February  2019, 13(1): 159-175. doi: 10.3934/ipi.2019009

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

* Corresponding author: Teemu Tyni

Received  February 2018 Revised  September 2018 Published  December 2018

Fund Project: This work was supported by the Academy of Finland (application number 250215, the Centre of Excellence in Inverse Problems Research (2014–2017) and application number 312123, the Centre of Excellence of Inverse Modelling and Imaging (2018–2025)).

We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.

Citation: Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
References:
[1]

T. Aktosun and V. G. Papanicolaou, Time evolution of the scattering data for a fourth-order linear differential operator, Inverse Problems, 24 (2008), 055013, 14 pp. doi: 10.1088/0266-5611/24/5/055013.  Google Scholar

[2]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[3]

L. Hörmander, The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[4]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.  doi: 10.4099/math1924.14.1.  Google Scholar

[5]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.  doi: 10.4099/math1924.14.1.  Google Scholar

[6]

R. Kanwal, Generalized Functions: Theory and Technique, Academic Press, New York, 1983.  Google Scholar

[7]

B. Pausader, Scattering for the beam equation in low dimensions, preprint, arXiv: 0903.3777v2. Google Scholar

[8]

V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and General, 42 (2009), 332002, 7pp. doi: 10.1088/1751-8113/42/33/332002.  Google Scholar

[9]

V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of mathematical physics, 53 (2012), 123522, 16pp. doi: 10.1063/1.4769825.  Google Scholar

[10]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[11]

T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20pp. doi: 10.1088/1361-6420/aa873e.  Google Scholar

[12]

T. Tyni, M. Harju and V. Serov, Recovery of singularities in fourth-order operator on the line from limited data, Inverse Problems, 32 (2016), 075001, 22pp. doi: 10.1088/0266-5611/32/7/075001.  Google Scholar

[13]

T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Prob. Imag., 12 (2018), 205-227.  doi: 10.3934/ipi.2018008.  Google Scholar

[14]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.  Google Scholar

show all references

References:
[1]

T. Aktosun and V. G. Papanicolaou, Time evolution of the scattering data for a fourth-order linear differential operator, Inverse Problems, 24 (2008), 055013, 14 pp. doi: 10.1088/0266-5611/24/5/055013.  Google Scholar

[2]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[3]

L. Hörmander, The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[4]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅰ, Japan. J. Math., 14 (1988), 1-57.  doi: 10.4099/math1924.14.1.  Google Scholar

[5]

K. Iwasaki, Scattering theory for 4th order differential operators: Ⅱ, Japan. J. Math., 14 (1988), 59-96.  doi: 10.4099/math1924.14.1.  Google Scholar

[6]

R. Kanwal, Generalized Functions: Theory and Technique, Academic Press, New York, 1983.  Google Scholar

[7]

B. Pausader, Scattering for the beam equation in low dimensions, preprint, arXiv: 0903.3777v2. Google Scholar

[8]

V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and General, 42 (2009), 332002, 7pp. doi: 10.1088/1751-8113/42/33/332002.  Google Scholar

[9]

V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of mathematical physics, 53 (2012), 123522, 16pp. doi: 10.1063/1.4769825.  Google Scholar

[10]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[11]

T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20pp. doi: 10.1088/1361-6420/aa873e.  Google Scholar

[12]

T. Tyni, M. Harju and V. Serov, Recovery of singularities in fourth-order operator on the line from limited data, Inverse Problems, 32 (2016), 075001, 22pp. doi: 10.1088/0266-5611/32/7/075001.  Google Scholar

[13]

T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Prob. Imag., 12 (2018), 205-227.  doi: 10.3934/ipi.2018008.  Google Scholar

[14]

E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.  Google Scholar

Figure 1.  Coefficients $q_1$ and $q_0$
Figure 2.  Numerical reconstruction (red) and the unknown combination (black)
[1]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090

[4]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[5]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[6]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[7]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[8]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

[9]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

[10]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315

[11]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[12]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[13]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279

[14]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[15]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

[16]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[17]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020389

[18]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[19]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006

[20]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (161)
  • HTML views (202)
  • Cited by (0)

Other articles
by authors

[Back to Top]