The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed ¯-Riemann–Hilbert problem

Minghe Zhang, Zhenya Yan

Communications in Theoretical Physics ›› 2024, Vol. 76 ›› Issue (6) : 65002.

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Communications in Theoretical Physics ›› 2024, Vol. 76 ›› Issue (6) : 65002. DOI: 10.1088/1572-9494/ad361b
Mathematical Physics

The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed ¯-Riemann–Hilbert problem

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Abstract

In this paper, we investigate the Cauchy problem of the Sasa–Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the ¯-nonlinear steepest descent method of the mixed ¯-Riemann–Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.

Key words

Sasa–Satsuma equation / inverse scattering / ¯-Riemann–Hilbert problem / ¯ steepest descent method / soliton resolution

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Minghe Zhang, Zhenya Yan. The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed ¯-Riemann–Hilbert problem[J]. Communications in Theoretical Physics, 2024, 76(6): 65002 https://doi.org/10.1088/1572-9494/ad361b

1. Introduction

In 1993, based on the scheme of the Riemann–Hilbert (RH) problem, Deift and Zhou discussed the long-time asymptotic behaviors of the solutions of the mKdV equation by combining classical Fourier analysis and the steepest descent method [1]. Based on this method, the long-time asymptotic behaviors of many integrable equations were explored, such as the sine-Gordon equation [2], the Korteweg-de Vries equation [3], the Camassa–Holm equation [4], the short pulse equation [5, 6], the Fokas–Lenells equation [7], the extended mKdV equation [8, 9], and so on.
As a development of the Deift–Zhou nonlinear steepest descent method, a powerful tool called the ¯-steepest descent method was first proposed by Mclaughlin and miller to analyze the asymptotic behaviors of orthogonal polynomials [10, 11]. Later, this method was successfully used to analyze the long-time behaviors of solutions to integrable nonlinear wave equations, such as the focusing NLS equation [12, 13], the defocusing NLS equation [14, 15], the derivative NLS equation [16], the mKdV equation [17, 18], the fifth-order mKdV equation [19], the complex short pulse equation [20], the modified Camassa–Holm equation [21], the Novikov equation [22], etc.
As a new-type integrable high-order equation of the nonlinear Schrödinger equation, the Sasa–Satsuma (SS) equation was presented [23]
qt+qxxx+3|q|2qx+3(|q|2q)x=0,(x,t)R×R+,
(1.1)
which admits a 3 × 3 Lax pair
Φx+ikσΦ=U(x,t;k)Φ,Φt+4ik3σ=W(x,t;k)Φ,
(1.2)
where Φ = Φ(x, t; k) is a matrix function of x, t and iso-spectral parameter kC,
U(x,t;k)=(02×2q(x,t)q(x,t)01×2),q(x,t)=(q(x,t)q(x,t)),σ=(I2×202×101×21),W(x,t;k)=4k2U+2ikσ(UxU2)+2U3Uxx+[Ux,U],
(1.3)
with ‘*' and ‘†' denoting the complex conjugation and Hermite transformation, respectively. In fact, the SS equation (1.1) can also be regarded as the special reduction (r = q*) of the two-component integrable complex modified KdV equations [24, 25]
qt+qxxx+6|q|2qx+3(qr)xr=0,rt+rxxx+6|r|2rx+3(qr)xq=0.
(1.4)
The SS equation admits many other integrable properties, such as N-soliton solutions, infinite conservation laws, nonlocal symmetries, Painlevé property, dark soliton solutions, and rogue wave solutions [2630]. Recently, Liu et al studied the long-time asymptotic behaviors of the SS equation via the Deift–Zhou nonlinear steepest descent method [31]. Recently, Xun and Fan used this method to study the long-time and Painlevé-type asymptotics of the SS equation under the assumption of scattering data admitting only finitely simple zeros [32].
Based on the above-mentioned situations, in this paper, we focus on the long-time asymptotic behaviors of solutions for the Cauchy problem of the integrable SS equation (1.1) with the initial data:
q(x,0)=q0(x)S(R),
(1.5)
under the assumption of the scattering data possessing finitely double zeros, where S(R) denotes the Schwartz space. We then obtain the long-time behaviors of the potential q(x, t).
The rest of this paper is organized as follows. In section 2, we review the direct and inverse scattering transforms about the 3 × 3 Lax pair of equation (1.1) and deduce the analytic region about the Jost functions. Furthermore, we set up the original RH problem. Based on the RH problem, in section 3, using the ideas from [13, 32], we give a series of the transformation of the RH problem to make it a model RH problem whose solution is a parabolic function. In section 4, through the transformations of the RH problem, the potential of RHP1 can be reconstructed by three parts. One is the double-pole soliton solutions by solving the RHP in the reflectionless case, and the other terms are provided by the error function E(r) and the pure ¯-problem.

