On an ill-posed problem for system of coupled sinh-Gordon equations
- Authors
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- Keywords:
- ill-posed problem, Sinh-Gordon equation, regularization
- Abstract
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The aim of this paper is considering the initial value problem for a system of coupled nonlinear sinh-Gordon equations by the association between two regularization methods: filter and truncation Fourier. Firstly, we give an example to show that the problem does not satisfy the third property which is called ill-posed in the sense of Hadamard. Secondly, under some a priori assumptions, we propose the stable regularization methods to regularize the system, i.e. the corresponding regularized solution converge to the exact solution in \(L^2\)-norm. Finally, to illustrate the proposed efficiency in the theoretical part, we show some numerical tests to check the convergence of the regularized solution and the regularized errors.
- References
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[1] M. M. Alves and A. Leitão, On level set type methods for elliptic Cauchy problems, Inverse Problems 23 (2007), no. 5, 2207–2222.
[2] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, Journal of Functional Analysis (2017), in press.
[3] C. Cao, M. A. Rammaha, and E. S. Titi, The Navier–Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom, Zeitschrift für Angewandte Mathematik und Physik 50 (1999), 341–360.
[4] E. N. Dancer and S. Yan, Multipeak solutions for an elliptic system of FitzHugh–Nagumo type, Mathematische Annalen 335 (2006), no. 3, 527–569.
[5] D. G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM Journal on Mathematical Analysis 17 (1986), no. 4, 836–849.
[6] Lawrence C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010.
[7] Jacques Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953.
[8] B. Kaltenbacher, A. Kirchner, and B. Vexler, Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems, Inverse Problems 27 (2011).
[9] B. Kaltenbacher and W. Polifke, Some regularization methods for a thermoacoustic inverse problem, Journal of Inverse and Ill-Posed Problems 18 (2011), 997–1011 (Special Issue: M. V. Klibanov).
[10] B. Kaltenbacher, F. Schoepfer, and Th. Schuster, Convergence of some iterative methods for the regularization of nonlinear ill-posed problems in Banach spaces, Inverse Problems 25 (2009), 19 pp.
[11] V. A. Khoa, M. T. N. Truong, N. H. M. Duy, and N. H. Tuan, The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing, Computers & Mathematics with Applications 73 (2017), no. 1, 141–162.
[12] A. Leitão and P. Kugler, Mean value iterations for nonlinear elliptic Cauchy problems, Numerische Mathematik 96 (2003), no. 2, 269–293.
[13] L. D. Long, H. D. Binh, D. Kumar, N. H. Luc, and N. H. Can, Stability of fractional order of time nonlinear fractional diffusion equation with Riemann–Liouville derivative, Mathematical Methods in the Applied Sciences 45 (2022), no. 10, 6194–6216.
[14] A. T. Nguyen, L. D. Long, D. Kumar, and V. T. Nguyen, Regularization of a final value problem for a linear and nonlinear biharmonic equation with observed data in Lq space, AIMS Mathematics 7 (2022), no. 12, 20660–20683.
[15] N. D. Phuong, L. D. Long, D. Kumar, and H. D. Binh, Determine unknown source problem for time fractional pseudo-parabolic equation with Atangana–Baleanu–Caputo fractional derivative, AIMS Mathematics 7 (2022), no. 9, 16147–16170.
[16] T. N. Thach, D. Kumar, N. H. Luc, and N. D. Phuong, On a semilinear fractional reaction-diffusion equation with nonlocal conditions, Alexandria Engineering Journal 60 (2021), no. 6, 5511–5520.
[17] N. H. Tuan, T. T. Binh, T. Q. Viet, and D. Lesnic, On the Cauchy problem for semilinear elliptic equations, Journal of Inverse and Ill-Posed Problems 24 (2016), no. 2, 123–138.
- Additional Files
- Published
- 28-03-2025
- Issue
- Vol. 1 No. 1 (2023)
- Section
- Research Article
- License
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Copyright (c) 2023 Devendra Kumar, Jagdev Singh

This work is licensed under a Creative Commons Attribution 4.0 International License.
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