On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation

Guoqiang Zhang; Weifang Weng; Zhenya Yan

REVIEW 4 major objections 4 minor 97 references

On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper derives explicit theta-function limits, with O(1/|x|) and O(1/t) error rates, for sine-Gordon kink-soliton gases under two generalized reflection coefficients.

desk verdict First rigorous sG kink-soliton gas asymptotics, but the load-bearing small-norm estimates are largely asserted; send back for full proofs. read the letter →

arxiv 2501.03493 v1 pith:IK3VECFL submitted 2025-01-07 nlin.SI math-phmath.APmath.MPphysics.optics

Guoqiang Zhang , Weifang Weng , Zhenya Yan This is my paper · ORCID

classification nlin.SImath-phmath.APmath.MPphysics.optics MSC 35Q5137K4035Q1537K1037K15

keywords sine-Gordonequationkink-solitongasRiemann-HilbertproblemsteepestdescentmethodJacobithetafunctionmodifiedBesselparametrixconfluenthypergeometriclong-timeasymptotics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Kink-soliton gases are limits of N-kink solutions as N→∞, and this paper gives their first rigorous large-space and long-time asymptotics for the sine-Gordon equation. For two classes of generalized reflection coefficients—one with a power-type singular point inside the spectral interval, one with a jump discontinuity—the paper proves that as x→−∞ the initial profile u(x,0) is a Jacobi theta-function expression plus O(1/|x|), and as t→∞ the solution is the same type of theta function plus O(1/t) in the slow sectors while decaying exponentially in the fast sector. A sympathetic reader would care because this converts a many-soliton limit from an open problem into an explicit formula whose phase is determined by an integral of the logarithm of the reflection coefficient.

What carries the argument

The load-bearing object is the scalar $g$-function used to conjugate the Riemann-Hilbert problem before any lens opening; for the initial-value problem it is $g_0(\lambda)=\lambda-\int_{\eta_2}^{\lambda}(\zeta^2-\rho)/R_0(\zeta)\,d\zeta$ on the two-cut surface $R_0=\sqrt{(\zeta^2-\eta_2^2)(\zeta^2-\eta_1^2)}$, and for long times it is a piecewise function $g=\theta-p$, where $p$ is an elliptic integral with branch cuts $(\alpha,\eta_2)\cup(-\eta_2,-\alpha)$ and $\alpha$ is fixed by the modulation equation $\xi=-\eta_2^{-2}W(\alpha/\eta_2)$. This $g$-function makes the lens jumps exponentially small and turns the spectral jumps into constant matrices of the form $e^{\Delta\sigma_3}$, so the outer parametrix is a Jacobi $\theta$ function while the local parametrices near $\eta_1,\eta_2$ are modified Bessel models and near $\eta_0$ they are either a second modified Bessel model or a confluent hypergeometric model. The argument proceeds through the standard $Y\mapsto T\mapsto S\mapsto E$ transformation chain, and the final error estimates for $E$ are what convert these model solutions into the stated asymptotic theorems.

What would settle it

Choose concrete values such as $\eta_1=1$, $\eta_0=2$, $\eta_2=3$, $\beta_0=\beta_1=\beta_2=0$, $\gamma\equiv1$ and compute the right-hand side of (1.17) for several large negative $x$; then solve the sine-Gordon initial-value problem numerically for a large-$N$ kink gas whose norming constants are spaced by the reflection-coefficient rule. If the difference between the numerical $u(x,0)$ and its $x$-derivative and the $\theta$ formula does not decay like a constant times $1/|x|$, Theorem 1 is false. The omitted proof of Proposition 3 can be checked directly by computing $P^{\eta}(P^\infty)^{-1}-I$ on the circle $\partial B(\eta_2)$ at leading order: it must be $O(|x|^{-1})$ (or $O(t^{-1})$) in the relevant regime.

