Magnus Expansion for the Direct Scattering Transform: High-Order Schemes

Mullyadzhanov, R. I.; Gelash, Andrey

doi:10.1007/s11141-021-10096-6

Cited by 6 publications

(7 citation statements)

References 65 publications

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“…Our derivation is based on the asymptotic behaviour of the wave function (2.8) for spatially localized KM and TW breathers. To evaluate the scattering matrix numerically, we use the standard second-order approach from [47] and the recently developed advanced high-order numerical schemes based on the Magnus expansion [37,48]. Our numerical tests demonstrate accurate recovery of the breather’s parameters and confirm the declared convergence orders of the developed schemes.…”

Section: Discussionmentioning

confidence: 80%

“…At the final stage, one retrieves numerically the scattering dataset falsefalse{λnnum,ρnnum,rnums,rnumMIfalsefalse} using the relations (4.4) and (4.5). To compute T^, we use the Boffetta–Osborne second-order scheme [47] together with recently proposed fourth- and sixth-order schemes based on the Magnus expansion [37,48]. We reproduce these numerical schemes in full in appendix §Ab.…”

Section: Direct Scattering Transform Algorithmmentioning

confidence: 99%

“…We consider the scattering problem for the ZS system with boundary conditions corresponding to the non-vanishing background field and find analytical relations between scattering coefficients and transfer matrix elements in order to identify breathers instead of solitons. To compute the transfer matrix numerically, we use the standard second-order Boffetta–Osborne scheme [47] together with recently proposed high-order numerical schemes based on the Magnus expansion [37,48]. We construct localized single- and multi-breather solutions of the fNLSE using the dressing method and verify the developed numerical approach to be able to accurately recover a complete set of the scattering data providing the information about the amplitude, velocity, phase and position of breathers.…”

Section: Introductionmentioning

confidence: 99%

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Numerical direct scattering transform for breathers

Mullyadzhanov,

Gudko,

Mullyadzhanov

et al. 2024

Proc. R. Soc. A.

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We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.

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Section: Discussionmentioning

confidence: 80%

Section: Direct Scattering Transform Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical direct scattering transform for breathers

Mullyadzhanov,

Gudko,

Mullyadzhanov

et al. 2024

Proc. R. Soc. A.

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“…Numerical methods for solving the direct SP are relatively well developed by now; see the references in [11,25,32]. In general, the inverse SP reconstructs u(x) by the available set of the spectral data.…”

Section: Statements Of Right and Left Scattering Problems For The Zs ...mentioning

confidence: 99%

“…Now we multiply equation ( 26) by σ (note that σ 2 = 1) and find the left transition matrix by writing the equations (25)(26) in matrix form:…”

Section: On the Connection Between The Right And Left Scatter-ing Pro...mentioning

confidence: 99%

Right and left inverse scattering problems formulations for the Zakharov-Shabat system

Chernyavsky¹,

Фрумин²,

Gelash³

2022

Preprint

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We consider right and left formulations of the inverse scattering problem for the Zakharov-Shabat system and the corresponding integral Gelfand-Levitan-Marchenko equations. Both formulations are helpful for numerical solving of the inverse scattering problem, which we perform using the previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we establish general relations between the right and left scattering coefficients. Here, along with the known results, we introduce a relation between the left and right norming coefficients for the N -soliton solution.Then we propose an auxiliary kernel of the left Gelfand-Levitan-Marchenko equations, which allows one to solve the right scattering problem numerically. We generalize the TIB algorithm, initially proposed in the left formulation, to the right scattering problem case with the obtained formulas. The test runs of the TIB algorithm illustrate our results reconstructing the various nonsymmetrical potentials from their right scattering data.

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