A Class of Solutions of Initial-boundary Value Problems for the Korteweg-de Vries Equation on the Interval and Bäcklund Transformations
DOI:
https://doi.org/10.5890/JAND.2027.03.004Abstract
The initial-boundary value problems (IBVPs) for the Korteweg-de Vries (KdV) equation on the interval $(0, q)$, where $q$ is a large positive number, are solved by the Inverse Scattering Method (ISM). By virtue of the appropriate setup of the associated scattering problem (SP), the fundamental equation in this inverse SP (ISP) is reduced to a system of linear algebraic equations. The known function in this equation describes only the discrete spectrum of the SP. The solution of this system completely describes the whole family of recovered potentials $y(x, t)$ in the linear Schrödinger equation (LSEq). By the unique solvability of the ISP, the recovered potential $y(x, t)$ is a solution of the considered IBVP. A class of various scattering data (SD) sets is derived from various pairs of prescribed initial and boundary conditions. A class $G$ of solutions of IBVPs on the interval $(0, q)$ is constructed from the class of various SD sets of SPs. The class of Bäcklund transformations (BTs) linking each solution of the IBVP in the class $G$ with its corresponding common solution of linear equations of the Lax pair is derived.References
[1] Ablowitz, M.J. and Segur, H. (1981), Solitons and the Inverse Scattering Transform, SIAM, Studies in Applied Mathematics: Philadelphia.
[2] Bona, J. and Winther, R. (1983), The Korteweg--de Vries equation, posed in a quarter-plane, SIAM Journal on Mathematical Analysis, 14(6), 1056-1106.
[3] Debnath, L. and Bhatta, D. (2007), Integral Transforms and Their Applications, 2nd Edition, Chapman and Hall/CRC: New York.
[4] Fokas, A.S. (2002), Integrable nonlinear evolution equations on the half-line, Communications in Mathematical Physics, 230(1), 1-39.
[5] Fokas, A.S. and Its, A.R. (1992), An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates, Theoretical and Mathematical Physics, 92(3), 964-978.
[6] Fokas, A.S. and Its, A.R. (1996), The linearization of the initial-boundary value problem of the nonlinear Schrodinger equation, SIAM Journal on Mathematical Analysis, 27(3), 738-764.
[7] Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M. (1967), Method for solving the Korteweg-de Vries equation, Physical Review Letters, 19(19), 1095-1097.
[8] Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M. (1974), Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Communications on Pure and Applied Mathematics, 27(1), 97-133.
[9] Gardner, C.S. and Morikawa, G.K. (1960), Similarity in the Asymptotic Behaviour of Collision Free Hydrodynamic Waves and Water Waves, Courant Institute of Mathematical and Scientific Research, Report NYO-9082, New York University: New York.
[10] Grudsky, S.M., Kravchenko, V.V., and Torba, S.M. (2023), Realization of the inverse scattering transform method for the Korteweg–de Vries equation, Mathematical Methods in the Applied Sciences, 46(8), 9217-9251.
[11] Hirota, R. (1974), A new form of Backlund transformations and its relation to the inverse scattering problem, Progress of Theoretical Physics, 52(5), 1498-1512.
[12] Hirota, R. (1980), Direct methods in soliton theory, In Solitons. Springer-Verlag: Berlin, 157-176.
[13] Kravchenko, V.V. (2020), Direct and Inverse Sturm-Liouville Problems: A Method of Solution, Birkhauser: Cham.
[14] Kravchenko, V.V. (2019), On a method for solving the inverse scattering problem on the line, Mathematical Methods in the Applied Sciences, 42(4), 1321-1327.
[15] Levitan, B.M. (1984), Inverse Sturm-Liouville Problems, Nauka: Moscow. (English Translation VNU Science Press: Utrecht, 1987).
[16] Levitan, B.M. and Sargsjan, I.S. (1970), Introduction to Spectral Theory, Nauka: Moscow.
[17] Marchenko, V.A. (1972), Spectral Theory of Sturm-Liouville Operators, Nauk: Kiev. (English Translation: Sturm-Liouville Operators and Applications, Birkhauser: Basel, 1986).
[18] Miura, R.M. (1968), Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, Journal of Mathematical Physics, 9(8), 1202-1204.
[19] Miura, R.M. (1976), The Korteweg-de Vries equation: A survey of results, SIAM Review, 18(3), 412-459.
[20] Miura, R.M. (1976), Backlund transformations, the inverse scattering method, solitons, and their applications. In Lecture Notes in Mathematics 515, NSF Research Workshop on Contact Transformations, Springer-Verlag: Berlin-Heidelberg-New York.
[21] Segur, H. (1973), The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, Journal of Fluid Mechanics, 59(4), 721-736.
[22] Vu, P.L. (1997), Explicit complex-valued solutions of the Korteweg--de Vries equation on the half-line and on the whole-line, Acta Applicandae Mathematica, 49(2), 107-149.
[23] Vu, P.L. and Hoang, N.H. (2000), On the degree of normalization polynomials of the scattering data for constructing solutions of the Korteweg-de Vries equation, Southeast Asian Bulletin of Mathematics, 24, 631-641.
[24] Vu, P.L. (2001), Some problems for a system of nonlinear equations on a half-line, Inverse Problems, 17(6), 1889-1906.
[25] Vu, P.L. (2004), Some problems for cubic nonlinear equations on a half-line, Acta Applicandae Mathematica, 84(1), 97-120.
[26] Vu, P.L. (2005), The Dirichlet initial-boundary-value problems for sine and sinh-Gordon equations on a half-line, Inverse Problems, 21(4), 1225-1248.
[27] Vu, P.L. (2007), The initial-boundary value poblem for the Korteweg-de Vries equation on the positive quarter-plane, Journal of Nonlinear Mathematical Physics, 14(1), 28-43.
[28] Vu, P.L. (2014), An initial-boundary value problem for the Korteweg-de Vries equation with dominant surface tension, Acta Applicandae Mathematicae, 129(1), 41-59.
[29] Vu, P.L. (2018), The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations, Journal of Nonlinear Mathematical Physics, 25(3), 399-432.
[30] Vu, P.L. (2024), The Bäcklund transformation between a common solution of both linear equations in the Lax pair and solution of the associated initial-boundary value problem for nonlinear equations on the half-line, Journal of Mathematical Sciences, 289, 67-94.
[31] Whitham, G.B. (1974), Linear and Nonlinear Waves, John Wiley: New York.
[32] Zakharov, V.E., Manakov, S.V., Novikov, S. and Pitaievskii, L.P. (1980), Theory of Solitons, the Inverse Method, Nauka: Moscow.
Article Metrics
Usage tracked since September 1, 2026.