Rigorous analysis of the unified transform method and long-range instabilities for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane

Chatziafratis, Andreas; Ozawa, Tohru; Tian, Shou-Fu

doi:10.1007/s00208-023-02698-4

Cited by 4 publications

(3 citation statements)

References 101 publications

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“…This task is facilitated by the synergy between powerful complex-analytic ideas, which are innate in the well-known Unified Transform Method (UTM-also called the Fokas Method, since its introduction by Fokas in the 1990s; see, for instance, [7][8][9][10][11][12][13][14][15][16][17][18][19][20], in chronological order, and many more references included therein), and a novel analytical approach, introduced in ref. [21], to investigating rigorous aspects of the UTM as well as regularity properties of linear PDEs with polynomial dispersion relations on the half-line (see, e.g., [22] for an application of this rigorous approach to the classical heat equation, and, for example, [23][24][25][26] for recent developments and extensions towards more complicated situations). It is noted that the double-diffusivity system (1.3) has already been studied [27] through the novel rigorously-minded analytical prism, while the analogous analyses for other important models containing the Cahn-Hilliard term too are underway and will appear in future articles.…”

Section: Introductionmentioning

confidence: 99%

On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

Chatziafratis,

Fokas,

Aifantis

2024

Z Angew Math Mech

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A novel technique is presented for explicitly solving inhomogeneous initial‐boundary‐value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half‐line, for a well‐known pseudo‐parabolic partial differential equation. This so‐called Barenblatt's equation arises in a plethora of important applications, ranging from heat‐mass transfer, solid‐fluid‐gas dynamics and materials science, to mechanical, chemical and petroleum engineering, as well as electron physics, radiation and diffusive processes. Our approach is based on the extension of the Fokas method, so that it can be applied to problems with mixed derivatives. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each type of IBVP, the novel formulae are utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain and the problem's well‐posedness. Furthermore, importantly, our solutions’ numerical advantages are demonstrated and highlighted by way of a concrete and illustrative example. Our rigorous approach can be extended to IBVPs for other significant models.

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Section: Introductionmentioning

confidence: 99%

On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

Chatziafratis,

Fokas,

Aifantis

2024

Z Angew Math Mech

Self Cite

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“…Recently, a new line of investigation and a rigorous analytical approach have been introduced in [18], applied in e.g. [26], and extended in [25], [27], [23], [19], [20], [21], [28], [22], in order to study miscellaneous properties (concerning e.g. regularity and qualitative theory, well-posedness and asymptotic analysis) of linear evolution PDE posed in the quarter-plane.…”

Section: Introductionmentioning

confidence: 99%

“…As in previous works [25], [27], we will not provide the initial construction of this formula by Fokas; we refer to [41] for a clear formal derivation. Here, we will rather completely and rigorously justify it a posteriori and prove that the function defined by this formula:…”

Section: Introductionmentioning

confidence: 99%

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

Chatziafratis,

Grafakos,

Kamvissis

2024

Dynamics of Partial Differential Equations

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We study the inhomogeneous Airy partial differential equation (also called Stokes or linearized Korteweg-de Vries equation with a negative sign) on the half-line with generic initial and boundary data in a classical smooth setting, via the formula provided by the Fokas unified transform method for linear evolution equations. We first present a suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in it, thus securing convergence in a strict sense. Writing the solution in an Ehrenpreis-Palamodov form, our analysis allows for rigorous a posteriori verification of the full initial-boundary-value problem and a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to prescribed values and the solution admits a smooth extension up to the boundary only under certain data compatibility conditions (with implications for well-posedness, control theory and efficient numerical computations). Importantly, based on this analysis, we perform an effective asymptotic study of far-field dynamics. This yields new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the 'corner' of the quadrant. In particular, the asymptotic behavior of the solution is sensitive to perturbations of the data at the origin. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property at soon as time becomes positive. Hereby, we report on the discovery of a novel type of a long-range instability phenomenon for linear dispersive differential equations. Our ideas are extendable to other Airy-like and more general problems for dispersive evolution equations.

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Integral representations for the double-diffusivity system on the half-line

Chatziafratis,

Aifantis,

Carbery

et al. 2024

Z. Angew. Math. Phys.

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Rigorous analysis of the unified transform method and long-range instabilities for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane

Cited by 4 publications

References 101 publications

On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

On Barenblatt's pseudoparabolic equation with forcing on the half‐line via the Fokas method

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

Integral representations for the double-diffusivity system on the half-line

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