Log in

Relevant bibliographies by topics / Gelfand–Levitan method / Journal articles

To see the other types of publications on this topic, follow the link: Gelfand–Levitan method.

Journal articles on the topic 'Gelfand–Levitan method'

Author: Grafiati

Published: 5 June 2025

Last updated: 24 June 2025

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the top 28 journal articles for your research on the topic 'Gelfand–Levitan method.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

Temirbekov, N. М., S. I. Kabanikhin, L. N. Тemirbekova, and Zh E. Demeubayeva. "Gelfand-Levitan integral equation for solving coefficient inverse problem." Bulletin of the National Engineering Academy of the Republic of Kazakhstan 85, no. 3 (2022): 158–67. http://dx.doi.org/10.47533/2020.1606-146x.184.

Full text

Abstract:

In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

APA, Harvard, Vancouver, ISO, and other styles

2

Bakanov, G. B., and S. K. Meldebekova. "Relationship between solutions of discrete inverse and auxiliary problems for a hyperbolic equation." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy) 31, no. 4 (2024): 19–27. https://doi.org/10.47526/2024-4/2524-0080.13.

Full text

Abstract:

One-dimensional and multidimensional methods for solving inverse problems for a hyperbolic equation by the Gelfand-Levitan method lead to the numerical solution of Fredholm integral equations of the first and second kind. This paper examines the connection between a discrete inverse problem and solutions to a discrete auxiliary problem for a hyperbolic equation studied by the Gelfand-Levitan method. First, we present the formulations of the discrete inverse problem and the discrete auxiliary problem for the hyperbolic equation, as well as the properties of the grid functions that are solutions to these problems. Lemmas are proved about the structure of the grid function determined by the solution of the auxiliary discrete problem, and the connection of this grid function with the desired grid function, which is the solution to the discrete inverse problem. When proving the lemmas, the properties of the solution to the discrete auxiliary problem and the discrete analogue of the Dirac delta function are taken into account. A theorem showing the existence of a discrete inverse problem solution and its uniqueness is proved.

APA, Harvard, Vancouver, ISO, and other styles

3

Kabanikhin, Sergey I., Nikita S. Novikov, and Maxim A. Shishlenin. "Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem." Journal of Physics: Conference Series 2092, no. 1 (2021): 012022. http://dx.doi.org/10.1088/1742-6596/2092/1/012022.

Full text

Abstract:

Abstract In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

APA, Harvard, Vancouver, ISO, and other styles

4

Kabanikhin, Sergey, Maxim Shishlenin, Nikita Novikov, and Nikita Prokhoshin. "Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations." Mathematics 11, no. 21 (2023): 4458. http://dx.doi.org/10.3390/math11214458.

Full text

Abstract:

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.

APA, Harvard, Vancouver, ISO, and other styles

5

Becker, Kyle M., and George V. Frisk. "Application of the Gelfand–Levitan method to geoacoustic inversion in shallow water." Journal of the Acoustical Society of America 108, no. 5 (2000): 2536. http://dx.doi.org/10.1121/1.4743394.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Chaturvedi, S., and K. Raghunathan. "Relation between the Gelfand-Levitan procedure and the method of supersymmetric partners." Journal of Physics A: Mathematical and General 19, no. 13 (1986): L775—L778. http://dx.doi.org/10.1088/0305-4470/19/13/004.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

HORVÁTH, MIKLÓS, and BARNABÁS APAGYI. "SOLUTION OF THE INVERSE SCATTERING PROBLEM AT FIXED ENERGY FOR POTENTIALS BEING ZERO BEYOND A FIXED RADIUS." Modern Physics Letters B 22, no. 23 (2008): 2137–49. http://dx.doi.org/10.1142/s0217984908016923.

Full text

Abstract:

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

APA, Harvard, Vancouver, ISO, and other styles

8

Novikov, Nikita, and Maxim Shishlenin. "Direct Method for Identification of Two Coefficients of Acoustic Equation." Mathematics 11, no. 13 (2023): 3029. http://dx.doi.org/10.3390/math11133029.

Full text

Abstract:

We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach.

APA, Harvard, Vancouver, ISO, and other styles

9

Romanov, V. G. "On Justification of the Gelfand–Levitan–Krein Method for a Two-Dimensional Inverse Problem." Siberian Mathematical Journal 62, no. 5 (2021): 908–24. http://dx.doi.org/10.1134/s003744662105013x.