Watch

Extended reading notes

Core claim

The paper's central claim is that the two generalized reflection coefficients $r_0(\lambda)=(\lambda-\eta_1)^{\beta_1}(\eta_2-\lambda)^{\beta_2}|\lambda-\eta_0|^{\beta_0}\gamma(\lambda)$ and $r_c(\lambda)=(\lambda-\eta_1)^{\beta_1}(\eta_2-\lambda)^{\beta_2}\chi_c(\lambda)\gamma(\lambda)$, with $0<\eta_1<\eta_0<\eta_2$, $\beta_j>-1$, and $\gamma$ continuous and strictly positive, give kink-soliton gases whose large-space and long-time behavior is explicit. As $x\to-\infty$, $\partial_x u(x,0)$, $\cos u(x,0)$ and $\sin u(x,0)$ are fixed ratios of Jacobi $\theta$ functions with modulus set by $\eta_1,\eta_2$ and argument shifted by $\Delta_1=\Omega_1(x+\varphi_1)$, where $\varphi_1$ is an integral of $\log r$ over $(\eta_1,\eta_2)$, with error $O(1/|x|)$. As $t\to\infty$, the same $\theta$ structure appears with $\alpha$ replacing $\eta_1$ in the sector $\xi_{\rm crit}<\xi<-\eta_2^{-2}$, with phase $\Delta_\alpha=\Omega_\alpha(x+t/(4\alpha\eta_2)+\varphi_\alpha)$ and error $O(1/t)$; for $\xi<\xi_{\rm crit}$ the $\eta_1$-$\theta$ function governs with $\Delta_1=\Omega_1(x+t/(4\eta_1\eta_2)+\varphi_1)$. For $\xi>-\eta_2^{-2}$ the solution decays exponentially. The proof constructs a piecewise $g$-function adapted to the sine-Gordon phase $\theta=(\xi\lambda+\lambda^{-1})/4$ and matching local parametrices at the endpoints and the internal singularity.

Load-bearing premise

Everything rests on the asserted but unproved small-norm estimates: Propositions 3–6 claim that after all transformations the error matrix E is within O(1/|x|) (or O(1/t)) of the identity, but Proposition 3 says its proof is omitted and Section 6 assumes local properties of T, S, E without proving them; if these bounds fail, the stated asymptotics do not follow.

Editorial extensions

If this is right

For the initial value, the kink-soliton gas approaches a theta-function profile as $x\to-\infty$ with error $O(1/|x|)$; for $x\to+\infty$ it is exponentially close to the vacuum state $u=0$.
For long times and speeds $\xi$ in $(\xi_{\rm crit},-\eta_2^{-2})$, the gas approaches the same theta-profile with $O(1/t)$; in the complementary fast sector $\xi>-\eta_2^{-2}$ decay is exponential.
The internal singularity $\eta_0$ controls the boundary between two long-time sectors: $r_0$ requires the second-kind modified Bessel model, $r_c$ requires a confluent hypergeometric model, and both produce the same leading theta function.
The phase shifts $\varphi_1$ and $\varphi_\alpha$ are determined by an integral of $\log r$ over the spectral interval, so the detailed shape of the reflection coefficient is remembered in the position and phase of the theta oscillations rather than in the leading amplitude.
When $r=r_0$ and $\beta_0=0$, the formula extends across the whole sector $(\xi_{\rm crit},-\eta_2^{-2})$, so the theta-function description is continuous through the point $\eta_0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

By the method's own logic, adding finitely many internal singularities $\{\eta_{0,j}\}$ should produce a theta-function limit with several phase contributions; a numerical check with large $N$ would test whether the phases add as the formula suggests.
An implication of the stated $O(1/t)$ decay is that there is no radiative tail at leading order in the slow sectors; that would distinguish kink-soliton gases from pure-radiation sine-Gordon solutions, whose dispersive tails decay at different rates.
The step reflection coefficient $r_c$ amounts to a discontinuous spectral density at $\eta_0$; one could measure the phase jump across the discontinuity in simulations of the $N$-kink gas and compare with the confluent hypergeometric transition-layer prediction.
The linear-in-$(x,t)$ argument of the theta profile predicts a measurable wavefront speed in the gas; tracking level sets of $\cos u$ in numerics should recover the modulation speed $\alpha$ and the sector boundaries $\xi_{\rm crit}$, $\xi_0$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is the first rigorous treatment of large-space and long-time asymptotics for sine-Gordon kink-soliton gases, and the main formulas look plausible and are derived rather than fitted. The transformation chain is spelled out in detail, the g-function construction is genuinely new (sG differs from KdV at the origin), and the parametrices for the interior singularity are thoughtfully chosen. The paper deserves a serious referee.

