The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

Boling Guo; Nan Liu

doi:10.1080/14029251.2019.1613055

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Volume 26, Issue 3, May 2019, Pages 483 - 508

The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

Authors

Boling Guo, Nan Liu^*

Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China,gbl@iapcm.ac.cn(Boling Guo);liunan16@gscaep.ac.cn(Nan Liu)

^*Corresponding author.

Corresponding Author

Nan Liu

Received 17 October 2018, Accepted 25 March 2019, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1613055 How to use a DOI?
Keywords: Coupled nonlinear Schrödinger equation; Riemann–Hilbert problem; Initial-boundary value problem; Long-time asymptotics
Abstract: We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrödinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a nonlinear steepest descent analysis of an associated 3 × 3 matrix Riemann–Hilbert problem, we can give the precise asymptotic formulas for the solution of the coupled nonlinear Schrödinger equation on the half-line.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

The well-known nonlinear steepest descent method first introduced by Deift and Zhou in [14] provides a powerful technique for determining asymptotics of solutions of nonlinear integrable evolution equations. This approach since then has been successfully applied in analyzing the long-time asymptotics of initial-value problems (IVPs) for a number of nonlinear integrable partial differential equations (PDEs) associated with 2 × 2 matrix spectral problems including the modified Kortewegde Vries (mKdV) equation [14], the defocusing nonlinear Schrödinger (NLS) equation [15], the KdV equation [19], the derivative NLS equation [25], the Fokas–Lenells equation [28], the short pulse equation [10] and the Kundu–Eckhaus equation [27]. Moreover, by combining the ideas of [14] with the so-called “g-function mechanism” [13], it is also possible to study asymptotics of solutions of the IVPs with shock-type oscillating initial data [11], nondecaying step-like initial data [6,22], nonzero boundary conditions at infinity [4] for various integrable equations. There also exists some meaningful papers [8, 9, 17] about the study of long-time asymptotics for the IVPs of integrable nonlinear evolution equations associated with 3 × 3 matrix spectral problems. For the large-time asymptotic analysis of the initial-boundary value problems (IBVPs) of integrable nonlinear PDEs, Lenells et al. derived some interesting asymptotic formulas for the solutions of mKdV equation [23] and derivative NLS equation [3] by using the steepest descent method. Furthermore, the long-time asymptotics for the focusing NLS equation with t-periodic boundary condition on the half-line is analyzed in [5]. We also have done some work about determining the long-time asymptotics for integrable equations on the half-line, see [20, 21]. However, there is only a little of literature [7] to consider the asymptotic behaviors for integrable nonlinear PDEs with Lax pairs involving 3 × 3 matrices on the half-line. Thus, it is necessary and important to consider the large-time asymptotic behaviors for the IBVPs of integrable equations with 3 × 3 Lax pairs on the half-line.

In particular, the purpose of this paper is aim to consider the long-time asymptotics for the IBVP of the coupled nonlinear Schrödinger (CNLS) equation

{iut+uxx+2(|u|2+|v|2)u=0,ivt+vxx+2(|u|2+|v|2)v=0,(1.1)

posed in the quarter-plane domain

Ω={0≤x<∞,0≤t<∞}.

We will denote the initial data, Dirichlet and Neumann boundary values of (1.1) as follows:

u(x,0)=u0(x), v(x,0)=v0(x), x≥0,u(0,t)=g0(t), ux(0,t)=g1(t), v(0,t)=h0(t), vx(0,t)=h1(t), t≥0.(1.2)

We also suppose that {u₀(x), v₀(x)} and {gj(t),hj(t)}01 belong to the Schwartz class S(ℝ₊). Equation (1.1) was also called Manakov model, which can be used to describe the propagation of an optical pulse in a birefringent optical fiber [26]. Subsequently, this system also arises in the context of multi-component Bose-Einstein condensates [12]. Due to its physical interest, equation (1.1) has been widely studied. It is noted that the IBVP for (1.1) on the half-line has been investigated via the Fokas method [16], where it was shown that the solution {u(x, t), v(x, t)} can be expressed in terms of the unique solution of a 3 × 3 matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane (see [18]). Meanwhile, the leading-order long-time asymptotics of the Cauchy problem of equation (1.1) was obtained in [17].

Our goal here is to derive the long-time asymptotics of the solution of (1.1) on the half-line by performing a nonlinear steepest descent analysis of the associated RH problem. Compared with other integrable equations, the asymptotic analysis of (1.1) presents some distinctive features: (1) Since the RH problem associated with (1.1) involves 3 × 3 jump matrix J(x, t, k), we first introduce two 2 × 1 vector-valued spectral functions r₁(k), h(k) and a 1 × 2 vector-valued spectral function r(k), and then rewrite our main RH problem as a 2 × 2 block one. This procedure is more convenient for the following long-time asymptotic analysis. (2) As we all known, the important step of the steepest descent method is to split the jump matrix J(x, t, k) into an appropriate upper/lower triangular form. This immediately leads to construct a δ(k) function to remove the middle matrix term, however, the function δ satisfies a 2 × 2 matrix RH problem in our present problem. The unsolvability of the 2 × 2 matrix function δ(k) is a challenge when we perform the scaling transformation to reduce the RH problem to a model RH problem. Fortunately, we can follow the idea introduced in [18] to use the available function detδ(k) which can be explicitly solved by the Plemelj formula to approximate the function δ(k) by error control. (3) The relevant RH problem for the Cauchy problem (1.1) considered in [17] only has a jump across ℝ, whereas the RH problem for the IBVP also has a jump across iℝ, and the jump across this line involves the spectral function h(k). Moreover, during the asymptotic analysis, one should find an analytic approximation h_a(t, k) of h(k). (4) Recalling the meaningful work about analyzing the long-time asymptotics for the Degasperis–Procesi equation on the half-line, the analysis presented in [7] shows that the structure of the jump matrix, which is 3 × 3, is essentially 2 × 2 (under an appropriate change of basis), whereas the analysis given in our present paper is more general.

The main result of this paper is stated as the following theorem.

Theorem 1.1.

