## Iterative reconstruction algorithms for solving the Schrödinger equations on conical spaces.(English)Zbl 1481.35146

Summary: We consider a system of Schrödinger equations on conical spaces. We first rewrite the iterative reconstruction algorithms for two kinds of average Schrödinger functionals and prove their convergence. Then the asymptotic pointwise error estimates are presented for both algorithms under the case that the average samples are corrupted by noise.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35J47 Second-order elliptic systems 35J61 Semilinear elliptic equations
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### References:

 [1] Afrouzi, G., Mirzapour, M., Rǎdulescu, V. D.: Variational analysis of anisotropic Schrödinger equations without Ambrosetti-Rabinowitz-type condition. Z. Angew. Math. Phys. 69(1, Art. 9), 17, (2018) · Zbl 1393.35048 [2] Aktosun, T.; Papanicolaou, VG, Inverse problem with transmission eigenvalues for the discrete Schrödinger equation, J. Math. Phys., 56, 082101 (1995) · Zbl 1323.81024 [3] Alves, R., Reis, M.: About existence and regularity of positive solutions for a quasilinear Schrödinger equation with singular nonlinearity. Electron. J. Qual. Theory Differ. Equ 23 (2020) (Paper No. 60) · Zbl 1474.35318 [4] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc., 348, 305-330 (1996) · Zbl 0858.35083 [5] Azzollini, A., The Kirchhoff equation in $${\mathbb{R}}^3$$ perturbed by a local nonlinearity, Differ. Integ. Equ., 25, 543-554 (2012) · Zbl 1265.35069 [6] Bertsekas, DP, Nonlinear Programming (1995), Belmont: Athena Scientific, Belmont · Zbl 0935.90037 [7] Case, KM; Kac, M., A discrete version of the inverse scattering problem, J. Math. Phys., 14, 594-603 (1973) [8] Cavalcanti, MM; Domingos, VN; Soriano, JA, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differ. Equ., 6, 701-730 (2001) · Zbl 1007.35049 [9] Chadan, K.; Sabatier, PC, Inverse Problems in Quantum Scattering Theory (1989), New York: Springer, New York · Zbl 0681.35088 [10] Chaharlang, MM; Razani, A., Infinitely many solutions for a fourth order singular elliptic problem, Filomat, 32, 14, 5003-5010 (2011) [11] Choque-Rivero, AE; Papanicolaou, VG, Bound states of the discrete Schrödinger equation with compactly supported potentials, Electron. J. Differ. Equ., 12, 23, 1-19 (1999) [12] Courant, R.; Hilbert, D., Methods of Mathematical Physics (1962), New York: Interscience, New York · Zbl 0099.29504 [13] Damanik, D.; Killip, R., Half-line Schrödinger operators with no bound states, Acta Math., 193, 31-72 (2004) · Zbl 1081.47027 [14] Antoine, X.; Besse, C.; Klein, P., Absorbing boundary conditions for general nonlinear Schrödinger equations, SIAM J. Sci. Comput., 33, 1008-1033 (2012) · Zbl 1231.35223 [15] D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247-262 (1992) · Zbl 0785.35067 [16] Damanik, D.; Teschl, G., Bound states of discrete Schrödinger operators with super-critical inverse square potentials, Proc. Am. Math. Soc., 135, 1123-1127 (2007) · Zbl 1110.47019 [17] Darboux, G., Leçons sur la théorie général des Surfaces, 2nd Part (1915), Paris: Gauthier-Villars, Paris · JFM 45.0881.05 [18] Deift, P.; Trubowitz, E., Inverse scattering on the line, Commun. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005 [19] Dijk, W.: Wytse Numerical Time-Dependent Solutions of the Schrödinger Equation with Piecewise Continuous Potentials. Phys. Rev. E. 93(6), 063307 (2016) (8 pp) [20] Faddeev, LD, The inverse problem in the quantum theory of scattering, J. Math. Phys., 4, 72-104 (1963) · Zbl 0112.45101 [21] Gel’fand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Am. Math. Soc. Transl. 1(ser. 2), 253-304 (1955) [22] Gomez-Ruggiero, M.; Martinez, JM; Moretti, A., Comparing algorithms for solving sparse nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 23, 459-483 (1992) · Zbl 0752.65039 [23] Guariglia, E., Silvestrov, S.: Fractional-wavelet analysis of positive definite distributions and wavelets on $$D^{\prime }(C)$$. In: Silvestrov, S., Rančić, M., (eds.), Engineering Mathematics II, pp. 337-353. Springer. Berlin (2016) · Zbl 1365.65294 [24] Hale, JK, Theory of Functional Differential Equations (2003), New York: Springer, New York [25] Huang, L., Tian, Z., Cai, Y.: Compact local structure-preserving algorithms for the nonlinear Schrödinger equation with wave operator. Math. Probl. Eng. (2020) (Art. ID 4345278, 12 pp) [26] Huang, L.; Hu, L., Fourth-order compact difference method for the linear Boussinesq equation, Math. Pract. Theory, 47, 7, 146-151 (2017) · Zbl 1399.65153 [27] Huang, L., Split-step multi-symplectic pseudo-spectral scheme for the nonlinear fourth-order Schrödinger equation, Numer. Math. J. Chinese Univ., 36, 2, 183-192 (2014) · Zbl 1324.65128 [28] Kassay, G.; Rǎdulescu, VD, Equilibrium Problems and Applications (2018), London: Mathematics in Science and Engineering. Elsevier/Academic Press, London · Zbl 1448.47005 [29] Kermack, W.O., M’Kendrick, A.D.