2. The direct scattering problem

2.1. Jost solutions of the Lax pair and scattering data

Based on the boundary-value condition lim|x|q0(x)=0, the eigenfunction of the Lax pair (1.2) has the following asymptotic form
Φ(k,x,t)ei(kx+4k3t)σ,|x|.
(2.1)
To change the large-space asymptotics of the eigenfunction of the Lax pair (1.2) into a unit matrix, let
Ψ(k,x,t)=Φ(k,x,t)ei(kx+4k3t)σ.
Then Ψ(k, x, t) satisfies the following modified Lax pair
Ψx+ik[σ,Ψ]=UΨ,Ψt+4ik3[σ,Ψ]=WΨ,
(2.2)
which can be written as a fully differential form
d(ei(kx+4k3t)σ^Ψ)=ei(kx+4k3t)σ^(UΨdx+WΨdt),
(2.3)
from which the Jost solutions Ψ+(k, x, t) and Ψ(k, x, t) can be rewritten as follows:
Ψ±(k,x,t)=Ix±eik(ξx)σ^U(ξ,t)Ψ±(k,y,t)dξ.
(2.4)
Let Ψ±=(Ψ±1(k,x,t),Ψ±2(k,x,t)), where Ψ±1(k, x, t) and Ψ±2(k, x, t) represent their first two columns and third column, respectively. It follows from equation (2.4) that Ψ−1, Ψ+2 are analytic in C+, and Ψ+1, Ψ−2 are analytic in C. Moreover,
(Ψ1(k,x,t),Ψ±2(k,x,t))=I+O(k1),kC±.
(2.5)
By using Abel's lemma and tr(U)=tr(W)=0, one knows that detΨ±(k,x,t) are independent of variable x and detΨ±=1. Furthermore, Ψ±ei(kx+4k3t)σ are linearly dependent to lead to
Ψ(k)ei(kx+4k3t)σ=Ψ+(k)ei(kx+4k3t)σS(k),
(2.6)
where S(k) is a 3 × 3 scattering matrix. Moreover, together with det(S)=1, one can know that Ψ± and S(k) admit the symmetries:
Ψ±(k;x,t)=Ψ1(k;x,t),Ψ±(k;x,t)=ϱΨ±(k,x,t)ϱ,
(2.7)
S(k)=S1(k),S(k)=ϱS(k)ϱ,
(2.8)
based on the two symmetries of U
U=U,ϱUϱ=U,ϱ=ϱ1=(σ1001),σ1=(0110).
(2.9)
One can further rewrite S(k) as
S(k)=(A(k)adj[A(k)]B(k)B(k)det[A(k)])=limxeikxσ^Ψ(k;x,0),
(2.10)
where A(k)=σ1A(--k)σ1=(aij(k))2×2, adj[A(k*)] denotes the adjoint matrix of A(k*), and B(k) = B*(–k*)σ1 = (B1(k), B2(k)), and
A(k)=I+Rq(x,0)Ψ12(k;x,0)dx,B(k)=Rq(x,0)Ψ11(k;x,0)e2ikxdx,
(2.11)
which imply that A(k) is analytic in C+ by virtue of the analyticity of Ψ−12(k; x, 0).
For the convenience of the following analysis, an assumption of scattering data is that the functions A(k) and detA(k) have no zeros on R and A(k) has finite double zeros in CR, γ(k):=B(k)A1(k)H1,1(R).

2.2. The Riemann–Hilbert problem with higher-order poles

Let A(k) have 2N double zeros k1, k2,…,k2N in C+ with kN+j=kj,j=1,2,,N since there is the symmetry A(k) = σ1A*(− k*)σ1, that is, A(kj)=A(kj)=0,A(kj)0(j=1,2,,2N). To establish a RH problem, we define the following sectionally meromorphic matrix M(k; x, t) with the aid of the analyticity of Jost functions
M(k;x,t)={(Ψ1(k)A1(k),Ψ+2(k)),kC+,(Ψ+1(k),Ψ2(k)detA(k)),kC,
(2.12)
such that M(k; x, t) has 2N double poles K = {kj, j = 1,…,2N} in C+ and 2N double poles K¯={kj,j=1,,2N} in C. According to equations (2.6) and (2.12), one can find that the matrix-valued function M(k; x, t) satisfies the following RH problem:
RHP-1. Find a matrix-valued function solution M(k; x, t) satisfying the following conditions:

Analyticity : M(k; x, t) is a meromorphic function in CR and has double poles at kjK and kjK¯;

Jump relation: M(k; x, t) has continuous boundary values M±(k; x, t) on R, and

M+(k)=M(k)V(k;x,t),kR,
(2.13)
where the jump matrix is
V(k;x,t)=(I2×2+γ(k)γ(k)γ(k)e2itθ(k;x,t)γ(k)e2itθ(k;x,t)1),θ(k;x,t)=k(xt+4k2).
(2.14)

Asymptotics:

M(k;x,t)=I+O(1k),k.
(2.15)

