Abstract : The study of physical phenomena by means of mathematical models often leads to integrable systems, which admit rich solutions including the solitons. The study on interactions of solitary waves is an important part of the modern theory of nonlinear waves. Various methods have been developed to build their solutions. In addition, stability and long-time asymptotics of solutions to integrable systems are interesting topics and attracts much attentions in the past years.
The proposed minisymposium aims at bringing together the researchers in the fields and at offering an overview of some of the current research activities in this area.
[05581] Recent results on the Fractional Nonlinear Schroedinger Equation
Format : Talk at Waseda University
Author(s) :
Alejandro Aceves (Southern Methodist University)
Abstract : The Fractional Nonlinear Schroedinger Equation (fNLSE) has been a topic of recent interest as it may have applications to nononlinear photonics. In this work we will present motivation for the models considered and recent results on discrete and continuous fNLSE.
In particular we will discuss the existence of discrete localized modes and their properties. Rigorous and numerical results on existence for the continuous fNLSE will also be presented.
[05588] Duality of positive and negative integrable hierarchiesvia relativistically invariant fields
Format : Online Talk on Zoom
Author(s) :
Senyue Lou (Ningbo University)
Xing-Biao Hu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Qingping Liu (China University of Mining and Tenchnology (Beijing))
Abstract : This talk concerns the relativistic invariance in integrable systems. Using the invariant sine-Gordon, Tzitzeica, Toda fields and second heavenly equations as dual relations, some well-known continuous and discrete integrable positive hierarchies are converted to the negative hierarchies. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the mastersymmetry method and the relativistic invariance of the duality relations. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach.
[04900] Integrable Deep Learning--PINN based on Miura transformations and discovery of new localized wave solutions
Format : Talk at Waseda University
Author(s) :
Yong Chen (East China Normal University)
Abstract : We put forth two physics-informed neural network (PINN) schemes based on Miura transformations. The novelty of this research is the incorporation of Miura transformation constraints into neural networks to solve nonlinear PDEs, which is an implementation method of unsupervised learning. The most noteworthy advantage of our method is that we can simply exploit the initial-boundary data of a solution of a certain nonlinear equation to obtain the data-driven solution of another evolution equation with the aid of Miura transformations and PINNs. In the process, the Miura transformation plays an indispensable role of a bridge between solutions of two separate equations. It is tailored to the inverse process of the Miura transformation and can overcome the difficulties in solving solutions based on the implicit expression. Moreover, two schemes are applied to perform abundant computational experiments to effectively reproduce dynamic behaviors of solutions for the well-known KdV equation and mKdV equation. Significantly, new data-driven solutions are successfully simulated and one of the most important results is the discovery of a new localized wave solution: kink-bell type solution of the defocusing mKdV equation and it has not been previously observed and reported to our knowledge. It provides a possibility for new types of numerical solutions by fully leveraging the many-to-one relationship between solutions before and after Miura transformations. Performance comparisons in different cases as well as advantages and disadvantages analysis of two schemes are also discussed. Based on the performance of two schemes and no free lunch theorem, they both have their own merits and thus more appropriate one should be chosen according to specific cases.
[04206] Drinfeld-Sokolov hierarchies and diagram automorphisms of affine Kac-Moody algebras
Format : Talk at Waseda University
Author(s) :
Chaozhong Wu (Sun Yat-Sen University)
Abstract : For a diagram automorphism of an affine Kac-Moody algebra such that the folded diagram is still an affine Dynkin diagram, we show that the associated Drinfeld-Sokolov hierarchy also admits an induced automorphism. We also show how to obtain the Drinfeld-Sokolov hierarchy associated to the affine Kac-Moody algebra that corresponds to the folded Dynkin diagram from the invariant sub-hierarchy of the original Drinfeld-Sokolov hierarchy. This is based on a joint work with Si-Qi Liu, Youjin Zhang and Xu Zhou.
[04907] On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions
Format : Online Talk on Zoom
Author(s) :
Engui Fan (Fudan University)
Abstract : We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation in the solitonic regions. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale (y,t). Further using the ∂ generalization of the Deift-Zhou steepest descent method, we derive different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ = y/t. We divide the half-plane {(y,t) : −∞ 0} into four asymptotic regions: The phase function θ(z) has no stationary phase point on the jump contour in the space-time solitonic regions ξ ∈ (−∞, −1/4) ∪ (2, +∞), corresponding asymptotic approximations can be characterized with an N(Λ)-solitons with diverse residual error order O(t−1+2ρ); The phase function θ(z) has four phase points and eight phase points on the jump contour in the space-time solitonic regions ξ ∈ (0, 2) and ξ ∈ (−1/4, 0), respectively. The corresponding asymptotic approximations can be characterized with an N(Λ)-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order O(t−3/4). Our results also confirm the soliton resolution conjecture and asymptotically stability of the N-soliton solutions for the mCH equation.