Full text

APA, Harvard, Vancouver, ISO, and other styles

10

Romanov, V. G. "On justification of the Gelfand–Levitan–Krein method for a two-dimensional inverse problem." Sibirskii matematicheskii zhurnal 62, no. 5 (2021): 1124–42. http://dx.doi.org/10.33048/smzh.2021.62.513.

Full text

APA, Harvard, Vancouver, ISO, and other styles

11

Medvedev, S. B., I. A. Vaseva, and M. P. Fedoruk. "High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand–Levitan–Marchenko equation." Communications in Nonlinear Science and Numerical Simulation 138 (November 2024): 108255. http://dx.doi.org/10.1016/j.cnsns.2024.108255.

Full text

APA, Harvard, Vancouver, ISO, and other styles

12

Mardona, Ganjayeva, and Berganova Feruza. "Integration of the Loaded Cordeveg-De Fries Equation in a Class of Fast Decreasing Functions." International Journal for Research in Applied Science and Engineering Technology 10, no. 7 (2022): 1621–30. http://dx.doi.org/10.22214/ijraset.2022.45627.

Full text

Abstract:

Annotation: This article is devoted to the integration of the loaded source Korteweg-de Vries equation in the class of rapidly decreasing functions. In this work, the Cauchy problem imposed on the Korteweg-de Vries equation was solved using the inverse problem method of the Sturm-Liouville operator scattering theory. Their Yost solutions are defined and integral Levin images are obtained for them. The givens of the scattering theory were described and some of their necessary properties were given, the Gelfand-Levitan-Marchenko integral equation, which is the main integral equation of the inverse problems of the scattering theory, was derived.

APA, Harvard, Vancouver, ISO, and other styles

13

Tanirbergenov, M. "INVERSE SCATTERING PROBLEM FOR A SYSTEM OF EQUATIONS DIRAC ON THE WHOLE LINE." Danish scientific journal, no. 71 (April 24, 2023): 36–50. https://doi.org/10.5281/zenodo.7878609.

Full text

Abstract:

<strong>Abstract</strong> In this paper, we study the inverse scattering problem for the Dirac operator on the whole line with real continuous coefficients  and , which tend to zero  rather quickly as and the Dirac operator with these coefficients considered on the semiaxis  has a purely discrete spectrum. For the Dirac operator under consideration, the –function is introduced, its properties are studied, the Gelfand–Levitan–Marchenko integral equation is derived, and the procedure for restoring the coefficients  and  is given.

APA, Harvard, Vancouver, ISO, and other styles

14

Avdonin, S. A., B. P. Belinskiy, and John V. Matthews. "Inverse problem on the semi-axis: local approach." Tamkang Journal of Mathematics 42, no. 3 (2011): 275–93. http://dx.doi.org/10.5556/j.tkjm.42.2011.916.

Full text

Abstract:

We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

APA, Harvard, Vancouver, ISO, and other styles

15

Tataris, Andreas, and Tristan van Leeuwen. "A Regularised Total Least Squares Approach for 1D Inverse Scattering." Mathematics 10, no. 2 (2022): 216. http://dx.doi.org/10.3390/math10020216.

Full text

Abstract:

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense.

APA, Harvard, Vancouver, ISO, and other styles

16

Doikou, Anastasia, and Iain Findlay. "Solitons: Conservation laws and dressing methods." International Journal of Modern Physics A 34, no. 06n07 (2019): 1930003. http://dx.doi.org/10.1142/s0217751x19300035.

Full text

Abstract:

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

APA, Harvard, Vancouver, ISO, and other styles

17

Belaiа, O. V., L. L. Frumin, and A. E. Chernyavsky. "Algorithms for solving the inverse scattering problem for the Manakov model." Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki 64, no. 3 (2024): 486–98. http://dx.doi.org/10.31857/s0044466924030091.

Full text

Abstract:

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

APA, Harvard, Vancouver, ISO, and other styles

18

Kincanon, Eric. "Modelling silicon etching using inverse methods." International Journal of Applied Mathematical Research 11, no. 1 (2022): 8–10. http://dx.doi.org/10.14419/ijamr.v11i1.31930.

Full text

Abstract:

This paper considers a real-world application of a recently presented alternative form of the Gelfand-Levitan equation. Here is considered the case of potential in the plasma above silicon during the etching process. It is shown that although standard methods have significant challenges, the alternative form of the Gelfand-Levitan equation gives a straightforward way to determine the reflection coefficient from an assumed potential.Â Â

APA, Harvard, Vancouver, ISO, and other styles

19

Kincanon, Eric. "Alternative form of the Gelfand Levitan equation." International Journal of Applied Mathematical Research 10, no. 2 (2021): 28. http://dx.doi.org/10.14419/ijamr.v10i2.31842.