Therefore, M(k; x, t) has double poles at each point in KK¯ with:
P2k=kjM(k;x,t)=limkkjM(k;x,t)(00Aje2itθ(k)0),
(2.16)
Resk=kjM(k;x,t)=limkkjM(k;x,t)(00Bje2itθ(k)0)+M(k;x,t)(00Aje2itθ(k)0),
(2.17)
P2k=kjM(k;x,t)=limkkjM(k;x,t)(0Aje2itθ(k)00),
(2.18)
Resk=kjM(k;x,t)=limkkjM(k;x,t)(0Bje2itθ(k)00)+M(k;x,t)(0Aje2itθ(k)00),
(2.19)
where
Aj=2B(kj)adj[A(kj)]det¨[A(kj)],Bj=(2itθ(kj)+23det[A(kj)]det¨[A(kj)])Aj+Aj.
(2.20)
We now give the reconstruction formula for the solution of (1.1). Let Ψ have the following Laurent expansion
Ψ=Ψ(0)+Ψ(1)k+Ψ(2)k2+
(2.21)
Then, by substituting (2.21) into (2.2) and comparing the k−2 in t-part and k0 in the x part, one can obtain
4i[σ,Ψ(1)]=4UΨ(0),Ψx(0)+i[σ,Ψ(1)]=UΨ(0).
(2.22)
From the above two equations, we have
U=i[σ,Ψ(1)],
(2.23)
i.e.
q(x,t)=(q(x,t),q(x,t))T=2iΨ12(1)=2ilimk(kM(k;x,t))12,
(2.24)
where q(x, t) solves the SS equation (1.1).
In the following, we mainly consider the solution of M(k; x, t) of RHP-1.

3. The mixed ¯-RH problem and its decomposition

3.1. Two factorizations of jump matrix V(k)

Since the jump matrix V(k; x, t) given by equation (2.14) admits the two different oscillatory terms for t > 0
O±=e±2itθ(k)=e±2it(4k3+xtk),θ(k)=kxt+4k3=4(k33k02k),
(3.1)
where θ(k) admits two phase points k=±k0,k0=x12t with xt < 0. To analyze their properties, one needs to consider the properties of Re[iθ(k)] of O±
Re[iθ(k)]=4Imk(Im2k3Re2k+3k02),
(3.2)
whose signature table is given in figure 1.
Figure 1. The signature table of Reiθ(k)=4Imk(Im2k3Re2k+3k02) with ±k0 being phase points.

Full size|PPT slide

To analyze the long-time asymptotics of RHP-1, we first divide all the poles into two parts:
Δ={k|Re2(k)13Im2(k)<k02},Δ+={k|Re2(k)13Im2(k)>k02}.
(3.3)
We assume that there are no poles corresponding to the region Δ for simplicity.
The jump matrix V(k, x, t) has two different decompositions of upper and lower triangular matrices:
V={(Iγ(k)e2itθ(k)01)(I0γ(k)e2itθ(k)1),kR[k0,k0],(I0γ(k)1+γ(k)γ(k)e2itθ(k)1)(I+γ(k)γ(k)0011+γ(k)γ(k))(Iγ(k)1+γ(k)γ(k)e2itθ(k)01),k(k0,k0).
(3.4)
To offset the influence of the diagonal matrix of the second decomposition, one needs to introduce the 2 × 2 matrix function δ(k) satisfying the following property:

The matrix function δ(k) and scalar function detδ(k) satisfy the following properties:

δ(k) and det(δ(k)) are analytic, and δ(k)δ(k)=I, det(δ(k))det(δ(k))=1 in C[k0,k0].

For k(k0,k0),

δ+(k)=δ(k)(1+γ(k)γ(k)),det(δ+(k))=det(δ(k))(1+|γ(k)|2);
(3.5)

|δ+(k)|2={|γ(k)|2+2,k(k0,k0),2,otherwise.
(3.6)
|δ(k)|2={2|γ(k)|21+|γ(k)|2,k(k0,k0),2,otherwise.
(3.7)

As |k| with |arg(k)|c<π,

δ(k)=I+O(k1),det(δ(k))=1+ik[12πk0k0log(1+|γ(ξ)|21+|γ(k0)|2)dξ2νk0]+O(k2);
(3.8)

det(δ(k))=(kk0k+k0)iν(k0)eX(k),
(3.9)
where
ν(k0)=12πlog(1+|γ(k0)|2),X(k)=12πik0k0log(1+|γ(ξ)|21+|γ(k0)|2)dξξk.
(3.10)

Along the ray k=±k0+R+eiϕ with |ϕ|c<π, as k±k0,

|det(δ(k))(kk0k+k0)iν(k0)eX(±k0)||kk0|1/2.
(3.11)

The proof of the above properties is similar to the proof of proposition 3.1 in [12].

