Special Session 34: Recent advances on integrable systems and related topics

Rational solutions for algebraic solitons in the massive Thirring model

Cheng He School of Mathematics and Statistics, Ningbo University Peoples Rep of China Co-Author(s):    Zhen Zhao, Baofeng Feng, Dmitry E. Pelinovsky

Abstract: We present an algebraic soliton of the massive Thirring model (MTM) as the simplest rational solution with spatial decay $\mathcal{O}(x^{-1})$; the corresponding potential relates to an embedded eigenvalue in the Kaup-Newell spectral problem. The hierarchy of rational solutions is studied: the $N$th member describes a nonlinear superposition of $N$ algebraic solitons of equal mass and corresponds to an embedded eigenvalue of algebraic multiplicity $N$. Using double-Wronskian determinants, we rigorously prove that each solution is defined by a polynomial of degree $N^2$ with $2N$ free parameters, admitting $\frac{N(N-1)}{2}$ poles in the upper half-plane and $\frac{N(N+1)}{2}$ in the lower half-plane. Numerical root analysis shows that the $N$th member describes the slow scattering of $N$ algebraic solitons on the time scale $\mathcal{O}(\sqrt{t})$.

On a generalized nonlocal shallow-water equation

Lili Huang Chongqing Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider a quasilinear nonlocal shallow-water model equation for moderate-amplitude equatorial waves with effects of the weak Coriolis force and equatorial undercurrent. This mathematical modeling is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and rotational Euler equations in the equatorial region. Moreover, we investigate the influences and interactions of the weak Coriolis force caused by the Earth rotation, vorticity and nonlocal higher nonlinearities on the wave-breaking phenomena. In certain cases, by applying the method of characteristics and extremal property of the solution to the Riccati-type differential inequality, we demonstrate that wave breaking phenomena occur only depending on a shape of wave initially in the local spacial point.

Point of constancy of the periodic linear Schr$\mathrm{\ddot{o}}$dinger equation

Jing Kang Northwest University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we investigate the points of constancy in the piecewise constant solution profiles of the periodic linear Schr$\mathrm{\ddot{o}}$dinger equation with step-function initial data at rational times. We characterize all points of constancy of the solution $u$ and the square of the modulus $|u|^2$ of $u$, respectively. We employ number theoretic techniques, including quadratic Gauss sums and half-Gauss sums. These results establish an intriguing connections between the revival phenomena of dispersive evolution equations on a periodic domain and the classical number theory.

Painlev\\`{e} XXXIV asymptotics for the defocusing nonlinear Schr\\{o}dinger equation with a finite-genus algebro-geometric background

Gaozhan Li Tsinghua University Peoples Rep of China Co-Author(s):

Abstract: In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlev\`e XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems.

Oscillatory solutions of Camassa-Holm equation

Ruomeng Li Zhengzhou University Peoples Rep of China Co-Author(s):    Xianguo Geng, Abdul-Majid Wazwaz, Manxue Liu

Abstract: A comprehensive and systematic method is introduced for deriving oscillatory $N$-breather solutions for the Camassa-Holm equation, which are a new class of solutions on the oscillatory backgrounds. The process of this method is divided into four distinct but interrelated stages: First, resorting to the B\acklund transformations in Hirota`s bilinear equations, a novel technique is devised to solve the spectral problems of a negative-order KdV equation involving theta-function potentials. Second, using these B\acklund transformations, an $N$-fold Darboux transformation for the Camassa-Holm equation is rigorously formulated. Third, reciprocal and Darboux transformations are applied to construct oscillatory $N$-breather solutions for the Camassa-Holm equation from the spectral function of a negative-order KdV equation. Finally, the reality, boundedness, and smoothness of these novel solutions are rigorously established by expanding the Wronskians into Hirota summations.

On the nonlinear waves for the coupled complex mKdV system

Xiaochuan Liu Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present some recent progress in the studies of nonlinear waves of the integrable coupled complex mKdV (ccmKdV) system. The general dark soliton solutions in pfaffian form are constructed and the link between the ccmKdV system and the BKP hierarchy is clarified. Furthermore, the multi-breather solutions are derived using the Kadomtsev-Petviashvili reduction method. This talk is based on the joint work with Professor Bao-Feng Feng and Dr. Chenxi Li.