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A 115, 700-721 (1997) · JFM 53.0517.01 [30] Kizin, P.: Stability of gap soliton complexes in the nonlinear Schrödinger equation with periodic potential and repulsive nonlinearity. Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 26(4), 591-602 (2016) [31] Ladde, GS; Lakshmikatham, V., Random Differential Inequalities (1980), New York: Academic Press, New York [32] Li, H., Local absorbing boundary conditions for two-dimensional nonlinear Schrödinger equation with wave operator on unbounded domain, Math. Methods Appl. Sci., 44, 18, 14382-14392 (2021) · Zbl 1479.35243 [33] Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North-Holland Math. Stud., vol. 30, pp. 284-346. North-Holland, Amsterdam, New York (1978) [34] Mastorakis, NE, Solution of the Schrodinger-Maxwell equations via finite elements and genetic algorithms with Nelder-Mead, WSEAS Trans. Math., 8, 4, 159-176 (2020) [35] Matveev, VB; Salle, MA, Darboux Transformations and Solitons (1991), Berlin: Springer-Verlag, Berlin · Zbl 0744.35045 [36] Mei, Y.; Wang, Y., Three types of solutions for a class of nonlinear Schrödinger equations, Acta Math. Sci. Ser. A., 39, 5, 1087-1093 (2019) · Zbl 1449.35396 [37] Nocedal, J.; Wright, SJ, Numerical Optimization (1999), New York: Spinger, New York · Zbl 0930.65067 [38] Ortega, JM; Rheinboldt, WC, Iterative Solution of Nonlinear Equations in Several Variables (1970), New York: Academic Press, New York · Zbl 0241.65046 [39] Papageorgiou, NS; Rǎdulescu, VD; Repovs, DD, Nonlinear Analysis-theory and Methods (2019), Springer, Cham: Springer Monographs in Mathematics, Springer, Cham · Zbl 1414.46003 [40] Rǎdulescu, V.; Repovš, D., Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (2015), Boca Raton FL: CRC Press, Taylor and Francis Group, Boca Raton FL · Zbl 1343.35003 [41] Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7, 26-33 (1997) · Zbl 0898.90119 [42] Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1978), New York: Analysis of operators. Academic Press, New York · Zbl 0401.47001 [43] Reichel, B.; Leble, S., On convergence and stability of a numerical scheme of coupled nonlinear Schrödinger equations, Comput. Math. Appl., 55, 745-759 (2008) · Zbl 1142.65074 [44] Simon, B., Analysis with weak trace ideals and the number of bound states of Schrödinger operators, Trans. Amer. Math. Soc., 224, 367-380 (1976) · Zbl 0348.47017 [45] Spiridonov, V.; Zhedanov, A., Discrete Darboux transformations, the discrete-time Toda lattice and the Askey-Wilson polynomials, Methods Appl. Anal., 2, 369-398 (1995) · Zbl 0859.33017 [46] Taniguchi, T., Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl., 16, 5, 965-975 (1998) · Zbl 0911.60054 [47] Wang, J.; Liu, J., Existence of sign-changing solutions for fractional Schrödinger equations, Acta Anal. Funct. Appl., 21, 4, 349-355 (2019) · Zbl 1463.35470 [48] Wang, L., Rǎdulescu, V.D., Zhang, B.: Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems. J. Math. Phys. 60(1), 011506 (2019) (18 pp) · Zbl 1410.35208 [49] Wang, Y.; Liu, C., Application of a new algebraic dynamical algorithm to cylindrical nonlinear Schrödinger equation, Far East J. Dyn. Syst., 25, 2, 95-98 (2014) · Zbl 1376.37110 [50] Xu, J.; Shan, S., Multi-symplectic Fourier pseudo-spectral algorithm for a nonlinear Schrödinger equation involving quintic term, J. Numer. Methods Comput. Appl., 31, 1, 55-63 (2010) · Zbl 1240.65308 [51] Yan, Z.; Park, J.; Zhang, W., A unified framework for asymptotic and transient behavior of linear stochastic systems, Appl. Math. Comput., 325, 31-40 (2018) · Zbl 1428.93125 [52] Yuan, Y., Trust region algorithm for nonlinear equations, Information, 1, 7-21 (1998) · Zbl 1006.65048 [53] Zhang, S., Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations, AIMS Math., 7, 1, 1015-1034 (2022) [54] Zheng, Q.; Wu, D., Multiple solutions for Schrödinger equations involving concave-convex nonlinearities without $$(AR)$$-type condition, Bull. Malays. Math. Sci. Soc., 44, 5, 2943-2956 (2021) · Zbl 1473.35144 [55] Zhu, S.; Zhang, J., Concentration of blow-up solutions for the nonlinear Schrödinger equation with a potential, Acta Math. Appl. Sin., 39, 6, 938-953 (2016) · Zbl 1374.35394 [56] Zouraris, GE, On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation, Modél. Math. Anal. Numér., 35, 389-405 (2001) · Zbl 0991.65088
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