In what follows, our aim is to find a transform of M(k; x, t) → M(1)(k; x, t) such that the jump matrix of M(1)(k; x, t) can be well decomposed. Let
M(1)(k;x,t)=M(k;x,t)T(k),
(3.12)
where
T(k)=(δ1(k)00det[δ(k)])=(T11(k)00T2(k))
(3.13)
with δ(k),det[δ(k)] are given by proposition 1, then the modified M(1)(k; x, t) satisfies the following RHP-2:
RHP-2. Find a matrix-valued function M(1)(k; x, t) satisfying

Analyticity: M(1)(k) is analytic in C(RKK¯).

Jump condition: M+(1)(k)=M(1)(k)V(1)(k),kR, where the jump matrix is

V(1)(k)={(IT1T2γ(k)e2itθ(k)01)(I0(T1T2)1γ(k)e2itθ(k)1),kR[k0,k0],(I0(T1T2)1γ(k)1+γ(k)γ(k)e2itθ(k)1)(IT1+T2+γ(k)1+γ(k)γ(k)e2itθ(k)01),k(k0,k0).
(3.14)

Asymptotics: M(1)(k)=I+O(k1),ask.

Moreover, M(1)(k; x, t) satisfies the following residue conditions at double poles kjK and kjK¯:
P2k=kjM(1)(k;x,t)=limkkjM(1)(k)(00AjT11T21e2itθ(k)0),
(3.15)
Resk=kjM(1)(k;x,t)=limkkjM(1)(k)×(00(BjT11T21+AjT11T21)e2itθ(k)0)+M(1)(k)(00AjT11T21e2itθ(k)0),
(3.16)
P2k=kjM(1)(k;x,t)=limkkjM(1)(0AjT1T2e2itθ00),
(3.17)
Resk=kjM(1)(k;x,t)=limkkjM(1)(k)×(0(BjT1T2AjT2T1)e2itθ(k)00)+M(1)(k)(0AjT1T2e2itθ(k)00).
(3.18)
To open the original jump curve R along the steepest descent lines arising from the phase points ±k0, let these contours be
Σ1±={k|kk0=R+ei(2±1)π4},Σ3±={k|kk0=dei(2±1)π4,d(0,2k0)},Σ2±={k|kk0=R+ei(2±1)π4},Σ4±={k|kk0=dei(21)π4,d(0,2k0)}.
(3.19)
Then the complex plane C is divided into ten open domains, denoted by Ωj±,j=1,2,3,4 and Ω5, Ω6 (see figure 2). In what follows, we will introduce the continuous functions related to the jump matrix V(1) in these regions.
Figure 2. Deformation of the jump countor from R to Σ(2).

Full size|PPT slide

Figure 3. Pole distribution. The red, green and yellow points generate the breather solutions. Moreover, the red points lie in the region K+(I), the green points lie in the region K(I), and the yellow points on the line Reiθ(k)=0.

Full size|PPT slide

Figure 4. The jump contour for the jump matrix V(in).

Full size|PPT slide

Figure 5. The jump contour Σ(E) for the error function E(k).

Full size|PPT slide

Let D=(k0,k0),D=(,k0),D+=(k0,+). Then there exists the continuous functions Rj±: Ω¯j±C, j=1,2,3,4 such that

R1±(k)={cos(2arctan(kk0))g1±+[1cos(2arctan(kk0))]f1±,kΩ¯1±,g1±=γ(k)T11(k)T21(k),kD±,f1±=γ(±k0)T11(k)eX(±k0)(kk0k+k0)iν(1XK(k)),kΣ1±,
(3.20)
R2±(k)={cos(2arctan(kk0))g2±+[1cos(2arctan(kk0))]f2±,kΩ¯2±,g2±=T1(k)T2(k)γ(k),kD±,f2±=T1(k)eX(±k0)(kk0k+k0)iνγ(±k0)(1XK(k)),kΣ2±,
(3.21)
R3±(k)={cos(2arctan(kk0))g3+[1cos(2arctan(kk0))]f3±,kΩ¯3±,g3=T1+(k)T2+(k)γ(k)1+γ(k)γ(k),kD,f3±=T1(k)eX(±k0)(kk0k+k0)iνγ(±k0)1+γ(±k0)γ(±k0)(1XK(k)),kΣ3±,
(3.22)
R4±(k)={cos(2arctan(kk0))g4+[1cos(2arctan(kk0))]f4±,kΩ¯4±,g4=γ(k)1+γ(k)γ(k)T11(k)T21(k),kD,f4±=γ(±k0)1+γ(±k0)γ(±k0)T11(k)eX(±k0)(kk0k+k0)iν(1XK(k)),kΣ4±,
(3.23)
and Rj±(k)(j=1,2,3,4) have these estimates
|Rj±(k)|1+Re(k)1/2,|¯Rj±(k)||¯χK(k)|+|hj(k)(Re(k))|+|kk0|1/2,¯Rj±(k)=0,forkΩ5Ω6     or    dist(k,KK¯)<ρ/3,
(3.24)
where
h1(k)=γ(k),h2(k)=γ(k),h3(k)=h2(k)1+h1(k)h2(k),h4(k)=h1(k)1+h1(k)h2(k).
and
ρ=12minλμKK¯|λμ|,XK(k)={1,dist(k,KK¯)<ρ3,0,dist(k,KK¯)>2ρ3.

