Uniqueness for a Cauchy problem for the generalized Schrödinger equation

İsmet Gölgeleyen; Özlem Kaytmaz; İsmet Gölgeleyen; Özlem Kaytmaz

doi:10.3934/math.2023287

AIMS Mathematics

2023, Volume 8, Issue 3: 5703-5724. doi: 10.3934/math.2023287

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Research article

Uniqueness for a Cauchy problem for the generalized Schrödinger equation

İsmet Gölgeleyen ,
Özlem Kaytmaz ^,

Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Zonguldak, 67100, Turkey

Received: 07 November 2022 Revised: 06 December 2022 Accepted: 15 December 2022 Published: 22 December 2022
MSC : 35A23, 35A32, 35Q40

In this work, we consider a Cauchy problem for the generalized Schrö dinger equation which has important applications in quantum kinetic theory, water wave problems and ferromagnetism. Due to its multidimensionality, it is important from the point of view of modern physics theories such as quantum field theory and string theory. We prove the uniqueness of the solution of the problem in an unbounded domain by using semigeodesic coordinates. The main tool is a pointwise Carleman estimate. To the authors' best knowledge, this is the first study which deals with the solvability of this problem.
- uniqueness,
- Cauchy problem,
- generalized Schrödinger equation,
- pointwise Carleman estimate
Citation: İsmet Gölgeleyen, Özlem Kaytmaz. Uniqueness for a Cauchy problem for the generalized Schrödinger equation[J]. AIMS Mathematics, 2023, 8(3): 5703-5724. doi: 10.3934/math.2023287

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Abstract

In this work, we consider a Cauchy problem for the generalized Schrö dinger equation which has important applications in quantum kinetic theory, water wave problems and ferromagnetism. Due to its multidimensionality, it is important from the point of view of modern physics theories such as quantum field theory and string theory. We prove the uniqueness of the solution of the problem in an unbounded domain by using semigeodesic coordinates. The main tool is a pointwise Carleman estimate. To the authors' best knowledge, this is the first study which deals with the solvability of this problem.