Assume the assumption 1 be valid. Then, for any positive constant N, as t → ∞, the solution (u(x, t) v(x, t)) of the IBVP for the CNLS equation (1.1) on the half-line satisfies the following asymptotic formula

(u(x,t)v(x,t))=(ua(x,t)va(x,t))t+O(lntt), t→∞, 0≤x≤Nt,(1.3)

where the error term is uniform with respect to x in the given range, and the leading-order coefficient (u_a(x, t) v_a(x, t)) is defined by

(ua(x,t) va(x,t))=e−π2νΓ(−iν)r(k0)2πeiα(ξ,t),(1.4)

where the 1 × 2 vector-valued spectral function r(k) is defined by (3.3), which determined by all the initial and boundary values,

ξ=xt, ν=12πln(1+r(k0)r†(k0)), k0=−ξ4,

and

α(ξ,t)=−π4+νln(8t)+4k02t+1π∫−∞k0ln(k0−s)dln(1+r(s)r†(s)).

Remark 1.1.

The asymptotic formula for the half-line problem obtained in Theorem 1.1 has the exact same functional form for the pure initial value problem. The only difference is that the definition of the spectral function r(k) for the half-line problem, which enters the asymptotic formula, involves not only the initial data but also the boundary values. In other words, the only effect of the boundary is to modify r(k). We can understand this as follows: since x grows faster than t in the given region, the distance to the boundary eventually gets so big that what happens at the boundary has a small effect on the solution (recall that we assume that the boundary values decay as t → ∞). Therefore, the boundary values and the initial data play similar roles and it is natural that they enter the asymptotic formula in similar ways.

The outline of the paper is following. In Section 2, we recall how the solution of CNLS equation (1.1) on the half-line can be expressed in terms of the solution of a 3 × 3 matrix RH problem. In section 3, we present the detailed derivation of the long-time asymptotics for the solution of CNLS equation, that is, we prove Theorem 1.1.

2. Preliminaries

In this section, we give a short review of the RH problem for (1.1) on the half-line, see [18] for further details. The Lax pair of equation (1.1) is

{μx(x,t,k)−ik[σ,μ(x,t,k)]=U(x,t)μ(x,t,k),μt(x,t,k)−2ik2[σ,μ(x,t,k)]=V(x,t,k)μ(x,t,k).(2.1)

where μ(x, t, k) is a 3 × 3 matrix-valued eigenfunction, k ∈ ℂ is the spectral parameter, and

σ=(−100010001), U(x,t)=i(0u(x,t)v(x,t)u¯(x,t)00v¯(x,t)00),(2.2)

V(x,t,k)=2kU(x,t)+iUx(x,t)σ+iU2(x,t)σ.(2.3)

Let {γj}13 denote contours in the (x, t)-plane connecting (x_j, t_j) with (x, t), and (x₁, t₁) = (0, ∞), (x₂, t₂) = (0, 0), (x₃, t₃) = (∞, t). The contours can be chosen to consist of straight line segments parallel to the x- or t-axis. We define three eigenfunctions {μj}13 of the Lax pair (2.1) by the solutions of the following integral equations

μj(x,t,k)=I+∫γjei(kx+2k2t)σ^wj(x′,t′,k), j=1,2,3,(2.4)

where the closed one-form w_j(x, t, k) is defined by

wj(x,t,k)=e−i(kx+2k2t)σ^[U(x,t)dx+V(x,t,k)dt]μj(x,t,k),(2.5)

and σ^ denote the operators which act on a 3 × 3 matrix X by σ^X=[σ,X], then eσ^X=eσXe−σ. It then can be shown that the functions {μj}13 are bounded and analytical for k ∈ ℂ while k belongs to

μ1:(D2,D3,D3), μ2:(D1,D4,D4), μ3:(ℂ−,ℂ+,ℂ+),(2.6)

where D_n denotes nth quadrant, 1 ≤ n ≤ 4, ℂ₊ and ℂ₋ denote the upper and lower half complex k-plane, respectively.

We define the matrix-valued spectral functions s(k) and S(k) by the relations

μ3(x,t,k)=μ2(x,t,k)ei(kx+2k2t)σ^s(k),μ1(x,t,k)=μ2(x,t,k)ei(kx+2k2t)σ^S(k).(2.7)

Evaluation of (2.7) at (x, t) = (0, 0) gives the following expressions

s(k)=μ3(0,0,k), S(k)=μ1(0,0,k).(2.8)

Hence, the functions s(k) and S(k) can be obtained respectively from the evaluations at x = 0 and at t = 0 of the functions μ₃(x, 0, k) and μ₁(0, t, k).

On the other hand, we can deduce from the properties of μ_j that s(k) and S(k) have the following properties:

(i)
s(k) is bounded and analytic for k ∈ (D₃ ∪ D₄, D₁ ∪ D₂,D₁ ∪ D₂), S(k) is bounded and analytic for k ∈ (D₂ ∪ D₄, D₁ ∪ D₃, D₁ ∪ D₃);
(ii)
dets(k) = 1 for k ∈ ℝ, detS(k) = 1 for k ∈ ℝ ∪ iℝ;
(iii)
s(k) = I + O(k⁻¹) and S(k) = I + O(k⁻¹) uniformly as k → ∞;
(iv)

s¯(k)=(s−1)T(k), S¯(k)=(S−1)T(k),(2.9)
where s¯(k)=s(k¯)¯ and S¯(k)=S(k¯)¯ denote the Schwartz conjugates.

The initial and boundary values of a solution of the CNLS equation (1.1) are not independent. It turns out that the spectral functions s(k) and S(k) must satisfy a surprisingly simple relation

S−1(k)s(k)=0, k∈(D¯4,D¯1,D¯1),(2.10)

which called the global relation.

For each n = 1,...,4, we define solution M_n(x, t, k) of (2.1) by the solution of following Fredholm integral equation:

(Mn)jl(x,t,k)=δjl+∫γjln(ei(kx+2k2t)σ^wn(x′,t′,k))jl, k∈Dn, j,l=1,2,3,(2.11)

where w_n is given by (2.5), and the contours γjln, n = 1, 2, 3, 4, j, l = 1, 2, 3 are defined by

γjln={γ1 if Reyj<Reyl and Rezj≥Rezl,γ2 if Reyj<Reyl and Rezj<Rezl,γ3 if Reyj≥Reyl,(2.12)

and we denote ikσ = diag(y₁, y₂, y₃), 2ik²σ = diag(z₁, z₂, z₃). For each n = 1,...,4, we define spectral functions S_n(k) by

Mn(x,t,k)=μ2(x,t,k)ei(kx+2k2t)σ^Sn(k), k∈Dn.(2.13)

According to (2.11), the {Sn(k)}14 can be computed from initial and boundary values alone as well as the spectral functions s(k) and S(k) (the calculation results can see [18]).