The proof is similar to [12, 32].

Let
M(2)(k;x,t)=M(1)(k;x,t)R(2)(k;x,t),
(3.25)
where R(2) is defined as
R(2)(k)={(I0Rj±(k)e2itθ(k)1),j=1,3,(IRj±(k)e2itθ(k)01),j=2,4,I3×3,otherwise
(3.26)
with Rj±(k) being defined by proposition 2 (notice that the transform causes the previous contour R to change into contour Σ(2)). Then M(2)(k; x, t) solves a mixed ¯-RH problem:
Mixed ¯-RHP. Find a matrix-valued function M(2)(k) = M(2)(k; x, t) solving

Continuity: M(2)(k) is continuous in C(Σ(2)KK¯).

Jump condition: M+(2)(k)=M(2)(k)V(2)(k),kΣ(2), where

V(2)(k)={(I0(1)jRj±e2itθ1),kΣj±,j=1,4,(I(1)jRj±e2itθ01),kΣj±,j=2,3,(I(R3+R3)e2itθ01),k(ik0tan(π12),ik0),(I0(R4+R4)e2itθ1),k(ik0tan(π12),ik0),I3×3,k(ik0tan(π12),ik0tan(π12)).
(3.27)

Asymptotics: M(2)(k) → I, k → ∞ ;

Moreover, for any kC(Σ(2)KK¯), one finds that
¯M(2)(k)=M(1)(k)¯R(2)(k),
(3.28)
where
¯R(2)(k)={(00¯Rj±(k)e2itθ(k)0),j=1,3,(0¯Rj±(k)e2itθ(k)00),j=2,4,03×3,otherwise,
(3.29)
and, M(2)(k; x, t) has double poles at kj and kj with
P2k=kjM(2)(k;x,t)=limkkjM(2)(k)(00AjT11T21e2itθ(k)0),
(3.30)
Resk=kjM(2)(k;x,t)=limkkjM(2)(k)×(00(BjT11T21+AjT11(T21))e2itθ(k)0)+(M(2))(k)(00AjT11T21e2itθ(k)0),
(3.31)
P2k=kjM(2)(k;x,t)=limkkjM(2)(0A^jT1T2e2itθ00),
(3.32)
Resk=kjM(2)(k;x,t)=limkkjM(2)(k)×(0(B^jT1T2+A^jT2(T1))e2itθ(k)00)+(M(2))(k)(0A^jT1T2e2itθ(k)00).
(3.33)

3.2. Analysis on a pure RH problem

Throughout this section, our aim is to decompose the above-mentioned mixed ¯-RHP into a pure RHP with ¯R(2)=0 and a pure ¯-problem with ¯R(2)0. The decomposition of M(2)(k; x, t) can be given as follows:
M(2)(k;x,t)={MRHP(2)(k;x,t),as¯R(2)=0,M(3)(k;x,t)MRHP(2)(k;x,t),as¯R(2)0,
(3.34)
where MRHP(2)(k;x,t) and M(3)(k; x, t) correspond to the pure RHP part and the pure ¯ part without jumps and poles of M(2)(k), respectively.
RHP-3. Find a matrix-valued function MRHP(2)(k) solving the following RHP

Analyticity: MRHP(2)(k) is analytic in C(Σ(2)KK¯).

Jump condition: MRHP+(2)(k)=MRHP(2)(k)V(2)(k),kΣ(2), where V(2)(k) is given by equation (3.27).

Asymptotics: MRHP(2)(k)I, k → ∞ .

The jump matrix V(2) has the following estimate:

V(2)(k;x,t)IL(Σ(2))={O(e6k0ρ2t),kΣj±U±k0,j=1,2,O(e8k02ρt),kΣj±U±k0,j=3,4,O((k0|kk0|)1t1/2),kΣ(2)U±k0,O(e14k03tan3(π12)t),k[±ik0,±ik0tan(π12)],0,k[ik0tan(π12),ik0tan(π12)],
(3.35)
where U±k0={k||k±k0|<ρ/2}.

3.2.1. Soliton solutions corresponding to discrete spectra

In order to analyze the leading term of the solution, we firstly consider RHP-1. RHP-1 reduces to the following RH problem:
RHP-4. A matrix-valued function M(k; x, tσd) with the scattering data σd={(kj,Aj,Bj)}k=12N and K={kj}j=12N and satisfies the following condition:

Analyticity: M(k; x, tσd) is analytical in C(Σ(2)KK¯).

Jump condition:

M+(k;x,t|σd)=M(k;x,t|σd)V(k),
(3.36)

Asymptotics: M(k;x,t|σd)=I+O(k1), k → ∞ .

