Abstract
We study the long-time asymptotic behavior of oscillatory Riemann–Hilbert problems (RHPs) arising in the mKdV hierarchy (reducing from the AKNS hierarchy). Our analysis is based on the idea of \({\bar{\partial }}\)-steepest descent. We consider RHPs generated from the inverse scattering transform of the AKNS hierarchy with weighted Sobolev initial data. The asymptotic formula for three regions of the spatial- and temporal-dependent variables is presented in details.
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Notes
Due to Zhou’s theorem (Zhou 1998), R(z) belongs to \(H^{1,n-1}(dz)\), then the time evolving reflection coefficient \(R(z)e^{\pm 2it\theta }\) will stay in \(H^{1,1}(\mathrm{d}z)\) since the degree of \(\theta \) is n.
This guarantees the time evolving of the initial data will stay in \(H^{1,1}\). Roughly speaking, from Zhou’s work, we know \(q(x,0)\in H^{n-1,1}\subset H^{1,1}\) is mapped to \(R(z)\in H^{1,n-1}\). Time evolution of the reflection coefficient gives \(R(z)e^{itz^n}\), which belongs to \(H^{1,1}\) due to the fact that \(R(z)\in H^{1,n-1}\), and then the inverse scattering leads to \(q(x,t)\in H^{1,n-1}\).
A good summary of this method can be found in Deift et al. (1993).
Here, \(R(z)\in C^1({\mathbb {R}})\) means R(z) is a function defined on the real line with continuous first-order derivative. While since O(z) is a matrix-valued function defined on the complex plan, so \(O(z)\in C^1({\mathbb {R}}^2)\) means all the entries have continuous first-order derivatives with respect to z and \({\bar{z}}\).
In the middle steps, c means a generic positive constant.
The existence and uniqueness will be discussed later.
Here the \(L^\infty (\Sigma )\) norm \(\Vert f(z)\Vert _{\infty }\) means \(\sup _{z\in \Sigma }|f(z)|\), where \(|f(z)|=\max _{i,j=1,2,z\in \Sigma }|f_{i,j}(z)|\).
\((w^-,w^+)\) will be called the factorization data for the jump matrix.
Here, \(w=w^++w^-\).
The fact is nontrivial, but the derivation is in Deift and Zhou (1993), see equation (4.47)
Surprisingly, the dependence on \(p_j,q_j\) will disappear.
Such a choice of \(\alpha \) guarantees that the new contours will stay within the regions where the corresponding exponential term will decay (considering Fig. 7).
As for the existence of the RHP \(m^{[3]}\), which is not completely trivial due to the fact that solutions to the Painlevé II equations have poles, we refer the readers to the book (Fokas et al. 2006) for the details
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Wang, F., Ma, WX. A \({\bar{\partial }}\)-Steepest Descent Method for Oscillatory Riemann–Hilbert Problems. J Nonlinear Sci 32, 10 (2022). https://doi.org/10.1007/s00332-021-09765-7
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DOI: https://doi.org/10.1007/s00332-021-09765-7