Then, equation (2.13) can be rewritten in the form of a 3 × 3 RH problem as follows:

M+(x,t,k)=M−(x,t,k)J(x,t,k), k∪ℝ∪iℝ,(2.14)

where the matrices M₊(x, t, k), M₋(x, t, k), and J(x, t, k) are defined by

M(x,t,k)={M+(x,t,k),k∈D1∪D3,M−(x,t,k),k∈D2∪D4,

and

J1=(100−s33S21−s23S31s¯11W¯11e2iθ10−s22S31−s32s21s¯11W¯11e2iθ01),J2=J1J4−1J3,J3=(1−s¯33S¯21−s¯23S¯31s11W11e−2iθ−s¯22S¯31−s¯32S¯21s11W11e−2iθ010001),J4=(1+|s12|2|s11|2+|s13|2|s11|2s12s11e−2iθs13s11e−2iθs¯12s¯11e2iθ10s¯13s¯11e2iθ01),(2.15)

W11=S¯11s11+S¯21s21+S¯31s31, θ(k)=kx+2k2t.(2.16)

The contour for this RH problem is depicted in Fig. 1.

In what follows, we will make the following simple assumptions.

Assumption 1. We assume that the following conditions hold:

the initial and boundary values lie in the Schwartz class.
the spectral functions s(k), S(k) defined in (2.7) satisfy the global relation (2.10).
s₁₁(k) and W₁₁(k) have no zeros in D¯3∪D¯4 and D¯3, respectively.
all the initial and boundary values are compatible with equation (1.1) to all orders at x = t = 0.

We have the following representation theorem (the proof can be found in [18]).

Theorem 2.1.

Let u₀(x), g₀(t), g₁(t), v₀(x), h₀(t), h₁(t) be functions in the Schwartz class 𝒮([0, ∞)), and the assumption 1 is satisfied. Then the RH problem (2.14) with the jump matrices given by (2.15) and the following asymptotics:

M(x,t,k)=I+O(1k), k→∞(2.17)

admits a unique solution M(x, t, k) for each (x, t) ∈ Ω. Define {u(x, t), v(x, t)} in terms of M(x, t, k) by

u(x,t)=2limk→∞(kM(x,t,k))12,v(x,t)=2limk→∞(kM(x,t,k))13.(2.18)

Then {u(x, t), v(x, t)} satisfies the CNLS equation (1.1). Furthermore, {u(x, t), v(x, t)} satisfies the initial and boundary value conditions

u(x,0)=u0(x), u(0,t)=g0(t), ux(0,t)=g1(t),v(x,0)=v0(x), v(0,t)=h0(t), vx(0,t)=h1(t).(2.19)

Remark 2.1.

In the case when s₁₁(k) and W₁₁(k) have no zeros, the unique solvability of above RH problem (2.14) is a consequence of the ‘vanishing’ lemma (the proof can be found in the appendix of [18]).

Remark 2.2.

For a well-posed problem, only a subset of the initial and boundary values can be independently prescribed. However, all boundary values are needed for the definition of S(k), and hence for the formulation of the main RH problem as well as the spectral function r(k) defined in (3.3). In general, the computation of the unknown boundary values, namely, the construction of the generalized Dirichlet-to-Neumann map, involves the solution of a nonlinear Volterra integral equation. We do not consider this parts in present paper since the detailed analysis has been studied in the paper [18], (see the Sections 5 and 6 in [18]). Our main concern in present paper is the derivation of the long-time asymptotics for the solution of the IBVP of the CNLS equation (1.1).

3. Long-time asymptotic analysis

Before we proceed to the following analysis, we will follow the ideas used in [2, 17] to rewrite our main 3 × 3 matrix RH problem as a 2 × 2 block one. This procedure is more convenient for the following long-time asymptotic analysis. More precisely, we rewrite a 3 × 3 matrix A as a block form

A=(A11A12A21A22),

where A₁₁ is scalar. Define the vector-valued spectral functions r₁(k) and h(k) by

r1(k)=(s¯12s¯11s¯13s¯11), k∈ℝ,(3.1)

h(k)=(s33S21−s23S31s¯11W¯11s22S31−s32S21s¯11W¯11), k∈D¯2,(3.2)

and let r(k) denote the sum given by

r(k)=r1†(k¯)+h†(k¯), k∈ℝ−.(3.3)

Then we can rewrite the RH problem (2.14) as

{M+(x,t,k)=M−(x,t,k)J(x,t,k),k∈Σ=ℝ∪iℝ,M(x,t,k)→I,k→∞,(3.4)

with the jump matrix J(x, t, k) is given by

J(x,t,k)={(10−h(k)etΦ(ξ,k)I),k∈iℝ+,(1−r(k)e−tΦ(ξ,k)−r†(k¯)etΦ(ξ,k)I+r†(k¯)r(k)),k∈ℝ−,(1−h†(k¯)e−tΦ(ξ,k)0I),k∈iℝ−,(1+r1†(k¯)r1(k)r1†(k¯)e−tΦ(ξ,k)r1(k)etΦ(ξ,k)I),k∈iℝ+,(3.5)

where

Φ(ξ,k)=4ik2+2iξk, ξ=xt,(3.6)

and ℝ₊ = [0, ∞) and ℝ₋ = (−∞, 0] denote the positive and negative halves of the real axis. In view of (2.18), we have

(u(x,t) v(x,t))=2limk→∞(kM(x,t,k))12.(3.7)

Moreover, we can deduce from the properties of s(k) and S(k) that the functions r₁(k), h(k), r(k) defined by (3.1), (3.2) and (3.3) possess the following properties:

r₁(k) is smooth and bounded on ℝ;
h(k) is smooth and bounded on D¯2 and analytic in D₂;
r(k) is smooth and bounded on ℝ₋;
There exist complex constants {r1,j}j=1∞ and {hj}j=1∞ such that, for any N ≥ 1,
r1(k)=∑j=1Nr1,jkj+O(1kN+1), |k|→∞, k∈ℝ,(3.8)

h(k)=∑j=1Nhjkj+O(1kN+1), k→∞, k∈D¯2.(3.9)

3.1. Transformations of the RH problem

Let N > 1 be given, and let ℐ denote the interval ℐ = (0, N]. The jump matrix J defined in (3.5) involves the exponentials e^±tΦ. It follows that there is a single stationary point located at the point where ∂Φ∂k=0, i.e., at k=k0=−ξ4. By performing a number of transformations in the following, we can bring the RH problem (3.4) to a form suitable for determining the long-time asymptotics.