Moreover, M(k; x, tσd) has double poles at kj and kj with
P2k=kjM(k;x,t|σd)=limkkjM(k;x,t|σd)(00Aje2itθ(k)0),
(3.37)
Resk=kjM(k;x,t|σd)=limkkjM(k;x,t)(00Bje2itθ(k)0)+M(k;x,t|σd)(00Aje2itθ(k)0),
(3.38)
P2k=kjM(k;x,t|σd)=limkkjM(k;x,t|σd)×(0Aje2itθ(k)00),
(3.39)
Resk=kjM(k;x,t|σd)=limkkjM(k;x,t|σd)×(0Bje2itθ(k)00)+M(k;x,t|σd)(0Aje2itθ(k)00).
(3.40)

Given scattering data σd=(kj,Aj,Bj))k=12N and discrete spectra K={kj}j=12N, the RH problem has a unique solution

qsol(x,t|σd)=(q(x,t),q(x,t))T=2ilimk(kM(k|σd))12.
(3.41)

The uniqueness of the solution can be guaranteed by Liouville's theorem. For the reflectionless case V(k)=I, it follows from equation (3.36) and the Plemelj formula that one has

M(k|σd)=I+j=12NResk=kjM(k|σd)kkj+j=12NResk=kjM(k|σd)kkj+j=12NP2k=kjM(k|σd)(kkj)2+P2k=kjM(k|σd)(kkj)2.
(3.42)
One can find that M(k|σd) has the following formulation of the sum of sparse matrices:
M(k)=I+l=12N((βl03×1)kkl+(03×2βl~)kkl+(αl03×1)(kkl)2+(03×2αl~)(kkl)2),
(3.43)
which, together with the residue condition, can further lead to the following equations:
(αj0)=(00ηj0)+l=12N((βl~ηj0)kjkl+(αlηj0)(kjkl)2),
(3.44)
(βj0)=(00ηj0)+l=12N((βlζj0)kjkl+(αlζj0)(kjkl)2(βlηj0)(kjkl)22(αlηj0)(kjkl)3),
(3.45)
(0αj)=(0ηj00)+l=12N((0βlηj)kjkl+(0αlηj)(kjkl)2),
(3.46)
(0βj)=(0ηj00)+l=12N((0βlζj)kjkl+(0αζj)kjkl+(0βlηj)(kjkl)2+2(0αlηj)(kjkl)3),
(3.47)
where ζj=Bje2itθ(kj),ηj=Aje2itθ(kj). Then αl,βl,αl~,βl~ can be solved from the above equations.

In what follows, we separate MRHP(2)(k) into two parts:
MRHP(2)(k)={E(k)M(out)(k),kCU±k0,E(k)M(LC)(k),M(LC)(k)=M(out)(k)M(in)(k),kU±k0,
(3.48)
where M(out) is used to find the pure solitions outside U±k0, which is defined in C and only admits discrete spectra without a jump. M(in) is defined in U±k0 without discrete spectra, and the model RHP considered by Liu [31]. Moreover, E(k) denotes the error between MRHP(2)(k) and M(out)(k) outside U±k0.
Let
M(out)(k;x,t|σd(out))=M(k;x,t|σd)(δ1(k)00detδ(k)),
(3.49)
with the scattering data
σd(out)={(kj,A~j,B~j,kjK}j=12N,{A~j,B~j}={Aj,Bj}δ1(kj)(detδ(kj))1.
(3.50)
Then M(out)(k|σd(out)) satisfies the following RH problem:
RHP-5. Find a matrix-valued function M(out)(k;x,t|σdout) without the jump condition solving the following problem:

Analyticity: M(out)(k|σdout) is analytic in C(Σ(2)KK¯).

Asymptotics: M(out)(k|σdout)I, k → ∞ ,

M(out)(k|σdout) has double poles at kj and kj with
P2k=kjM(1)(k;x,t)=limkkjM(1)(k)(00AjT11T21e2itθ(k)0),
(3.51)
Resk=kjM(1)(k;x,t)=limkkjM(1)(k)×(00(BjT11T21+AjT11(T21))e2itθ(k)0)+(M(1))(k)(00AjT11T21e2itθ(k)0),
(3.52)
P2k=kjM(1)(k;x,t)=limkkjM(1)(0A^jT1T2e2itθ00),
(3.53)
Resk=kjM(1)(k;x,t)=limkkjM(1)(k)×(0(B^jT1T2+A^jT2(T1))e2itθ(k)00)+(M(1))(k)(0A^jT1T2e2itθ(k)00).
(3.54)

RHP-5 has the uniqueness solution, and its potential is equivalent to one of the reflectionless cases of RHP-4, that is

qsol(x,t|σd(out))=qsol(x,t|σd)=(q(x,t),q(x,t))T=2ilimk(kM(k|σd))12.
(3.55)

According to the reconstruction formula (2.24), the proof is similar to [32].