References

[1]	A. K. Amirov, Integral geometry and inverse problems for Kinetic equations, The Netherlands: De Gruyter, 2001. https://doi.org/10.1515/9783110940947
[2]	A. Davey, K. Stewartson, On three-dimensional packets of surface waves, P. R. Soc. A-Math. Phy., 338 (1974), 101-110. https://doi.org/10.1098/rspa.1974.0076 doi: 10.1098/rspa.1974.0076
[3]	V. E. Zakharov, E. I. Schulman, Degenerative dispersion laws, motion invariants and kinetic equations, Physica D., 1 (1980), 192-202. https://doi.org/10.1016/0167-2789(80)90011-1 doi: 10.1016/0167-2789(80)90011-1
[4]	Y. Ishimori, A note on the Cauchy problem for Schrö dinger type equations on the Riemannian manifold, Math. Japonica, 72 (1984), 33-37.
[5]	C. Sulem, P. L.Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse, New York: Springer, 1999. https://doi.org/10.1515/9783110915549
[6]	X. Zhang, L. Liu, Y. Wu, Y. Cui, The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach, J. Math. Anal. Appl., 464 (2018), 1089-1106. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[7]	X. Zhang, L. Liu, Y. Wu, Y. Cui, Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach, Electron. J. Differ. Equ., 147 (2018), 1-15.
[8]	X. Zhang, J. Jiang, Y. Wu, Y. Cui, Existence and asymptotic properties of solutions for a nonlinear Schr ödinger elliptic equation from geophysical fluid flows, Appl. Math. Lett., 90 (2019), 229-237. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[9]	X. Zhang, L. Liu, Y. Wu, B. Wiwatanapataphee, Multiple solutions for a modified quasilinear Schrö dinger elliptic equation with a nonsquare diffusion term, Nonlinear Anal. Model., 26 (2021), 702-717. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[10]	X. Zhang, L. Liu, Y. Wu, B. Wiwatanapataphee, Y. Cui, Solvability and asymptotic properties for an elliptic geophysical fluid flows model in a planar exterior domain, Nonlinear Anal. Model., 26 (2021), 315-333. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[11]	H. Wang, Y. Zhang, A kind of nonisospectral and isospectral integrable couplings and their Hamiltonian systems, Commun. Nonlinear Sci., 99 (2021), 105822. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[12]	H. Wang, Y. Zhang, A kind of generalized integrable couplings and their Bi-Hamiltonian structure, Int. J. Theor. Phys., 60 (2021), 1797-1812. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[13]	H. Wang, Y. Zhang, A new multi-component integrable coupling and its application to isospectral and nonisospectral problems, Commun. Nonlinear Sci., 105 (2022), 106075. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[14]	H. Wang, Y. Zhang, Application of Riemann-Hilbert method to an extended coupled nonlinear Schrödinger equations, J. Comput. Appl. Math., 420 (2023), 114812. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[15]	C. E. Kenig, G. Ponce, L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. math., 134 (1998), 489-545. https://doi.org/10.1007/s002220050272 doi: 10.1007/s002220050272
[16]	C. E. Kenig, G. Ponce, C. Rolvung, L. Vega, The general quasilinear ultrahyperbolic Schrödinger equation, Adv. Math., 206 (2006), 402-433. https://doi.org/10.1016/j.aim.2005.09.005 doi: 10.1016/j.aim.2005.09.005
[17]	A. K. Amirov, M. Yamamoto, Inverse problem for a Schrödinger-type equation, Dokl. Math., 77 (2008), 212-214. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[18]	İ. Gölgeleyen, Ö. Kaytmaz, Conditional stability for a Cauchy problem for the ultrahyperbolic Schrödinger equation, Appl. Anal., 101 (2022), 1505-1516. https://doi.org/10.1080/00036811.2020.1781829 doi: 10.1080/00036811.2020.1781829
[19]	F. Gölgeleyen, Ö. Kaytmaz, A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation, Anal. Math. Phys., 9 (2019), 2171-2199. https://doi.org/10.1007/s13324-019-00326-6 doi: 10.1007/s13324-019-00326-6
[20]	T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 2 (1939), 1-9.
[21]	M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013. https://doi.org/10.1134/S1064562408020142 doi: 10.1134/S1064562408020142
[22]	M. V. Klibanov, A. Timonov, Carleman estimates for coefficient inverse problem and numerical applications, The Netherlands: VSP, 2004. https://doi.org/10.1515/9783110915549
[23]	C. E. Kenig, Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems, P. Int. Congr. Math., 1 (1986), 948-960.
[24]	A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Am. J. Math., 80 (1958), 16-36. https://doi.org/https://doi.org/10.2307/2372819 doi: 10.2307/2372819
[25]	L. Hörmander, Linear partial differential operators, Berlin: Springer, 1963.
[26]	L. Baudouin, J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Probl., 18 (2002), 1537. https://doi.org/10.1088/0266-5611/18/6/307 doi: 10.1088/0266-5611/18/6/307
[27]	A. Mercado, A. Osses, L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Probl., 24 (2008), 015017. https://doi.org/10.1088/0266-5611/24/1/015017 doi: 10.1088/0266-5611/24/1/015017
[28]	G. Yuan, M. Yamamoto, Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality, Chin. Ann. Math. B, 31 (2010), 555-578. https://doi.org/10.1007/s11401-010-0585-4 doi: 10.1007/s11401-010-0585-4
[29]	M. M. Lavrentiev, V. G. Romanov, S. P. Shishatskii, Ill-Posed problems of mathematical physics and analysis, Providence: American Mathematical Society, 1986.
[30]	F. Gölgeleyen, M. Yamamoto, Uniqueness of solution of an inverse source problem for ultrahyperbolic equations, Inverse Probl., 36 (2020), 035008. https://doi.org/10.1088/1361-6420/ab63a2 doi: 10.1088/1361-6420/ab63a2

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