The first transformation is to deform the vertical part of Σ so that it passes through the critical point k₀. Letting U₁ and U₂ denote the triangular domains shown in Fig. 2. Then the first transform is as follows:

M(1)(x,t,k)=M(x,t,k)×{(10−h(k)etΦ(ξ,k)I),k∈U1,(1h†(k¯)e−tΦ(ξ,k)0I)k∈U2,I,elsewhere.(3.10)

Then we obtain the RH problem

{M+(1)(x,t,k)=M−(1)(x,t,k)J(1)(x,t,k),k∈Σ(1),M(1)(x,t,k)→I,k→∞.(3.11)

The jump matrix J⁽¹⁾(x, t, k) is given by

J1(1)=(10−hetΦI),J2(1)=(1−retΦ−r†etΦI+r†r),J3(1)=(1−h†e−tΦ0I),J4(1)=(1+r1†r1r1†e−tΦr1etΦI),(3.12)

where Ji(1) denotes the restriction of J⁽¹⁾ to the contour labeled by i in Fig. 2. The next transformation is:

M(2)(x,t,k)=M(1)(x,t,k)Δ(k),(3.13)

where

Δ(k)=(1detδ(k)00δ(k)),(3.14)

and δ(k) satisfies the following 2 × 2 matrix RH problem across (−∞, k₀) oriented in Fig. 2:

{δ−(k)=(I+r†(k)r(k))δ+(k),k<k0,δ(k)→I,k→∞.(3.15)

Furthermore,

{detδ−(k)=(1+r(k)r†(k))detδ+(k),k<k0,detδ(k)→1,k→∞.(3.16)

Remark 3.1.

It is noted that we introduce a 2×2 matrix-valued function δ(k) to remove the middle matrix term while we split the jump matrix J2(1)(x,t,k) into an appropriate upper/lower triangular form. This may be a main difference compared with the long-time asymptotic analysis of integrable nonlinear evolution equations associated with 2 × 2 matrix spectral problems on the half-line [3,20,21].

Since the jump matrix I + r^†(k)r(k) is positive definite, the vanishing lemma [1] yields the existence and uniqueness of the function δ(k). By Plemelj formula, det δ(k) can be solved by

detδ(k)=exp{12πi∫−∞k0ln(1+r(s)r†(s))s−kds}=(k−k0)−iνeχk, k∈ℂ\(−∞,k0],(3.17)

where

ν=12πln(1+r(k0)r†(k0))>0,(3.18)

χ(k)=−12πi∫−∞k0ln(k−s)d ln(1+r(s)r†(s)).(3.19)

On the other hand, a direct calculation as in [17] shows that

|δ(k)|k≤const<∞, |detδ(k)|≤const<∞,(3.20)

for all k, where we define

|A|=(trA†A)12,for any matrix A,‖B(⋅)‖Lp=‖|B(⋅)|‖Lp,for any matrix function B(⋅).(3.21)

Then we find that M⁽²⁾(x, t, k) satisfies the following RH problem

{M+(2)(x,t,k)=M−(2)(x,t,k)J(2)(x,t,k),k∈Σ(2),M(2)(x,t,k)→I,k→∞.(3.22)

and the contour Σ⁽²⁾ = Σ⁽¹⁾, the jump matrix J(2)=Δ−−1J(1)Δ+ , namely,

J1(2)=(10−δ−1h(detδ)−1etΦI),J2(2)=(1−r2δ−detδ−e−tΦ0I)(10−δ+−1r2†(detδ+)−1etΦI),J3(2)=(1−h†δdetδetΦ0I),J4(2)=(1r1†δdetδe−tΦ0I)(10δ−1r1(detδ)−1etΦI),

where we define r₂(k) by

r2(k)=r(k)1+r(k)r†(k¯).(3.23)

Before processing the next deformation, we will follow the idea of [3, 23, 24] and decompose each of the functions h, r₁, r₂ into an analytic part and a small remainder because the spectral functions have limited domains of analyticity. The analytic part of the jump matrix will be deformed, whereas the small remainder will be left on the original contour.

Lemma 3.1.

There exist a decomposition

h(k)=ha(t,k)+hr(t,k), t>0, k∈iℝ+,

where the functions h_a and h_r have the following properties:

(i)
For each t > 0, h_a(t, k) is defined and continuous for k∈D¯1 and analytic for k ∈ D₁.
(ii)
For each ξ ∈ ℐ and each t > 0, the function h_a(t, k) satisfies
|ha(t,k)−h(0)|≤Cet4|ReΦ(ξ,k)|, |ha(t,k)|≤C1+|k|2et4|ReΦ(ξ,k)|,(3.24)
for k∈D¯1, where the constant C is independent of ξ, k, t.
(iii)
The L¹, L² and L^∞ norms of the function h_r(t,·) on iℝ₊ are O(t^−3/2) as t → ∞.

Proof.

Since h(k) ∈ C⁵(iℝ₊), we find that

h(n)(k)=dndkn(∑j=04h(j)(0)j!kj)+O(k5−n), k→0, k∈iℝ+, n=0,1,2.(3.25)

On the other hand, we have

h(n)(k)=dndkn(∑j=13h(j)k−j)+O(k−4−n), k→∞, k∈iℝ+, n=0,1,2.(3.26)

Let

f0(k)=∑j=29aj(k+i)j,(3.27)

where {aj}29 are complex such that

f0(k)={∑j=04h(j)(0)j!kj+O(k5),k→0,∑j=13hjk−j+O(k−4),k→∞.(3.28)

It is easy to verify that (3.28) imposes eight linearly independent conditions on the a_j, hence the coefficients a_j exist and are unique. Letting f = h − f₀, it follows that

(1)
f₀(k) is a rational function of k ∈ ℂ with no poles in D¯1;
(2)
f₀(k) coincides with h(k) to four order at 0 and to third order at ∞, more precisely,
dndknf(k)={O(k5−n),k→0,O(k−4−n),k→∞, k∈iℝ+, n=0,1,2.(3.29)

The decomposition of h(k) can be derived as follows. The map k ↦ ψ = ψ(k) defined by ψ(k) = 4k² is a bijection [0, i∞) ↦ (−∞, 0], so we may define a function F : ℝ → ℂ by

F(ψ)={(k+i)2f(k),ψ≤0,0,ψ>0.(3.30)

F(ψ) is C⁵ for ψ ≠ 0 and

F(n)(ψ)=(18k∂∂k)n((k+i)2f(k)), ψ≤0.