We now consider the long-time asymptotic behavior of soliton solutions. Firstly, we define a space-time region
D(v1,v2)={(x,t)R2|x=vt,v[v1,v2]},
(3.56)
where v2v1 < 0. Let
I=[v14,v24],K(I)={kjK|v14Kjv24},N(I)=|K(I)|,K(I)={kjK|Kj<v14},K+(I)={kjK|Kj>v24},Kj=3Re2kjIm2kjcj(I)=cjδ1(kj)ei2πk0k0log(1+|γ(ζ)|2)ζkjdζ.
(3.57)
See figure 3. Then we have the following proposition:

Given scattering data σd={(kj,Aj,Bj)}j=12N and σd(I)={(kj,cj(I))|kjK(I)}. At t+ with (x,t)D(v1,v2), we have

M(k;x,t|σd)=[I+O(e8μt)]MΔI(k;x,t|σd(I)),
(3.58)
where μ(I)=minkjK/K(I){Imkjdist(3Re2kjIm2kj,I)}.

The proof is similar to [13, 32].

Suppose that qsol is the soliton solution of the SS equation corresponding to its scattering data σd={(kj,Aj,Bj)}j=12N, then one has

qsol(x,t|σd(out))=qsol(x,t|σd(I))+O(e8μt),t+,
(3.59)
where qsol(x,t|σd(I)) is the soliton solution corresponding to the scattering data σd(I) of the SS equation.

3.2.2. The solvable local RH problem

RHP-6. Find a matrix-valued function M(in)(k; x, t) which satisfies

Analyticity: M(in)(k; x, t) is analytical in CΣ(2) with symmetry: M(in)(k) = ϱM(in)*(− k*)ϱ.

Jump condition: M(in)(k; x, t) has the jump condition

M+(in)(k)=M(in)(k)V(in)(k),kΣ(2).
(3.60)
where the jump matrix V(in)(k) = V(2)(k) is given by equation (3.27). See figure 4.

Asymptotics: M(in)(k) → I, k → ∞ .

RHP-6 is a solvable model for the SS equation. Here, we mainly adopt the final results for solving the model RHP (see [31] for more details), whose solution has the asymptotics:
M(in)(k)=I148tk0(k+k0)M1(in)+148tk0(kk0)ϱ(M1(in))ϱ+O(logtt),
(3.61)
with ∥M(in) ≲ 1, where
M1(in)=(0iϖ2β12iϖ2β210),
(3.62)
with
ϖ=(192τ)iν2eX(k0)8iτ,β12=β21=νΓ(iν)eπ(2νi)42πσ2γT(k0).
(3.63)
According to RHP-5 and RHP-6, one has the solvable local model RHP with M(LC)(k) = M(out)(k)M(in)(k) inside U±k0 which is a bounded function in U±k0 and has the same jump condition as MRHP(2)(k).

3.2.3. A small norm RH problem

In this section, we mainly consider the small norm RHP corresponding to the error matrix function E(k) given by equation (3.48). Firstly, according to the definition of MRHP(2)(k) and M(LC)(k), we can obtain that E(k) satisfies the following RHP:
RHP-7. Find a matrix-valued function E(k) solving

Analyticity: E(k) is continuous in CΣ(E), where Σ(E)=U±k0(Σ(E)U±k0).

Jump condition: E(k) has the following jump condition (see figure 5)

E+(k)=E(k)V(E)(k),kΣ(E),
(3.64)
where matrix V(E)(k) is defined by
V(E)(k)={M(out)(k)V(2)(k)M(out)(k)1,kΣ(2)U±k0,M(out)(k)M(in)(k)M(out)(k)1,kU±k0.
(3.65)

Asymptotic behaviors: E(k) → I, k → ∞ .

The jump matrix V(E)(k) has the following uniform estimate

|V(E)(k)I|={O(e6k0ρ2t),kΣj±U±k0,j=1,2,O(e8k02ρt),kΣj±U±k0,j=3,4,O(e14k03tan3(π/12)t),k[±ik0,±ik0tan(π12)],O(t1/2),kU±k0,0,k[ik0tan(π12),ik0tan(π12).
(3.66)

The proof can be seen in [32].

Based on the Beals–Coifman theorem, we can construct the solution of RHP-7 in the form
E(k)=I+12πiΣ(E)κE(ξ)[V(E)(ξ)I]ξkdξ,
(3.67)
where κEL2(E)) satisfies (ICωE)κE=I,
ωE=(ωE)++(ωE)=V(E)I,(ωE)=0,(ωE)+=V(E)I,CωEg=C(g(ωE)+)+C+(g(ωE))=C(g(V(E)I))
(3.68)
with C denoting the Cauchy projection operator
Cg(k)=limζkΣ(E)12πiΣ(E)g(ξ)ξζdξ,
(3.69)
and CL2 is a finite value.
Since
CωEL2(Σ(E))≲∥CL2(Σ(E))V(E)IL(ΣE)O(t1/2),
(3.70)
the matrix function κE exists and is unique, and the solution E(k) of RHP-7 exists and is unique.