Full text

Abstract:

This paper presents an alternative form of the Gelfand-Levitan Equation. By assuming a particular form of the spectral measure function and the potential kernel, an equation relating the potential and the reflection coefficient is found. This equation has an advantage over the Gelfand-Levitan Equation in that it can be solved without using iterative methods. The validity of the equation is demonstrated by looking at a singular and non-singular potential.

APA, Harvard, Vancouver, ISO, and other styles

20

Shubov, Marianna A. "Transformation operators for class of damped hyperbolic equations." Asymptotic Analysis 24, no. 3-4 (2000): 183–208. https://doi.org/10.3233/asy-2000-411.

Full text

Abstract:

We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.

APA, Harvard, Vancouver, ISO, and other styles

21

A. Kincanon, Eric. "Uniqueness of approximate solutions to the Gelfand Levitan equation." International Journal of Applied Mathematical Research 9, no. 1 (2020): 32. http://dx.doi.org/10.14419/ijamr.v9i1.30522.

Full text

Abstract:

This brief paper considers a potential issue of using iterative solutions for the Gelfand-Levitan equation. Iterative solutions require approx-imation methods and this could lead to a loss of uniqueness of solutions. The calculations in this paper demonstrate that this is not the case and that uniqueness is preserved.

APA, Harvard, Vancouver, ISO, and other styles

22

Treumann, Rudolf A., Wolfgang Baumjohann, and Yasuhito Narita. "Inverse scattering problem in turbulent magnetic fluctuations." Annales Geophysicae 34, no. 8 (2016): 673–89. http://dx.doi.org/10.5194/angeo-34-673-2016.

Full text

Abstract:

Abstract. We apply a particular form of the inverse scattering theory to turbulent magnetic fluctuations in a plasma. In the present note we develop the theory, formulate the magnetic fluctuation problem in terms of its electrodynamic turbulent response function, and reduce it to the solution of a special form of the famous Gelfand–Levitan–Marchenko equation of quantum mechanical scattering theory. The last of these applies to transmission and reflection in an active medium. The theory of turbulent magnetic fluctuations does not refer to such quantities. It requires a somewhat different formulation. We reduce the theory to the measurement of the low-frequency electromagnetic fluctuation spectrum, which is not the turbulent spectral energy density. The inverse theory in this form enables obtaining information about the turbulent response function of the medium. The dynamic causes of the electromagnetic fluctuations are implicit to it. Thus, it is of vital interest in low-frequency magnetic turbulence. The theory is developed until presentation of the equations in applicable form to observations of turbulent electromagnetic fluctuations as input from measurements. Solution of the final integral equation should be done by standard numerical methods based on iteration. We point to the possibility of treating power law fluctuation spectra as an example. Formulation of the problem to include observations of spectral power densities in turbulence is not attempted. This leads to severe mathematical problems and requires a reformulation of inverse scattering theory. One particular aspect of the present inverse theory of turbulent fluctuations is that its structure naturally leads to spatial information which is obtained from the temporal information that is inherent to the observation of time series. The Taylor assumption is not needed here. This is a consequence of Maxwell's equations, which couple space and time evolution. The inversion procedure takes advantage of a particular mapping from time to space domains. Though the theory is developed for homogeneous stationary non-flowing media, its extension to include flows, anisotropy, non-stationarity, and the presence of spectral lines, i.e. plasma eigenmodes like those present in the foreshock or the magnetosheath, is obvious.

APA, Harvard, Vancouver, ISO, and other styles

23

Kabanikhin, Sergey I., Karl K. Sabelfeld, Nikita S. Novikov, and Maxim A. Shishlenin. "Numerical solution of the multidimensional Gelfand–Levitan equation." Journal of Inverse and Ill-posed Problems 23, no. 5 (2015). http://dx.doi.org/10.1515/jiip-2014-0018.

Full text

Abstract:

AbstractThe coefficient inverse problem for the two-dimensional wave equation is solved. We apply the Gelfand–Levitan approach to transform the nonlinear inverse problem to a family of linear integral equations. We consider the Monte Carlo method for solving the Gelfand–Levitan equation. We obtain the estimation of the solution of the Gelfand–Levitan equation in one specific point, due to the properties of the method. That allows the Monte Carlo method to be more effective in terms of span cost, compared with regular methods of solving linear system. Results of numerical simulations are presented.