By (3.29), F ∈ C²(ℝ) and F⁽ⁿ⁾(ψ) = O(|ψ|⁻¹⁻ⁿ) as |ψ| → ∞ for n = 0, 1, 2. In particular,

‖dnFdψn‖L2(ℝ)<∞, n=0,1,2,(3.31)

that is, F belongs to H²(ℝ). By the Fourier transform F^(s) defined by

F^(s)=12π∫ℝF(ψ)e−iψsdψ(3.32)

where

F(ψ)=∫ℝF^(s)eiψsds,(3.33)

it follows from Plancherel theorem that ‖s2F^(s)‖L2(ℝ)<∞. Equations (3.30) and (3.33) imply

f(k)=1(k+i)2∫ℝF^(s)eiψsds, k∈iℝ+.(3.34)

Writing

f(k)=fa(t,k)+fr(t,k), t>0, k∈iℝ+,

where the functions f_a and f_r are defined by

fa(k)=1(k+i)2∫−t4∞F^(s)e4ik2sds, t>0, k∈D¯1,(3.35)

fr(t,k)=1(k+i)2∫−∞−t4F^(s)e4ik2sds, t>0, k∈iℝ+,(3.36)

we infer that f_a(t,·) is continuous in D¯1 and analytic in D₁. Moreover, since |Re4ik²| ≤ |ReΦ(ξ, k)| for k∈D¯1 and ξ ∈ ℐ, we can get

|fa(t,k)|≤1|k+i|2‖F^(s)‖L1(ℝ)sups>−t4esRe4ik2≤C1+|k|2et4|ReΦ(ξ,k)|, t>0, k∈D¯1, ξ∈ℐ.(3.37)

Furthermore, we have

|fr(t,k)|≤1|k+i|2∫−∞−t4s2|F^(s)|s−2ds≤C1+|k|2‖s2F^(s)‖L2(ℝ)∫−∞−t4s−4ds,≤C1+|k|2t−3/2, t>0, k∈iℝ+, ξ∈ℐ.(3.38)

Hence, the L¹, L² and L^∞ norms of f_r on iℝ₊ are O(t^−3/2). Letting

ha(t,k)=f0(k)+fa(t,k), t>0, k∈D¯1,hr(t,k)=fr(t,k), t>0, k∈iℝ+,(3.39)

we find a decomposition of h with the properties listed in the statement of the lemma.

We next introduce the open subsets {Ωj}18, as displayed in Fig. 3. The following lemma describes how to decompose r_j, j = 1, 2 into an analytic part r_j, a and a small remainder r_j, r.

Lemma 3.2.

There exist decompositions

r1(k)=r1,a(x,t,k)+r1,r(x,t,k), k>k0,r2(k)=r2,a(x,t,k)+r2,r(x,t,k), k<k0,(3.40)

where the functions {rj,a,rj,r}12 have the following properties:

(1)
For each ξ ∈ ℐ and each t > 0, r_j, a(x, t, k) is defined and continuous for k∈Ω¯j and analytic for Ω_j, j = 1, 2.
(2)
The functions r_1,a and r_2,a satisfy, for ξ ∈ ℐ, t > 0, j = 1, 2,
|rj,a(x,t,k)−rj(k0)|≤C|k−k0|et4|ReΦ(ξ,k)|, |rj,a(x,t,k)|≤C1+|k−k0|2et4|ReΦ(ξ,k)|, k∈Ω¯j,(3.41)
where the constant C is independent of ξ, k, t.
(3)
The L¹, L² and L^∞ norms of the function r_1,r(x, t, ·) on (k₀, ∞) are O(t^−3/2) as t → ∞ uniformly with respect to ξ ∈ ℐ.
(4)
The L¹, L² and L^∞ norms of the function r_2,r(x,t, ·) on (−∞, k₀) are O(t^−3/2) as t → ∞ uniformly with respect to ξ ∈ ℐ.

Proof.

Analogous to the proof of Lemma 3.1. One can also see [17, 24].

The purpose of the next transformation is to deform the contour so that the jump matrix involves the exponential factor e^−tΦ on the parts of the contour where ReΦ is positive and the factor e^tΦ on the parts where Re Φ is negative according to the signature table for Re Φ. More precisely, we put

M(3)(x,t,k)=M(2)(x,t,k)G(k),(3.42)

where

G(k)={(10−δ−1r1,a(detδ)−1etΦI),k∈Ω1,(1−r2,aδdetδe−tΦ0I),k∈Ω2,(10δ−1r2,a†(detδ)−1etΦI),k∈Ω3,(1r1,a†δdetδe−tΦ0I)k∈Ω4,(1−ha†δdetδe−tΦ0I),k∈Ω5,(10δ−1ha(detδ)−1etΦI),k∈Ω6,I,k∈Ω7∩Ω8.(3.43)

Then the matrix M⁽³⁾(x, t, k) satisfies the following RH problem

{M+(3)(x,t,k)=M−(3)(x,t,k)J(3)(x,t,k),k∈Σ(3),M(3)(x,t,k)→I,k→∞.(3.44)

with the jump matrix J(3)=G−−1(k)J(2)G+(k) given by

J1(3)=(10δ−1(r1,a+h)(detδ)−1etΦI),J2(3)=(1−r2,aδdetδe−tΦ0I),J3(3)=(10−δ−1r2,a†(detδ)−1etΦI),J4(3)=(1(r1,a†+h†)δdetδe−tΦ0I),J5(3)=(1(r1,a†+ha†)δdetδe−tΦ0I),J6(3)=(10δ−1(r1,a+ha)(detδ)−1etΦI),J7(3)=(1r1,r†δdetδe−tΦ0I)(10δ−1r1,r(detδ)−1etΦI),J8(3)=(1−r2,rδ−detδ−e−tΦ0I)(10−δ+−1r2,r†(detδ+)−1etΦI),J9(3)=(10−δ−1hr(detδ)−1etΦI),J10(3)=(1−hr†δdetδe−tΦ0I),(3.45)

with Ji(3) denoting the restriction of J⁽³⁾ to the contour labeled by i in Fig. 3.

Remark 3.2.