V(E) and κE admit the following important estimates

κEIL2(Σ(E))=O(t1/2),V(E)ILp=O(t1/2),p[1,+),k0.
(3.71)

The matrix function E(k) has the following asymptotics

E(k;x,t)=I+E1(x,t)k+O(k2),k,
(3.72)
where
E1(x,t)=i2πΣ(E)κE(ξ)(VEI)dξ.
(3.73)
Moreover, E1(x,t) is given by
E1(x,t)=148tk0M(out)(k0)M1(in)(k0)M(out)1(k0)+148tk0M(out)(k0)M1B0(k0)M(out)1(k0)+O(t1logt).
(3.74)

The proof is similar as [32].

3.3. Analysis on a pure ¯-problem

Here we consider the pure ¯-problem which is obtained by removing the pure RHP part with ¯R(2)=0. Let
M(3)(k)=M(2)(k)MRHP(2)(k)1.
(3.75)
Then we know that M(3) is continuous and has no jumps in the complex plane, and solves a pure ¯-problem. Pure ¯-problem. Find a matrix-valued function M(3)(k, x, t) solving

Continuity: M(3)(k) is continuous in CΣ(2).

Jump condition: ¯M(3)(k)=M(3)(k)W(3)(k), kC, where W(3)=MRHP(2)(k)¯R(2)(k)MRHP(2)(k)1.

Asymptotic behaviors: M(3)(k) → I, k → ∞ .

The solution of the above pure ¯-problem can be given by the following integral equation
(IF)M(3)(k)=I,
(3.76)
where F is the Cauchy operator defined as
F[f](k)=1πCf(ξ)W(3)(ξ)ξkdA(ξ)
(3.77)
with dA(ξ) being the Lebesgue measure.

For large time t, there exists the estimate for F:

FLL(k0t)1/4,
(3.78)
which implies that the operator (IF)1 is invertible and the solution of the pure ¯-problem exists and is unique.

M(3)(k) has the following asymptotic expansion
M(3)=I+M1(3)k+O(1k2),M1(3)=1πCM(3)(ξ)W(3)(ξ)dA(ξ),k.
(3.79)
According to proposition 10 and the asympotics of M1(3), one has

For large time t, there exists the estimate for M1(3)

|M1(3)|(k0t)3/4.
(3.80)

4. Long-time asymptotics in the region x<0,|x/t|=O(1) and x>0,|x/t|=O(1)

Based on the above discussions, our main result is summarized as follows:

Let σd={(kj,Aj,Bj),kjK}j=12N denote the scattering data generated by initial data q0(x)S(R) with the second-order discrete spectra. For fixed v2v1R, define I=[v1/4,v2/4] and a space-time cone D(v1,v2) for variables x and t. Let qsol(x,t,σd(I)) be the N(I) solution corresponding to the modified scattering data σd(I)={(kj,cj(I)),kjK(I)}. When x<0, as t+ with (x,t)D(v1,v2), we have the long-time asymptotics of the SS equation

q(x,t)=qsol(x,t|σd(I))+pt1/2+O(t3/4),
(4.1)
where
p=148k0(M(out)(k0)M1(in)(k0)M(out)1(k0)M(out)(k0)ϱ(M1(in)(k0))ϱM(out)1(k0))12.
(4.2)
Similarly, when x>0, as t+ with (x,t)D(v1,v2), we have
q(x,t)=qsol(x,t|σd(I))+O(t1).
(4.3)

Based on a series of transformations (3.12), (3.25), (3.34) and (3.48), we find

M(k)=M(3)(k)E(k)M(out)(k)R(2)1(k)T1(k).
In particular, by considering k along the imaginary axis (i.e. in Ω5,Ω6), we have
M=(I+M1(3)k+...)(I+E1k+...)(I+M1(out)k+...)×(I+T1k+...)=I+M1k+...,
which generates
M1=M1(out)+E1+M1(3)+T1.
(4.4)
According to the reconstruction formula (2.24) and proposition 11, the following estimate holds
q(x,t)=2i(M1(out))12+2i(E1)12+O(t3/4).
(4.5)
Notice that
2i(M1(out))12=qsol(x,t|σd(out)),
(4.6)
which, together with proposition 9, yields
(E1)12=pt1/2+O(t1logt),
(4.7)
where p is given by equation (4.2). Substituting equations (4.6) and (4.7) into (4.5) yields
q(x,t)=qsol(x,t|σd(out))+pt1/2+O(t3/4).
(4.8)
Based on equation (3.59), we find the final asymptotic expression (4.1) with (x,t)D(v1,v2).

Though the large-time asymptotics of the potential given by equation (4.1) has the same form as in [32], they have different meanings. In (4.1), qsol(x,t|σd(I)) denotes the soliton solutions generated by the double poles of the scattering data of the spectral problem, while it denotes the simple poles case in [32].

Theorem 1 did not consider the Painlevé asymptotics in the Painlevé region, in which the main term of the potential has no soliton solution. Thus, one need not consider the order of the discrete spectra such that the formula of the corresponding asymptotic behavior is the same as the one in [32].

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