Obliquely interacting solitary waves and wave wakes in free-surface flows

Xudan Luo Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: We investigate the weakly nonlinear isotropic bi-directional Benney-Luke (BL) equation, with a particular focus on soliton dynamics. The associated modulation equations are derived that describe the evolution of soliton amplitude and slope. By analyzing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine-Hugoniot conditions, which help characterize the Mach expansion and Mach reflection phenomena. We also derive analytical formulas for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev-Petviashvili (KP) equation. Furthermore, as a far-field approximation for the forced BL equation -- which models wave and flow interactions with local topography -- the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.

Algebro-geometric solutions for the Ito hierarchy

Jiao Wei Zhengzhou University Peoples Rep of China Co-Author(s):    Minxin Jia, Xianguo Geng

Abstract: This talk is about the Riemann theta function solutions for the Ito hierarchy. The Lax pair of the Ito hierarchy is derived from a 4*4 matrix spectral problem using the zero-curvature equation and Lenard equations. Then, we introduce the corresponding tetragonal curve and its Riemann theta function through the characteristic polynomial of the Lax matrix, and also discuss the construction of three kinds of Abelian differentials. Building on the theory of tetragonal curves, we investigate algebro-geometric properties of Baker-Akhiezer functions and fundamental meromorphic functions. Finally, Riemann theta function solutions for the entire Ito hierarchy are derived via asymptotic analysis.

On the algebraic structure of Kadomtsev-Petviashvili hierarchies with constraints

ZHIWEI WU Sun Yat-sen University Peoples Rep of China Co-Author(s):    Song Li, Kelei Tian

Abstract: In this talk, we will discuss the construction of KP hierarchies with constraints under the scheme of loop group. Matrix representations of corresponding pseudo-differential operators are given. To illustrate this process, we will give an example which is a generation of the constraint KP. Darboux transformation is constructed and formulas connecting Tau functions and formal inverse scattering solutions are given.

Critical asymptotics for the semiclassical Camassa-Holm equation at the point of gradient catastrophe

Yiling Yang Chongqing Unversity Peoples Rep of China Co-Author(s):    Taiyang Xu and Lun Zhang

Abstract: We investigate the Cauchy problem for the Camassa-Holm (CH) equation with a small dispersion parameter $\epsilon>0$. Under a negative analytic initial data, we prove that before a certain time, the solution could be well approximated by the solution to Hopf equation which corresponds to $\epsilon=0$. Near the point of gradient catastrophe to the Hopf equation, we show that the solution of CH equation is approximated by a particular Painlev\`e transcendent in the double scaling limit. This proves the validity of the Dubrovin conjecture concerning the critical asymptotics for a broad class of Hamiltonian perturbative hyperbolic equations. We established our results by performing the steepest descent analysis to an associated Riemann-Hilbert problem.

Controllable rogue waves on the Jacobi-periodic background for the higher-order nonlinear Schrodinger equation

Yunfei Yue Chongqing Technology and Business University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we have systematically studied rogue wave solutions on elliptic function backgrounds for the higher-order nonlinear Schrodinger equation. By utilizing the Darboux transformation and the nonlinearization method of Lax pairs, we have derived first and second-order rogue wave solutions on cn- and dn-periodic backgrounds. Their dynamical behavior have analyzed in detail and illustrated them through graphics.

Statistical mean and variance analysis for the dynamical behaviors of stochastic Boussinesq equations

Haiqiong Zhao Shanghai University of International Business and Economics Peoples Rep of China Co-Author(s):    Hai-qiong Zhao

Abstract: Some stochastic Boussinesq equations under Brown motion are investigated. The integrability of the stochastic Boussinesq equations are established by demonstrating the existence of a Lax pair. By using the Darboux transformation, we construct a variety of exact solutions, including multi-soliton solutions, periodic solutions and rational solutions, under integrated Brownian motion. Most importantly, the dynamic behaviors of the stochastic soliton solutions are analyzed through two key probabilistic characteristics: the statistical mean and variance. Specifically, the long-time asymptotic expressions for these characteristics are rigorously derived, revealing a high degree of consistency with the statistical mean and variance obtained from sampled solutions.