[05585] Local and global analyticity for a generalized Camassa-Holm system
Format : Talk at Waseda University
Author(s) :
Hideshi Yamane (Kwansei Gakuin University)
Abstract : We solve the analytic Cauchy problem for the generalized two-component Camassa-Holm system introduced by R. M. Chen and Y. Liu.
We show the existence of a unique local/global-in-time analytic solution under certain conditions.
This is the first result about global analyticity for a Camassa-Holm-like system.
The proof is based the technique by Barostichi, Himonas and Petronilho.
[04390] Long-time asymptotics for the defocusing NLS equation with step-like boundary conditions
Format : Talk at Waseda University
Author(s) :
Deng-Shan Wang (Beijing Normal University)
Abstract : The long-time asymptotics for the defocusing NLS equation with step-like boundary conditions is investigated by the Riemann-Hilbert formulation. Whitham modulation theory shows that there are six cases for this initial discontinuity problem according to the orders of the Riemann invariants. We formulate the leading-order terms and the corresponding error estimates for each region of the six cases by Deift-Zhou nonlinear steepest-descent method. It is demonstrated that the asymptotic solutions match very well with the results from Whitham modulation theory and the direct numerical simulations.
[05490] New revival phenomena for bidirectional dispersive hyperbolic equations
Format : Talk at Waseda University
Author(s) :
Jing Kang (Northwest University)
Abstract : In this talk, the dispersive revival and fractalisation phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study the periodic initial-boundary problem of the linear beam equation with step function initial data, and analyze the manifestation of the revival phenomenon for the corresponding solutions at rational times. Next, we extend the investigation to the periodic initial-boundary problems of more general bidirectional dispersive equations. We prove that, if the initial functions are of bounded variation, the dynamical evolution of such periodic initial-boundary problem depend essentially upon the large wave number asymptotics of the associated dispersion relations. Integral polynomial or asymptotically integral polynomial dispersion relations produce dispersive revival/fractalisation rational/irrational dichotomy effects, whereas those with non-polynomial growth results in fractal profiles at all times. Finally, numerical experiments are used to demonstrate how such effects persist into the nonlinear regime, in the concrete case of the nonlinear beam equation. This is a joint work with Peter J. Olver, Xiaochuan Liu and Changzheng Qu.
[05580] Darboux transformation and soliton solutions for a generalized Sasa-Satsuma equation
Format : Talk at Waseda University
Author(s) :
Zuo-nong Zhu (Shanghai Jiao Tong University)
Hongqian Sun (Shanghai Jiao Tong University)
Abstract : Sasa-Satsuma equation is an important integrable equation. In this talk, we will investigate a generalized Sasa-Satsuma equation introduced by Geng and Wu. Darboux transformation and soliton solutions including hump-type, breather-type solitons for the generalized Sasa-Satsuma equation are constructed.
[05584] Rogue waves and solitons of nonlinear integrable/nearly integrable systems
Format : Talk at Waseda University
Author(s) :
Zhenya Yan (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : In this talk, we mainly discuss some properties of rogue waves and solitons of some nonlinear integrable/nearly integrable systems, which include stability, interactions and excitations of solitons, and rogue wave structures.
[05579] The modified KdV equation on the background of elliptic function solutions
Format : Talk at Waseda University
Author(s) :
Liming Ling (South China University of Technology)
Abstract : In this talk, we first introduce the spectral stability and orbital stability of the elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. On the other hand, by using the Darboux-Backlund transformation, we construct multi-elliptic-localized wave solutions. The asymptotic analysis of these multi-elliptic-localized wave solutions is also involved in this talk.
[04134] Pattern Transformation in Higher-Order Lumps of the Kadomtsev-Petviashvili I Equation
Format : Talk at Waseda University
Author(s) :
Bo Yang (Ningbo University)
Abstract : Pattern formation in higher-order lumps of the Kadomtsev-Petviashvili I equation at large time is analytically studied. For a broad class of these higher-order lumps, we show that two types of solution patterns appear at large time. The first type of patterns comprises fundamental lumps arranged in triangular shapes, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. As time evolves from large negative to large positive, this triangular pattern reverses itself along the x-direction. The second type of patterns comprise fundamental lumps arranged in non-triangular shapes in the outer region, which are described analytically by nonzero-root structures of the Wronskian-Hermit polynomials, together with possible fundamental lumps arranged in triangular shapes in the inner region, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. When time evolves from large negative to large positive, the non-triangular pattern in the outer region switches its x and y directions, while the triangular pattern in the inner region, if it arises, reverses its direction along the x-axis. Our predicted patterns at large time are compared to true solutions, and excellent agreement is observed.
