# American Institute of Mathematical Sciences

June  2022, 11(3): 837-867. doi: 10.3934/eect.2021028

## On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

 Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Received  January 2021 Revised  April 2021 Published  June 2022 Early access  May 2021

In this paper we consider the Schrödinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces $X^k( \mathbb{R})$ and in $\phi+H^k( \mathbb{R})$. Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.

Citation: Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028
##### References:
 [1] F. Béthuel, P. Gravejat and D. Smets, Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation, Ann. Inst. Fourier (Grenoble), 64 (2014), 19-70.  doi: 10.5802/aif.2838. [2] T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010. [3] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998. [4] A. de Laire, Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity, Comm. Partial Differential Equations, 35 (2010), 2021-2058.  doi: 10.1080/03605302.2010.497200. [5] C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538. [6] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations, 33 (2008), 729-771.  doi: 10.1080/03605300802031614. [7] P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004. [8] P. Gérard, The Gross-Pitaevskii equation in the energy space, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2008, pages 129–148. doi: 10.1090/conm/473/09226. [9] M. Hayashi and T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differential Equations, 261 (2016), 5424-5445.  doi: 10.1016/j.jde.2016.08.018. [10] N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P. [11] N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129. [12] S. Le Coz, Standing waves in nonlinear Schrödinger equations, In Analytical and Numerical Aspects of Partial Differential Equations, Walter de Gruyter, Berlin, 2009, pages 151–192. [13] E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, Journal of Plasma Physics, 16 (1976), 321-334.  doi: 10.1017/S0022377800020249. [14] M. Murai, K. Sakamoto and S. Yotsutani, Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl.), 2015,878–900. doi: 10.3934/proc.2015.0878. [15] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, volume 139 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. [16] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277. [17] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. II, Funkcial. Ekvac., 24 (1981), 85-94. [18] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, volume 1756 of Lecture Notes in Mathematics., Springer-Verlag, Berlin, 2001.

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##### References:
 [1] F. Béthuel, P. Gravejat and D. Smets, Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation, Ann. Inst. Fourier (Grenoble), 64 (2014), 19-70.  doi: 10.5802/aif.2838. [2] T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010. [3] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998. [4] A. de Laire, Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity, Comm. Partial Differential Equations, 35 (2010), 2021-2058.  doi: 10.1080/03605302.2010.497200. [5] C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538. [6] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations, 33 (2008), 729-771.  doi: 10.1080/03605300802031614. [7] P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004. [8] P. Gérard, The Gross-Pitaevskii equation in the energy space, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2008, pages 129–148. doi: 10.1090/conm/473/09226. [9] M. Hayashi and T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differential Equations, 261 (2016), 5424-5445.  doi: 10.1016/j.jde.2016.08.018. [10] N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P. [11] N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129. [12] S. Le Coz, Standing waves in nonlinear Schrödinger equations, In Analytical and Numerical Aspects of Partial Differential Equations, Walter de Gruyter, Berlin, 2009, pages 151–192. [13] E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, Journal of Plasma Physics, 16 (1976), 321-334.  doi: 10.1017/S0022377800020249. [14] M. Murai, K. Sakamoto and S. Yotsutani, Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl.), 2015,878–900. doi: 10.3934/proc.2015.0878. [15] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, volume 139 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. [16] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277. [17] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. II, Funkcial. Ekvac., 24 (1981), 85-94. [18] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, volume 1756 of Lecture Notes in Mathematics., Springer-Verlag, Berlin, 2001.
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