Recalling the meaningful work [3], begin performing the asymptotic analysis, the authors introduced a new spectral variable which led to a new phase function Φ(ζ, λ) = 4iλ² +2iζλ only possed a single critical point, which has the same form as our phase function (3.6). Thus, it is not surprising that the figures of deformation contours similar to [3]. However, our jump matrices involved are all 3 × 3 (that is, the spectral functions r₁(k), r₂(k), h(k) are vectors) and the function δ(k) is a 2 × 2 matrix, which have the essential difference compared with [3]. As a result, the analysis in the next section will present some new skills, see Lemma 3.3.

3.2. Local model near k₀

It is easy to check that the jump matrix J⁽³⁾ decays to identity matrix I as t → ∞ everywhere except near k₀. Thus, the main contribution to the long-time asymptotics should come from a neighborhood of the stationary phase point k₀. To focus on k₀, we make a scaling transformation by

Nk0:k↦z8t+k0.(3.46)

Let D_ε(k₀) denote the open disk of radius ε centered at k₀ for a small ε > 0. Then, the map k ↦ z is a bijection from D_ε(k₀) to the open disk of radius 8tɛ centered at the origin. Since the function δ(k) satisfying a 2 × 2 matrix RH problem (3.15) can not be solved explicitly, to proceed the next step, we will follow the idea developed in [17] to use the available function detδ(k) to approximate δ(k) by error estimate. More precisely, we rewrite the (12) entry of J2(3) as

(r2,aδdetδe−tΦ)(k)=(r2,a(detδ)2e−tΦ)(k)+(r2,a(δ−detI)detδe−tΦ)(k).(3.47)

For the first part in the right-hand side of (3.47), we have

Nk0(r2,a(detδ)2e−tΦ)(z)=η2ρ2r2,a(z8t+k0),(3.48)

where

η=(8t)iν2e2ik02t+χ(k0), ρ=z−iνe−iz24eχ(z8t+k0)−χ(k0).(3.49)

Let δ˜(k)=r2,a(k)δ(k)−detδ(k)Ie−tΦ(ξ,k), then we have the following estimate.

Lemma 3.3.

For z∈L^={z=αe3iπ4:−∞<α<+∞}, as t → ∞, the following estimate for δ˜(k) hold:

|(Nk0δ˜)(z)|≤Ct−1/2,(3.50)

where the constant C > 0 independent of z, t.

Proof.

The idea of the proof comes from [17]. It follows from (3.15) and (3.16) that δ˜ satisfies the following RH problem across (−∞, k₀) oriented in Fig. 2:

{δ˜−(k)=(1+r(k)r†(k))δ˜+(k)+e−tΦ(ξ,k)f(k),k<k0,δ˜(k)→0,k→∞,(3.51)

where f(k) = [r_2,a(r^†r − rr^†I)δ₊](k). Then the function δ˜(k) can be expressed by

δ˜(k)=X(k)∫−∞k0e−tΦ(ξ,s)f(s)X−(s)(s−k)ds,X(k)=exp{12πi∫−∞k0ln(1+r(s)r†(s))s−kds}.(3.52)

It follows from r_2,a(r^†r − rr^†I) = r_2,r(rr^†I − r^†r) that f(k) = O(t^−3/2). Similar to Lemma 3.2, f(k) can be decomposed into two parts: f_a(x, t, k) which has an analytic continuation to Ω₂ and f_r(x, t, k) which is a small remainder. In particular,

|fa(x,t,k)|≤C1+|k−k0+1t|2et4|ReΦ(ξ,k)|,k∈Lt,|fr(x,t,k)|≤C1+|k−k0+1t|2t−3/2,k∈(−∞,k0),(3.53)

where Lt={k=k0−1t+αe3iπ4:0≤α<∞}. Therefore, for z∈L^, we can find

(Nk0δ˜)(z)=X(k0+z8t)∫k0−1tk0etΦ(ξ,s)f(s)X−(s)(s−k0−z8t)ds+X(k0+z8t)∫−∞k0−1te−tΦ(ξ,s)fa(x,t,s)X−(s)(s−k0−z8t)ds+X(k0+z8t)∫−∞k0−1te−tΦ(ξ,s)fr(x,t,s)X−(s)(s−k0−z8t)ds=I1+I2+I3.

Then,

|I1|≤∫k0−1tk0|f(s)||s−k0−z8t|ds≤Ct−3/2|ln|1−22zt1/2||≤Ct−2,|I3|≤∫−∞k0−1t|fr(x,t,s)||s−k0−z8t|ds≤Ct−1/2.

By the Cauchy’s theorem, we can evaluate I₂ along the contour L_t instead of the interval (−∞, k0−1t). Using the fact ReΦ(ξ, k) > 0 in Ω₂, we can obtain |I₂| ≤ Ce^−ct. This completes the proof of the lemma.

Remark 3.3.

The estimate (3.50) also holds if r_2,a is replaced with r1,a†+h† or r1,a†+ha†. There is a similar estimate

|(Nk0δ^)(z)|≤Ct−1/2, t→∞,(3.54)

for z∈L^¯, where δ^(k)=[δ−1(k)−(detδ)−1I]ρ(k)etΦ(ξ,k), ρ=r2,a†, r_1,a + h or r_1,a + h_a.

Remark 3.4.

As mentioned above, here we use function detδ(k) which can be explicitly written down to approximate the unsolvable function δ(k) by error control. This procedure has never appeared in the long-time asymptotic analysis of the integrable nonlinear evolution equations associated with 2 × 2 matrix spectral problems.

In other words, we have the following important relation:

(Nk0Ji(3))(x,t,z)=J˜(x,t,z)+O(t−1/2), i=1,…,6,(3.55)

where J˜(x,t,z) is given by

J˜(x,t,z)={(10(r1,a+ha)η−2ρ−2I),k∈𝒳1∩D1,(10(r1,a+h)η−2ρ−2I),k∈𝒳1∩D2,(1−r2,aη2ρ20I),k∈𝒳2,(10−r2,a†η−2ρ−2I),k∈𝒳3,(1(r1,a†+h†)η2ρ20I),k∈𝒳4∩D3,(1(r1,a†+ha†)η2ρ20I),k∈𝒳4∩D4,(3.56)

where 𝒳 = X +k₀ denote the cross X defined by (3.57) centered at k₀, and X = X₁ ∪ X₂ ∪ X₃ ∪ X₄ ∪ ℂ be the cross defined by

X1={leiπ4|0≤l<∞}, X2={le3iπ4|0≤l<∞},X3={le−3iπ4|0≤l<∞}, X4={le−iπ4|0≤l<∞},(3.57)

and oriented as in Fig. 4.

