[02083] Integrable Aspects of Nonlinear Wave Equations, Solutions and Asymptotics
Session Date & Time :
02083 (1/3) : 4C (Aug.24, 13:20-15:00)
02083 (2/3) : 4D (Aug.24, 15:30-17:10)
02083 (3/3) : 4E (Aug.24, 17:40-19:20)
Type : Proposal of Minisymposium
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.
Liming Ling (South China University of Technology)
Bo Yang (Ningbo University)
Talks in Minisymposium :
[04134] Pattern Transformation in Higher-Order Lumps of the Kadomtsev-Petviashvili I Equation
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.
[04206] Drinfeld-Sokolov hierarchies and diagram automorphisms of affine Kac-Moody algebras
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.
[04390] Long-time asymptotics for the defocusing NLS equation with step-like boundary conditions
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.
[04900] Integrable Deep Learning--PINN based on Miura transformations and discovery of new localized wave solutions
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.
[04907] On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions
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.
