Grudsky, Kravchenko and Torba (2023) turned the inverse scattering transform (IST) for the Korteweg–de Vries (KdV) equation into a practical numerical scheme: both the direct and the inverse scattering problem for the associated Schrödinger operator are reduced to Fourier–Laguerre-type series coefficients and a system of linear algebraic equations solved by the finite section method, with the potential recovered by two differentiations of the first coefficient. The authors gave a rigorous justification of the method — including applicability of the finite section method and an explicit algebraic decay rate for the Hankel matrix entries in dependence on the decay exponent α\alpha of the potential — and demonstrated accuracy and speed on examples, but the proved rates are entrywise and are never converted into quantitative, end-to-end error bounds for the reconstructed potential, and no rule for choosing the truncation size is given. The open problem arising from this gap is to give explicit computable a priori error bounds for the recovered potential that combine all error sources of the pipeline — truncation of the linear system, quadrature of the oscillatory Fourier-type integrals, errors in the computed series coefficients, and numerical double differentiation — to derive from them a practical truncation-selection rule guaranteeing a prescribed accuracy, and to settle whether and how far the paper's hypotheses can be weakened while keeping such bounds valid, and where they genuinely break down. Follow-up work by the same school (Zakharov–Shabat 2025, AKNS 2026, complex potentials 2026) extends the approach to other systems while leaving this error analysis open.

Background

The Korteweg–de Vries (KdV) equation ut−6uux+uxxx=0u_t-6uu_x+u_{xxx}=0 models shallow-water waves and admits soliton solutions. The inverse scattering transform (IST), introduced by Gardner, Greene, Kruskal and Miura in 1967, solves its initial value problem by encoding the initial profile q(x)q(x) into scattering data of the one-dimensional Schrödinger operator −y′′+q(x)y=ρ2y-y''+q(x)y=\rho^2 y (the reflection coefficient, the bound-state eigenvalues, and the norming constants — the weights fixing the normalization of the bound-state eigenfunctions), evolving these data trivially in time, and recovering the solution by solving the Gelfand–Levitan–Marchenko (GLM) integral equation — the classical linear integral equation of inverse scattering theory that reconstructs the potential from scattering data. In theory elegant, the IST was rarely used for general (non-solitonic) numerical evolution because solving the direct and inverse scattering problems on the whole line numerically was considered difficult; practitioners instead used finite-interval discretizations with artificial boundary conditions.

Grudsky, Kravchenko and Torba (2023) proposed a practical realization. Their method rests on representations of the Jost solutions (the Schrödinger eigenfunctions normalized by decay conditions at ±∞\pm\infty) and of the kernels of transmutation operators (operators mapping free Schrödinger solutions onto solutions for the potential qq) as Fourier–Laguerre series, i.e. series in Laguerre functions whose first coefficient a0(x)a_0(x) or b0(x)b_0(x) alone determines everything relevant: a0a_0 gives the Jost solution at one distinguished spectral point, and the potential is recovered from it by two differentiations, q=(a0′′−a0′)/(a0+1)q=(a_0''-a_0')/(a_0+1). The direct problem reduces to computing the series coefficients by a recurrent integration procedure (given the Jost solutions at the spectral point ρ=i/2\rho=i/2, themselves obtained from a Cauchy problem solved e.g. by the spectral parameter power series method — a representation of the needed solution as a power series in the spectral parameter whose coefficients are computed by repeated quadratures of the potential); the inverse problem reduces to a system of linear algebraic equations for the coefficients, derived both from the GLM equation and, equivalently, from a Hankel-operator equation (a Hankel operator is the integral operator on the half-line whose kernel depends on the sum of the arguments; the reduction of the KdV inverse problem to Hankel operators is due to Grudsky and Rybkin). The infinite system is solved approximately by the finite section method — truncating to indices m,n≤Nsm,n\le N_s.

What the paper proves stays at the level of the Hankel operator and its matrix entries, not of the reconstructed potential: under the decay assumption q∈L1(R,(1+∣x∣)α)q\in L^1(\mathbb{R},(1+|x|)^\alpha), α≥1\alpha\ge 1, together with a factorization condition on the reflection coefficient and a dominance inequality on an antiderivative of qq (their (7.3)–(7.7)), the Hankel operator is compact and I+H(φ)I+H(\varphi) invertible in ℓ1\ell^1, and the finite section method is applicable (Theorems 7.3–7.4); moreover, the entries of the Hankel matrix are proved to satisfy the explicit algebraic decay estimate ∣hp∣=O(1/∣p∣l−1/2)|h_p|=O(1/|p|^{l-1/2}) with l=⌊45α+1⌋l=\lfloor\tfrac{4}{5}\alpha+1\rfloor, and the finite section method is applicable in the Wiener class WβW^\beta for any 0≤β<l−320\le\beta<l-\tfrac{3}{2} (Theorems 7.6–7.7). These entrywise rates are never turned into a bound for the reconstructed potential. Numerically, the matrix entries are computed from oscillatory Fourier-type integrals over the real line (their (6.4)–(6.7)) by a change of variable to the unit circle and the trapezoidal rule with 10410^4 points; the final recovery applies spline double differentiation. The truncation size NsN_s (5 or 9 equations in the examples) is chosen ad hoc, and accuracies such as 1.5⋅10−31.5\cdot 10^{-3} are reported empirically, not derived. Truncation error estimates exist only for the Jost-solution series representations themselves (Delgado–Khmelnytskaya–Kravchenko 2020; Kravchenko's 2020 book, Theorem 10.2), not for the full direct-plus-inverse pipeline.