For any fixed z ∈ X and ξ ∈ ℐ, we have

(r1,a+h)(z8t+k0)→r†(k0), r2,a(z8t+k0)→r(k0)1+r(k0)r†(k0),ρ2→e−iz22z−2iν,(3.58)

as t → ∞. This implies that the jump matrix J˜ tend to the matrix J^(k₀) for large t, where

J(k0)(x,t,z)={(10η−2eiz22z2iνr†(k0)I),z∈X1,(1−η−2e−iz22z−2iνr(k0)1+r(k0)r†(k0)0I),z∈X2,(10−η−2eiz22z2iνr†(k0)1+r(k0)r†(k0)I),z∈X3,(1η−2e−iz22z−2iνr(k0)0I),z∈X4.(3.59)

Theorem 3.1.

The following RH problem:

{M+X(x,t,k)=M−X(x,t,k)JX(x,t,k),z∈X,MX(x,t,k)→I,z→∞,(3.60)

with the jump matrix JX(x,t,z)=ησ^J(k0) has a unique solution M^X (x, t, z). This solution satisfies

MX(x,t,z)=I−iz(0βX(βX)†0)+O(1z2), z→∞,(3.61)

where the error term is uniform with respect to argz ∈ [0, 2π] and the function β^X is given by

βX=eiπ4−πν2νΓ(−iν)2πr(k0),(3.62)

where Γ(·) denotes the standard Gamma function. Moreover, for each compact subset 𝒟 of ℂ,

supr(k0)∈𝒟supz∈ℂ\X|MX(x,t,z)|<∞.(3.63)

Proof.

The proof relies on deriving an explicit formula for the solution M^X in terms of parabolic cylinder functions. One can see [17].

Let D_ε(k₀) denote the open disk of radius ε centered at k₀ for a small ε > 0 and 𝒳^ε = 𝒳 ∩ D_ε(k₀). We can approximate M⁽³⁾ in the neighborhood D_ε(k₀) of k₀ by

M(k0)(x,t,z)=η−σ^MX(x,t,z).(3.64)

Lemma 3.4

For each ξ ∈ ℐ and t > 0, the function M^(k₀)(x, t, k) defined in (3.64) is an analytic function of k ∈ D_ε(k₀) \ 𝒳^ε. Furthermore,

|M(k0)(x,t,z)−I|≤C, t>3, ξ∈ℐ, k∈Dɛ(k0)¯\𝒳ɛ.(3.65)

Across 𝒳^ε, M^(k₀) satisfied the jump condition M+(k0)=M−(k0)J(k0), where the jump matrix J^(k₀) satisfies the following estimates for 1 ≤ p ≤ ∞:

‖J(3)−J(k0)‖Lp(𝒳ɛ)≤Ct−12−12plnt, t>3, ξ∈ℐ,(3.66)

where C > 0 is a constant independent of t, ξ, z. Moreover, as t → ∞,

‖(M(k0))−1(x,t,k)−I‖L∞(∂Dɛ(k0))=O(t−1/2),(3.67)

and

−12πi∫∂Dɛ(k0)((M(k0))−1(x,t,k)−I)dk=η−σ^M1X8t+O(t−1),(3.68)

where M1X is defined by

M1X=−i(0βX(βX)†0).(3.69)

Proof.

The analyticity of M^(k₀) is obvious. Since |η| = 1, thus, the estimate (3.65) follows from the definition of M^(k₀) in (3.64) and the estimate (3.63).

On the other hand, we have J(3)−J(k0)=J(3)−J˜+J˜−J(k0). However, according to the Lemma 89 in [15], we conclude that

‖J˜−J(k0)‖L∞(𝒳1ɛ)≤C|eiγ2z2|t−1/2lnt, 0<γ<12, t>3, ξ∈ℐ,

for k∈𝒳1ɛ, that is, z=8tueiπ4, 0 ≤ u ≤ ε. Thus, for ε small enough, it follows from (3.55) that

‖J(3)−J(k0)‖L∞(𝒳1ɛ)≤C|eiγ2z2|t−1/2lnt.(3.70)

Then we have

‖J(3)−J(k0)‖L1(𝒳1ɛ)≤Ct−1lnt, t>3, ξ∈ℐ.(3.71)

By the general inequality ‖f‖Lp≤‖f‖L∞1−1/p‖f‖L11/p, we find

‖J(2)−J(k0)‖Lp(𝒳1ɛ)≤Ct−1/2−1/2plnt, t>3, ξ∈ℐ.(3.72)

The norms on 𝒳jɛ, j = 2, 3, 4, are estimated in a similar way. Therefore, (3.66) follows.

If k ∈ ∂D_ε(k₀), the variable z=8t(k−k0) tends to infinity as t → ∞. It follows from (3.61) that

MX(x,t,z)=I+M1X8t(k−k0)+O(1t), t→∞, k∈∂Dɛ(k0).(3.73)

Since

M(k0)(x,t,k)=η−σ^MX(x,t,z),

thus we have

(M(k0))−1(x,t,z)−I=−η−σ^M1X8t(k−k0)+O(1t), t→∞, k∈∂Dɛ(k0).(3.74)

The estimate (3.67) immediately follows from (3.74) and |M1X|≤C. By Cauchy’s formula and (3.74), we derive (3.68).

3.3. Derivation of the asymptotic formula

Define the approximate solution M^(app)(x, t, k) by

M(app)={M(k0),k∈Dɛ(k0),I,elsewhere.(3.75)

Let M^(x,t,k) be

M^=M(3)(M(app))−1,(3.76)

then M^(x,t,k) satisfies the following RH problem

{M^+(x,t,k)=M^−(x,t,k)J^(x,t,k),k∈Σ^,M^(x,t,k)→I,k→∞.(3.77)

where the jump contour Σ^=Σ(3)∪∂Dɛ(k0) is depicted in Fig. 5, and the jump matrix J^(x,t,k) is given by

J^={M−(k0)J(3)(M+(k0))−1,k∈Σ^∩Dɛ(k0),(M(k0))−1,k∈∂Dɛ(k0),J(3),k∈Σ^\Dɛ(k0)¯.(3.78)

Let W^=J^−I, and we rewrite Σ^ as follows:

Σ^=∂Dɛ(k0)∪𝒳ɛ∪Σ^1∪Σ^2,

where

Σ^1=∪16Σj(3)/Dɛ(k0), Σ^2=∪710Σj(3),

and {Σj(3)}110 denoting the restriction of Σ⁽³⁾ to the contour labeled by j in Fig. 3. Then the following inequalities are valid.