What's new: previous rigorous soliton-gas asymptotics covered KdV and mKdV, and pure-radiation sG asymptotics were known, but nobody had done kink-soliton gases for sG with these generalized reflection coefficients. The theta-function leading terms and the phase integrals are concrete and checkable.

Where it's soft: the load-bearing small-norm estimates in Propositions 3-6 are not actually proved. Proposition 3 says 'proof follows standard arguments and is omitted for brevity.' Propositions 4-6 have abbreviated proofs, but they just assert the matching P^ηj(P^∞)^{-1} = I2 + O(t^{-1}) on the boundaries of the parametrix neighborhoods. That matching is exactly what turns the steepest-descent analysis into the stated O(1/t) and O(1/|x|) errors. Section 6 also says local properties of T, S, E are 'assumed' and leaves the case r=r0, beta0=0 as an exercise, even though Theorem 2 claims that case. These are specific, addressable gaps, not signs of a wrong result—but as written, the theorems are not fully supported. The authors themselves flag the omissions, so they know where the missing work is.

The citation pattern is fine; the paper builds on Deift-Zhou, Girotti-Grava-Jenkins-McLaughlin, and the sG review [83], and the open-problem framing matches what the review actually says.

Bottom line: this is a serious paper with a narrow but real hole in the proof. For readers in integrable systems and RH asymptotics, it will be a useful reference once the proof is completed. I'd want the authors to provide the missing small-norm estimates and sort out the beta0=0 case before I'd cite it. But I'd send it to a knowledgeable referee—conditional acceptance, not desk reject.

Referee Report

4 major / 4 minor

Summary. The paper studies the large-space (x -> +-infinity) and long-time (t -> +infinity) asymptotic behavior of kink-soliton gases for the sine-Gordon equation in light-cone coordinates. The gas is described by a Riemann-Hilbert problem whose jumps are determined by two generalized reflection coefficients: r_0(lambda) = (lambda-eta_1)^{beta_1} (eta_2-lambda)^{beta_2} |lambda-eta_0|^{beta_0} gamma(lambda) and r_c(lambda) = (lambda-eta_1)^{beta_1} (eta_2-lambda)^{beta_2} chi_c(lambda) gamma(lambda), with chi_c a step function at eta_0. The authors implement the Deift-Zhou steepest descent chain Y -> T -> S -> E, construct a piecewise g-function with a Whitham-determined parameter alpha, and use Airy, modified Bessel, and confluent hypergeometric parametrices. Their main results, Theorems 1 and 2, give theta-function asymptotic formulas with O(1/|x|) or O(1/t) error terms in the sectors xi < xi_crit, xi_crit < xi < xi_0, and xi_0 < xi < -eta_2^{-2}, and exponential decay in the complementary sector.

Significance. If the proofs are completed, this would be a genuinely valuable contribution: it extends the rigorous Deift-Zhou analysis of soliton gases from KdV to the sine-Gordon equation, handles reflection coefficients with endpoint zeros/singularities and an internal singular or discontinuous point, and provides explicit, falsifiable theta-function asymptotics. The construction of the g-function for the negative-flow phase theta = xi lambda + lambda^{-1}/4 and the use of the second-kind Bessel and confluent hypergeometric parametrices at eta_0 are plausible and go beyond a routine adaptation of prior work. The paper is not circular: the asymptotic constants are derived from the reflection coefficients and the Whitham parameter, not fitted to the output. However, several load-bearing small-norm and parametrix-matching estimates are only asserted, and one case claimed in Theorem 2 is explicitly left as an exercise. The significance of the paper can only be assessed after those gaps are filled.

major comments (4)

