
The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed
Minghe Zhang, Zhenya Yan
Communications in Theoretical Physics ›› 2024, Vol. 76 ›› Issue (6) : 65002.
The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed
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
Sasa–Satsuma equation /
inverse scattering /
• | Analyticity : M(k; x, t) is a meromorphic function in |
• | Jump relation: M(k; x, t) has continuous boundary values M±(k; x, t) on |
• | Asymptotics: |
The matrix function
• | |
• | For |
• |
|
• | As |
• | where |
• | Along the ray |
The proof of the above properties is similar to the proof of proposition 3.1 in [12].
• | Analyticity: M(1)(k) is analytic in |
• | Jump condition: |
• | Asymptotics: |
Figure 3. Pole distribution. The red, green and yellow points generate the breather solutions. Moreover, the red points lie in the region |
Let
• | Continuity: M(2)(k) is continuous in |
• | Jump condition: |
• | Asymptotics: M(2)(k) → I, k → ∞ ; |
• | Analyticity: |
• | Jump condition: |
• | Asymptotics: |
The jump matrix
• | Analyticity: M(k; x, t∣σd) is analytical in |
• | Jump condition: |
• | Asymptotics: |
Given scattering data
The uniqueness of the solution can be guaranteed by Liouville's theorem. For the reflectionless case
• | Analyticity: |
• | Asymptotics: |
RHP-5 has the uniqueness solution, and its potential is equivalent to one of the reflectionless cases of RHP-4, that is
According to the reconstruction formula (
Suppose that qsol is the soliton solution of the SS equation corresponding to its scattering data
• | Analyticity: M(in)(k; x, t) is analytical in |
• | Jump condition: M(in)(k; x, t) has the jump condition where the jump matrix V(in)(k) = V(2)(k) is given by equation ( |
• | Asymptotics: M(in)(k) → I, k → ∞ . |
• | Analyticity: E(k) is continuous in |
• | Jump condition: E(k) has the following jump condition (see figure 5) where matrix V(E)(k) is defined by |
• | Asymptotic behaviors: E(k) → I, k → ∞ . |
The jump matrix
The proof can be seen in [32].
The matrix function E(k) has the following asymptotics
where Moreover,The proof is similar as [32].
• | Continuity: M(3)(k) is continuous in |
• | Jump condition: |
• | Asymptotic behaviors: M(3)(k) → I, k → ∞ . |
For large time t, there exists the estimate for F:
which implies that the operatorFor large time t, there exists the estimate for
Let
Based on a series of transformations (
Though the large-time asymptotics of the potential given by equation (
Theorem
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