Lemma 3.5.

For 1 ≤ p ≤ ∞, the following estimates hold for t > 3 and ξ ∈ ℐ,

‖W^‖Lp(∂Dɛ(k0))≤Ct−12,(3.79)

‖W^‖Lp(𝒳ɛ)≤Ct−12−12plnt,(3.80)

‖W^‖Lp(Σ^1)≤Ce−ct,(3.81)

‖W^‖Lp(Σ^2)≤Ct−32.(3.82)

Proof.

The inequality (3.79) is a consequence of (3.67) and (3.78). For k ∈ 𝒳^ε, we find

W^=M−(k0)(J(3)−J(k0))(M+(k0))−1.

Therefore, it follows from (3.65) and (3.66) that the estimate (3.80) holds. For k∈D2∩(𝒳1\Dɛ(k0)¯), Ŵ only has a nonzero δ⁻¹(r_1,a + h)(detδ)⁻¹e^tΦ in (21) entry. Hence, for t ≥ 1, by (3.20), we get

|W^21|≤C|r1,a+ha|etReΦ≤C1+|k|2e−3|k−k0|2t≤Ce−3ɛ2t.

In a similar way, the estimates on 𝒳j\Dɛ(k0)¯, j = 2, 3, 4 hold. This proves (3.81). Since the matrix Ŵ on Σ^2 only involves the small remainders h_r, r_1,r and r_2,r, thus, by Lemmas 3.1 and 3.2, the estimate (3.82) follows.

The results in Lemma 3.5 imply that:

‖W^‖L∞(Σ^)≤Ct−1/2lnt,‖W^‖L1∩L2(Σ^)≤Ct−1/2, t>3, ξ∈ℐ.(3.83)

This uniformly vanishing bound on Ŵ shows that the RH problem (3.77) is a small-norm RH problem, for which there is a well known existence and uniqueness theorem. In fact, we may write

M^(x,t,k)=I+12πi∫Σ^(μ^W^)(x,t,ζ)ζ−kdζ, k∈ℂ\Σ^,(3.84)

where the 3 × 3 matrix-valued function μ^(x,t,k) is the unique solution of

μ^=I+C^W^μ^,(3.85)

and

‖μ^(x,t,⋅)−I‖L2(Σ^)=O(t−1/2), t>∞, ξ∈ℐ.(3.86)

The singular integral operator C^W^:L2(Σ^)+L∞(Σ^)→L2(Σ^) is defined by

C^W^f=C^−(fW^)(C^−f)(k)=limk→Σ^−∫Σ^f(ζ)ζ−kdζ2πi,

where Ĉ₋ is the well-known Cauchy operator. Moreover, by (3.83), we find

‖C^W^‖B(L2(Σ^))≤C‖W^‖L∞(Σ^)≤Ct−1/2lnt,(3.87)

where B(L2(Σ^)) denotes the Banach space of bounded linear operators L2(Σ^)→L2(Σ^). Therefore, the resolvent operator (I−C^W^)−1 is existent and thus of both μ^ and M^ for large t.

It follows from (3.84) that

limk→∞k(M^(x,t,k)−I)=−12πi∫Σ^(μ^W^)(x,t,k)dk.(3.88)

Using (3.80) and (3.86), we have

∫𝒳ɛ(μ^W^)(x,t,k)dk=∫𝒳ɛW^(x,t,k)dk+∫𝒳ɛ(μ^(x,t,k)−I)W^(x,t,k)dk≤‖W^‖L1(𝒳ɛ)+‖μ^−I‖L2(𝒳ɛ)‖W^‖L2(𝒳ɛ)≤Ct−1lnt, t→∞.

Similarly, by (3.81) and (3.86), the contribution from Σ^1 to the right-hand side of (3.88) is

O(‖W^‖L1(Σ^1)+‖μ^−I‖L2(Σ^1)‖W^‖L2(Σ^1))=O(e−ct), t→∞.

By (3.82) and (3.86), the contribution from Σ^2 to the right-hand side of (3.88) is

O(‖W^‖L1(Σ^2)+‖μ^−I‖L2(Σ^2)‖W^‖L2(Σ^2))=O(t−3/2), t→∞.

Finally, by (3.68), (3.79) and (3.86), we can get

−12πi∫∂Dɛ(k0)(μ^W^)(x,t,k)dk=−12πi∫∂Dɛ(k0)W^(x,t,k)dk−12πi∫∂Dɛ(k0)(μ^(x,t,k)−I)W^(x,t,k)dk=−12πi∫∂Dɛ(k0)((M(k0))−1(x,t,k)−I)dk+O(‖μ^−I‖L2(∂Dɛ(k0))‖W^‖L2(∂Dɛ(k0)))=η−σ^M1X8t+O(t−1), t→∞.

Thus, we obtain the following important relation

limk→∞k(M^(x,t,k)−I)=η−σ^M1X8t+O(t−1lnt), t→∞.(3.89)

Taking into account that (3.7), (3.10), (3.13), (3.42), (3.76) and (3.89), for sufficient large k∈ℂ\∑^, we get

(u(x,t) v(x,t))=2limk→∞(kM(x,t,k))12=2limk→∞(kM^(x,t,k)−I)12=(η−σ^M1X)122t+O(lntt)=−iβXη22t+O(lntt), t→∞.(3.90)

In view of (3.49) and (3.62), we obtain our main results stated as the Theorem 1.1.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 3
Pages: 483 - 508
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1613055 How to use a DOI?
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TY  - JOUR
AU  - Boling Guo
AU  - Nan Liu
PY  - 2021
DA  - 2021/01/06
TI  - The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line
JO  - Journal of Nonlinear Mathematical Physics
SP  - 483
EP  - 508
VL  - 26
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1613055
DO  - 10.1080/14029251.2019.1613055
ID  - Guo2021
ER  -

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Journal of Nonlinear Mathematical Physics

The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

1. Introduction

Theorem 1.1.

Remark 1.1.

2. Preliminaries

Theorem 2.1.

Remark 2.1.

Remark 2.2.

3. Long-time asymptotic analysis

3.1. Transformations of the RH problem

Remark 3.1.

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Remark 3.2.

3.2. Local model near k0

Lemma 3.3.

Proof.

Remark 3.3.

Remark 3.4.

Theorem 3.1.

Proof.

Lemma 3.4

Proof.

3.3. Derivation of the asymptotic formula

Lemma 3.5.

Proof.

References

Cite this article

3.2. Local model near k₀