10] P. Gérard, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107. [11] A. Grunrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Notic., 41 (2005), 2525-2558.  doi: 10.1155/imrn.2005.2525. [12] B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, preprint, arXiv: 2003.05011. [13] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equation revisited, Int. Math. Res. Notic., (2005), 2815–2828. doi: 10.1155/imrn.2005.2815. [14] C. E. Kenig, Y. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005. [15] F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001. [16] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^nd$ edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. [17] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincare Anal. Non-linéaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0. [18] A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124. [19] S. Oh and A. Stefanov, On quadratic Schrödinger equations on $\mathbb{R}^{1+1}$: A normal form approach, J. London Math. Soc., 86 (2012), 499-519.  doi: 10.1112/jlms/jds016. [20] H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differ. Equ., 4 (1999), 561-580. [21] H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equations, (2001), 1–23. [22] T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035. [23] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. [24] K. Trulsen, I. Kliakhandler, K. Dysthe and M. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids, 12 (2000), 2432-2437.  doi: 10.1063/1.1287856.

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##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. [2] H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4. [3] J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6. [4] D. Cai, A. Majda, D. McLaughlin and E. Tabak, Dispersive wave turbulence in one dimension, Phys. D., 152/153 (2001), 551-572.  doi: 10.1016/S0167-2789(01)00193-2. [5] Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863. [6] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low-regularity ill-posedness of canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040. [7] A. Erdélyi, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954. [8] A. Esfahani, Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives, Z. Angew. Math. Phys., 66 (2015), 3345-3356.  doi: 10.1007/s00033-015-0586-y. [9] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math, 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591. [10] P. Gérard, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107. [11] A. Grunrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Notic., 41 (2005), 2525-2558.  doi: 10.1155/imrn.2005.2525. [12] B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, preprint, arXiv: 2003.05011. [13] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equation revisited, Int. Math. Res. Notic., (2005), 2815–2828. doi: 10.1155/imrn.2005.2815. [14] C. E. Kenig, Y. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005. [15] F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001. [16] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^nd$ edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. [17] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincare Anal. Non-linéaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0. [18] A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124. [19] S. Oh and A. Stefanov, On quadratic Schrödinger equations on $\mathbb{R}^{1+1}$: A normal form approach, J. London Math. Soc., 86 (2012), 499-519.  doi: 10.1112/jlms/jds016. [20] H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differ. Equ., 4 (1999), 561-580. [21] H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equations, (2001), 1–23. [22] T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035. [23] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. [24] K. Trulsen, I. Kliakhandler, K. Dysthe and M. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids, 12 (2000), 2432-2437.  doi: 10.1063/1.1287856.
The graphs of $K$ with negative values of $\beta$ are shown in the left and in the right figures with both signs
Plots of ground states of (1) with $\omega = 1$ and various values of $\beta$
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