Haie Long | Searcharxiv

arXiv2023

Simple and high-order $N$-solitons of the nonlocal generalized Sasa-Satsuma equation via an improved Riemann-Hilbert method

In this paper, we investigate the nonlocal generalized Sasa-Satsuma (ngSS) equation based on an improved Riemann-Hilbert method (RHM). Different from the traditional RHM, the $t$-part of the Lax pair plays a more important role rather than the $x$-part in analyzing the spectral problems. So we start from the $t$-part of the spectral problems. In the process of dealing with the symmetry reductions, we are surprised to find that the computation is much less than the traditional RHM. We can more easily derive the compact expression of $N$-soliton solution of the ngSS equation under the reflectionless condition. In addition, the general high-order $N$-soliton solution of the ngSS equation is also deduced by means of the perturbed terms and limiting techniques. We not only demonstrate different cases for the dynamics of these solutions in detail in theory, but also exhibit the remarkable features of solitons and breathers graphically by demonstrating their 3D, projection profiles and wave propagations. Our results should be significant to understand the nonlocal nonlinear phenomena and provide a foundation for fostering more innovative research that advances the theory.

math-ph↗

arXiv2022

Stochastic asymptotical regularization for nonlinear ill-posed problems

Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this paper, we extend the regularization theory of SAR for nonlinear inverse problems. By combining techniques from classical regularization theory and stochastic analysis, we prove the regularizing properties of SAR with regard to mean-square convergence. The convergence rate results under the canonical sourcewise condition are also studied. Several numerical examples are used to show the accuracy and advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can quantify the uncertainty in error estimates for ill-posed problems, improve accuracy by selecting the optimal path, escape local minima for nonlinear problems, and identify multiple solutions by clustering samples of obtained approximate solutions.

math.NA↗