[Section 5.3, Proposition 3] The proof of Proposition 3 is omitted ('The proof follows standard arguments and is omitted here for brevity'), but this proposition is the sole justification for E(0;x,0) = I_2 + O(|x|^{-1}) and E[1](x,0) = O(|x|^{-1}) in Eq. (5.135), which feed directly into Eq. (5.137) and hence into Theorem 1. The O(|x|^{-1}) estimate on the neighborhoods B(+-eta_2, +-eta_1, +-eta_0) requires a proof that each local parametrix P^{eta_j} matches the outer parametrix P^infinity to first order in 1/|x| on the boundary of the neighborhood. For the newly introduced M_mb and M_CH parametrices this is not a copy of the KdV case, and the estimates should be written out, especially for beta_j in (-1,0), where the local behaviors are singular.
[Section 6, opening paragraph and Theorem 2] Theorem 2 explicitly claims the long-time asymptotics for r = r_0 with beta_0 = 0 in the sector xi in (xi_crit, -eta_2^{-2}), but Section 6 states that this case is 'straightforward', is omitted, and is left 'as an exercise for the reader.' This is not a cosmetic omission: for beta_0 = 0 the jump is supported on the full interval (eta_1, eta_2) without a distinguished singular point at eta_0, so the f-function, the lens decomposition, and the small-norm estimates in Propositions 4 and 5 have to be re-derived. The manuscript provides no proof for this case.
[Section 6, opening paragraph and Sections 6.3.1-6.3.3] The sentence 'we assume that the necessary local properties of T, S, and E have already been established' is not justified by the preceding large-x analysis, because the g- and f-functions, and therefore the local parametrices at +-alpha and +-eta_0, change with the sector xi. The time-dependent local parametrices P^{eta_0} in Eqs. (6.165)-(6.170) differ from those in Section 5, and the boundary estimates P^{eta_j}(P^infinity)^{-1} = I_2 + O(t^{-1}) are asserted without detailed remainder estimates. These boundary estimates are exactly what is needed for E(0;x,t) = I_2 + O(t^{-1}) and E[1](x,t) = O(t^{-1}) in Eqs. (6.160)-(6.161), (6.172)-(6.173), and (6.180)-(6.181).
[Propositions 4-6] The O(t^{-1}) estimates on B(+-eta_2, +-alpha, +-eta_0) are stated in Eqs. (6.163), (6.175), and (6.183) without proof. In particular, the matching for the confluent hypergeometric parametrix M_CH in Eqs. (6.169)-(6.170) and for the second-kind modified Bessel parametrix M_mb in Eqs. (6.165)-(6.168) involves conformal maps that depend on xi through p(eta_0;xi); the order-t^{-1} remainder is not demonstrated. Since these estimates are load-bearing for the O(1/t) error terms in Theorem 2, they cannot be replaced by a citation to 'standard arguments' in a paper whose central novelty is precisely the local analysis at eta_0.

minor comments (4)

[Eq. (1.19)] The definition of the Jacobi theta function contains the expression '+-in^2 tau', which is not standard; it should presumably read 'pi i n^2 tau' or the sign convention should be stated precisely, since all later formulas in Theorems 1 and 2 use theta functions.
[Section 5.3, Eqs. (5.114)-(5.115)] The notation B(+-eta_2, +-eta_0, +-alpha) and B(+-eta_2, +-alpha) uses alpha, but in the large-x problem of Section 5 the relevant point is eta_1, not alpha; this appears to be a typo and should be corrected to avoid confusion.
[Section 6.3.3, proof of Proposition 6] The proof refers to 'C_2 \setminus {-eta_1, -alpha, -eta_2}' in a region where alpha = eta_1 and no separate alpha has been introduced; the set should be written in terms of eta_1 only.
[Section 6.1, Eqs. (6.140)-(6.141)] The jump conditions for the f-function are stated only on some subintervals; for the intervals (eta_1, alpha) and (-alpha, -eta_1) in the case xi in (xi_0, -eta_2^{-2}) it should be stated explicitly whether f is analytic there or what its jump is, since these intervals carry non-constant jumps in the S-problem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic formulas are derived from the stated reflection coefficients via an explicit Riemann-Hilbert steepest-descent construction, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained with respect to its inputs: the kink-soliton gas is defined through the reflection coefficients r0 and rc in (1.8)-(1.9), and the claimed asymptotic formulas in Theorems 1-2 are obtained by the Deift-Zhou steepest descent method, in which the g-function, f-function, outer parametrix, and local parametrices are all constructed explicitly from r, the endpoints η1, η2, and the Whitham-determined parameter α. The leading theta-function terms are evaluated from the outer parametrix P∞, whose entries are explicit functions of the phase integrals φ1 and φα defined in (1.18) and (1.22) in terms of log r; these phases are not fitted to the asymptotic output. The error terms O(|x|^-1) and O(t^-1) are traced to the small-norm estimates in Propositions 3-6, and those propositions derive the decay of E(0) and E[1] from bounds on the jump matrices V^E; while the proofs are omitted ("The proof follows standard arguments and is omitted here for brevity" in Proposition 3, and Section 6 leaves the case r = r0, β0 = 0 to the reader), an omitted proof is a rigor gap, not circularity, because the estimates are not obtained by assuming the asymptotic formulas they are used to prove. The paper's self-citations ([90], [96]) concern unrelated multi-soliton and rogue-wave constructions and are not load-bearing for the main asymptotic theorems. No step exhibits a definition of X in terms of Y followed by a derivation of Y from X, and no fitted parameter is relabeled as a prediction. The skeptical concern about unproved small-norm estimates is best classified as incompleteness or correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No physical entities or fitted constants are introduced; the theorem parameters are explicit model inputs. The main burden is mathematical: convergence of the soliton-gas limit, applicability of Deift-Zhou machinery, and the unproved small-norm estimates in Propositions 3-6.