Problem Statement

For the numerical inverse scattering transform scheme of Grudsky–Kravchenko–Torba (arXiv:2206.01317; Mathematical Methods in the Applied Sciences, 2023) for the KdV equation — in which the direct problem is solved by the recurrent integration procedure for the Fourier–Laguerre series coefficients (with the Jost solutions at the spectral point ρ=i/2\rho=i/2 obtained from a Cauchy problem), the inverse problem is reduced to the linear algebraic systems obtained from the truncated Hankel/GLM equation with matrix entries given by the oscillatory Fourier-type integrals (their (6.4)–(6.7)) evaluated under the unit-circle change of variable by the trapezoidal rule, and the potential is recovered from the first coefficient a0a_0 (or b0b_0) by the double differentiation q=(a0′′−a0′)/(a0+1)q=(a_0''-a_0')/(a_0+1) — establish a complete a priori error analysis. Specifically: (i) prove explicit, computable error bounds for the reconstructed potential q~\tilde q in at least one norm of interest (for example the uniform norm on a given spatial interval at a given time tt) that combine the finite-section truncation error of the linear system at truncation size NsN_s, the quadrature error of the oscillatory integrals, the errors propagated from the numerically computed series coefficients and Jost solutions, and the error of the numerical double differentiation, with all constants explicit or computable from the decay and smoothness parameters of the initial profile; (ii) derive from these bounds a practical rule for selecting the truncation size NsN_s (and the companion discretization parameters) that guarantees a prescribed target accuracy uniformly over a stated reconstruction domain and time; and (iii) determine whether and how far the hypotheses (7.3)–(7.7) used in the paper (the decay of qq, the factorization (7.4)–(7.5) of the reflection coefficient, and the dominance inequality (7.7)) can be weakened while keeping such bounds valid — how far strictly weaker assumptions than (7.3)–(7.7) still suffice for the bounds of (i) and where those bounds genuinely break down, whether as a sharp threshold within the decay family q∈L1(R,(1+∣x∣)α)q\in L^1(\mathbb{R},(1+|x|)^\alpha) or through explicit failure examples under strictly weaker assumptions, and, if no weakening of (7.3)–(7.7) is possible, why.

References

  1. Grudsky, S. M., Kravchenko, V. V., 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. DOI 10.1002/mma.9049. arXiv:2206.01317. https://doi.org/10.1002/mma.9049
  2. Delgado, B. B., Khmelnytskaya, K. V., Kravchenko, V. V. (2020). A representation for Jost solutions and an efficient method for solving the spectral problem on the half line. Mathematical Methods in the Applied Sciences 43(16), 9304–9319. DOI 10.1002/mma.5881. https://doi.org/10.1002/mma.5881
  3. Kravchenko, V. V. (2020). Direct and Inverse Sturm–Liouville Problems: A Method of Solution. Frontiers in Mathematics. Birkhäuser/Springer, Cham. DOI 10.1007/978-3-030-47849-0. https://doi.org/10.1007/978-3-030-47849-0
  4. Kravchenko, V. V. (2025). Spectral parameter power series for Zakharov–Shabat direct and inverse scattering problems. Mathematical Methods in the Applied Sciences 48(16), 14661–14671. DOI 10.1002/mma.11205. arXiv:2505.02169. https://doi.org/10.1002/mma.11205
  5. Kravchenko, V. V. (2026). Approximate direct and inverse scattering for the AKNS system. Journal of Inverse and Ill-Posed Problems 34(3), 395–418. DOI 10.1515/jiip-2025-0096. arXiv:2507.05525. https://doi.org/10.1515/jiip-2025-0096
  6. Khmelnytskaya, K. V., Kravchenko, V. V., Murcia-Lozano, L. E. (2026). Numerical method for the inverse scattering problem for a complex potential of compact support. Lobachevskii Journal of Mathematics 47(3), 1033–1042. DOI 10.1134/S1995080226616358. https://doi.org/10.1134/S1995080226616358
  7. Rybkin, A. (2024). Trace formulas revisited and a new representation of KdV solutions with short-range initial data. Nonlinearity 37(11), 115008. DOI 10.1088/1361-6544/ad7c10. arXiv:2308.13656. https://doi.org/10.1088/1361-6544/ad7c10
  8. Grudsky, S. M., Rybkin, A. (2015). Soliton theory and Hankel operators. SIAM Journal on Mathematical Analysis 47(3), 2283–2323. DOI 10.1137/151004926. https://doi.org/10.1137/151004926
  9. Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M. (1967). Method for solving the Korteweg-deVries equation. Physical Review Letters 19, 1095–1097. DOI 10.1103/PhysRevLett.19.1095. https://doi.org/10.1103/PhysRevLett.19.1095