APA, Harvard, Vancouver, ISO, and other styles

24

Medvedev, Sergey, Irina Vaseva, and Mikhail Fedoruk. "Block Toeplitz Inner-Bordering method for the Gelfand–Levitan–Marchenko equations associated with the Zakharov–Shabat system." Journal of Inverse and Ill-posed Problems, February 7, 2023. http://dx.doi.org/10.1515/jiip-2022-0072.

Full text

Abstract:

Abstract We propose a generalized method for solving the Gelfand–Levitan–Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.

APA, Harvard, Vancouver, ISO, and other styles

25

Ismailov, Mansur I. "Inverse scattering transform in two spatial dimensions for the N-wave interaction problem with a dispersive term." Journal of Inverse and Ill-posed Problems, July 28, 2021. http://dx.doi.org/10.1515/jiip-2020-0111.

Full text

Abstract:

Abstract A dispersive N-wave interaction problem ( N = 2 ⁢ n {N=2n} ), involving n velocities in two spatial and one temporal dimensions, is introduced. Explicit solutions of the problem are provided by using the inverse scattering method. The model we propose is a generalization of both the N-wave interaction problem and the ( 2 + 1 ) {(2+1)} matrix Davey–Stewartson equation. The latter examines the Benney-type model of interactions between short and long waves. Referring to the two-dimensional Manakov system, an associated Gelfand–Levitan–Marchenko-type, or so-called inversion-like, equation is constructed. It is shown that the presence of the degenerate kernel reads explicit soliton-like solutions of the dispersive N-wave interaction problem. We also present a discussion on the uniqueness of the solution of the Cauchy problem on an arbitrary time interval for small initial data.

APA, Harvard, Vancouver, ISO, and other styles

26

Kabanikhin, S. I., and G. B. Bakanov. "A discrete analog of the Gelfand-Levitan method in a two-dimensional inverse problem for a hyperbolic equation." Siberian Mathematical Journal 40, no. 2 (1999). http://dx.doi.org/10.1007/s11202-999-0007-6.

Full text

APA, Harvard, Vancouver, ISO, and other styles

27

Snieder, Roel. "How time-discretization can break the asymptotics of inverse scattering." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 479, no. 2275 (2023). http://dx.doi.org/10.1098/rspa.2023.0177.

Full text

Abstract:

Inverse scattering by the Marchenko or Gelfand–Levitan equations consists of two steps. The first step consists of solving an integral equation to retrieve the wavefield in the interior of a scattering medium from waves recorded outside of the scattering region. The second step consists of estimating the scattering potential from this wavefield. The estimation of the potential can be justified either by taking a high-frequency limit, or by evaluating the wavefield an infinitesimal time before or after the direct wave. Waveforms obtained in experiments are discretized in time with a sampling interval d t . We show an example that this time discretization precludes the extraction of the potential. The reason is that the frequencies of discretized waveforms are below the Nyquist frequency. This limitation precludes taking the high-frequency limit that is used for estimating the potential from the waveforms in the interior of the scattering region. We present an alternative method to estimate the potential from wavefields in the interior of the scattering region. This method can be used even when the waves are in the strong scattering regime.

APA, Harvard, Vancouver, ISO, and other styles

28

Ye, Rusuo, and Yi Zhang. "Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval." Discrete and Continuous Dynamical Systems - S, 2022, 0. http://dx.doi.org/10.3934/dcdss.2022111.

Full text

Abstract:

<p style='text-indent:20px;'>We apply the Fokas unified method to study initial-boundary value (IBV) problems for the two-component complex modified Korteweg-de Vries (mKdV) equation with a <inline-formula><tex-math id="M1">\begin{document}$ 4\times 4 $\end{document}</tex-math></inline-formula> Lax pair on the interval. The solution can be written by the solution of a <inline-formula><tex-math id="M2">\begin{document}$ 4\times 4 $\end{document}</tex-math></inline-formula> Riemann-Hilbert (RH) problem constructed in the complex <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-plane. The relevant jump matrices are explicitly expressed in terms of three matrix-valued spectral functions related to the initial values, and the Dirichlet-Neumann boundary values, respectively. Moreover, we get that these spectral functions satisfy a global relation and also study the asymptotic analysis of the spectral functions. By considering the global relation, we express the unknown boundary values in terms of the known initial and boundary values via a Gelfand-Levitan-Marchenko (GLM) representation.</p>

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!