assumptions (5)

domain assumption The discrete N-kink Riemann-Hilbert problems converge in the limit N1,N2 -> infinity to the RH problem (1.11) with jump matrices given by Cauchy integrals.
This is the kink-soliton gas construction in Section 1.1; convergence is asserted to follow from elementary calculus for beta_j > -1, and is not independently verified in the paper.
domain assumption The generalized reflection coefficients r0 and rc with beta_j > -1, continuous positive analytically extendable gamma, and c != 1 define admissible soliton gases.
This is the stated model class for Theorems 1 and 2; the results are conditional on this assumption.
standard math The Deift-Zhou steepest descent machinery (g-function, lenses, parametrices, small-norm estimates) applies to the sine-Gordon kink-soliton gas RH problem.
Borrowed from references [39,56,57]; the framework is standard, but its applicability to sG with an interior singularity at eta_0 is exactly what the paper must establish.
standard math The sign properties of p in Propositions 1 and 2 hold, giving exponential decay on the lens contours.
The proofs are sketched in the paper and rely on elliptic integral inequalities; if the signs were wrong, the exponential decay mechanism would fail.
ad hoc to paper The local parametrices match the outer parametrix to O(|x|^-1) or O(t^-1), and the error jump matrices are small in L1, L2, L-infinity (Propositions 3-6).
This is the load-bearing unproved step; Proposition 3's proof is explicitly omitted, and Section 6 assumes local properties without proof, making this the central fragility of the paper.

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Pith. "Pith review of On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation." pith.science (2026). https://pith.science/paper/IK3VECFL

@misc{pith2026250103493,
  author       = {Pith},
  title        = {Pith review of: On Large-Space and Long-Time Asymptotic Behaviors of Kink-Soliton Gases in the Sine-Gordon Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IK3VECFL}},
  note         = {Machine review of arXiv:2501.03493}
}

abstract

We conduct a comprehensive analysis of the large-space and long-time asymptotics of kink-soliton gases in the sine-Gordon (sG) equation, addressing an important open problem highlighted in the recent work [Phys. Rev. E 109 (2024) 061001]. We focus on kink-soliton gases modeled within a Riemann-Hilbert framework and characterized by two types of generalized reflection coefficients, each defined on the interval $[\eta_1, \eta_2]$: $r_0(\lambda) = (\lambda - \eta_1)^{\beta_1} (\eta_2 - \lambda)^{\beta_2} |\lambda - \eta_0|^{\beta_0} \gamma(\lambda)$ and $r_c(\lambda) = (\lambda - \eta_1)^{\beta_1} (\eta_2 - \lambda)^{\beta_2} \chi_c(\lambda) \gamma(\lambda)$, where $0 < \eta_1 < \eta_0 < \eta_2$ and $\beta_j > -1$, $\gamma(\lambda)$ is a continuous, strictly positive function defined on $[\eta_1, \eta_2]$. The function $\chi_c(\lambda)$ demonstrates a step-like behavior: it is given by $\chi_c(\lambda) = 1$ for $\lambda \in [\eta_1, \eta_0)$ and $\chi_c(\lambda) = c^2$ for $\lambda \in (\eta_0, \eta_2]$, with $c$ as a positive constant distinct from one. To rigorously derive the asymptotic results, we leverage the Deift-Zhou steepest descent method. A central component of this approach is constructing an appropriate $g$-function for the conjugation process. Unlike in the KdV equation, the sG presents unique challenges for $g$-function formulation, particularly concerning the singularity at the origin. The Riemann-Hilbert problem also requires carefully constructed local parametrices near endpoints $\eta_j$ and the singularity $\eta_0$. At the endpoints $\eta_j$, we employ a modified Bessel parametrix of the first kind. For the singularity $\eta_0$, the parametrix selection depends on the reflection coefficient: the second kind of modified Bessel parametrix is used for $r_0(\lambda)$, while a confluent hypergeometric parametrix is applied for $r_c(\lambda)$.

Figures

Figures reproduced from arXiv: 2501.03493 by the authors.

**Figure 1.** Left: Jump contours for Airy parametrix MmB; Right: Jump contours for the first type of modified Bessel parametrix MAi . 2.2 The first type of modified Bessel parametrix MmB (ζ; β) The matrix MmB (ζ; β) is constructed using the modified Bessel functions of the first and second kinds, denoted Iβ(ζ) and Kβ(ζ), respectively, where the index β lies within the range (−1, +∞). These functions provide solutions to the modi… view at source ↗

**Figure 2.** Left: Jump contour for modified Bessel parametrix [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Sign charts for ℜ (θ) with ξ = −1(Left) and ξ = 0(Right): ℜ (θ) > 0 in greay regions and ℜ (θ) < 0 in white regions Similarly, for λ ∈ {λ | ℜ(λ) ≤ 0} \ (−η2, 0], p(λ; ξ) is given by: p(λ; ξ) = Z λ −η2 Q(y; ξ) 4y 2R(y; ξ) dy, (3.75) where the integration path extends from ±η2 to λ. The functions R(y; ξ) and Q(y; ξ) are defined depending on the region ξ ∈ (ξcrit, −η −2 2 ) ∪ (−∞, ξcrit). The function R(y; ξ) represent… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Contour deformation by opening lenses for [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]

**Figure 5.** Figure 5: Contour deformation by opening lenses for [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]

**Figure 6.** Figure 6: Jump contours of the error matrix E(λ; x, 0) for r = r0, rc with β0 6= 0 with A η0 0l+ = [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]

**Figure 7.** Figure 7: Jump contours of the error vector E(λ; x, 0) for r = r0 with β0 = 0 • E(λ; x, 0) is analytic in λ for λ ∈ C \ (B(±η2, ±η0, ±η1) ∪ Cc 1 ∪ Cc 2 ); • It normalizes to the identity matrix I2 at infinity; • For λ ∈ B(±η2, ±η0, ±η1) ∪ Cc 1 ∪ Cc 2 , E(λ; x, 0) admits continuo…

**Figure 8.** Figure 8: Contour deformation in the region ξ ∈ [PITH_FULL_IMAGE:figures/full_fig_p040_8.png]

**Figure 9.** Figure 9: Contour deformation in the region ξ ∈ (ξcrit, ξ0) The 2 × 2 matrix-valued function S(λ; x, t) is defined as follows: S(λ; x, t) =    T(λ; x, t)U tp f [PITH_FULL_IMAGE:figures/full_fig_p040_9.png]

**Figure 10.** Figure 10: Jump contour of the error matrix E(λ; x, t) in the region ξ0 < ξ < −η −2 2 and p+ + p− < 0, for λ ∈ (−α, −η1], which give rise to the estimate of the jump matrix on C c . The estimate on B (±η2, ±α) is derived by P η2 (λ) (P ∞(λ))−1 = I2 + O [PITH_FULL_IMAGE:figures/…

**Figure 11.** Figure 11: Jump contour of the error matrix E(λ; x, t) in the region ξcrit < ξ < ξ0 Proposition 5 (Small norm estimate in the region ξ ∈ (ξcrit, ξ0)). For β2 > −1, β0 > −1, β1 ≥ 0, the jump matrices V E has the following small norm estimates [PITH_FULL_IMAGE:figures/full_fig_p0…

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