Special Session 51: Recent progress on the rogue waves and their applications

Numerical simulations of nonlinear evolution of modulational instabilities of Short-Crested Waves

Malek ABID Aix-Marseille Universit\`e/IRPHE France Co-Author(s):    Marc Francius, and Christian Kharif

Abstract: Understanding the formation of extreme waves in crossing seas remains a major challenge in ocean physics. In this work, we investigate the modulational instability of short-crested waves (SCWs), a fundamental mechanism through which energy localizes and potentially leads to rogue waves. Using the coupled nonlinear Schr\{o}dinger (CNLS) framework, we analyze how crossing angle and perturbation direction control the nature of unstable modes. A key result is the first explicit link between CNLS instability predictions and the resonance-based classification (Ia/Ib) derived from fully nonlinear theory. Through systematic numerical simulations, complemented by fully nonlinear Euler computations using a High-Order Spectral Method, we show that a single perturbation can trigger markedly different dynamical responses, including self-phase and cross-phase instabilities. These findings reveal strong sensitivity of instability growth and wave amplification to wave geometry. While the CNLS model provides an efficient and accurate description of early-stage dynamics, fully nonlinear effects become essential to capture long-term evolution and extreme wave amplification.

Onset of instability for the Alber equation, and applications for rogue waves

Agissilaos G Athanassoulis University of Dundee Scotland Co-Author(s):    Irene Kyza

Abstract: The Alber equation has been long used as a statistical model for ocean waves. Its bifurcation between a linearly stable and linearly unstable regime have long been linked with rogue waves. However, the nature of this bifurcation in the fully nonlinear problem (as opposed to the linearized one) has been much less understood. In this work, we use numerical simulation to explore that question. First of all, we focus on the lengthscales of the problem, as too short a computational domain can artificially suppress the instability. We show that, in the stable case, there is nonlinear Landau damping in the fully nonlinear problem. Moreover, we verify that inhomogeneities do start to grow exponentially in the unstable regime -- but, for intermediate intensities, their maxima plateau at values negligible with regard to the background. Thus, the presence of linear instability does not guarantee the appearance of localized maxima. A second bifurcation is observed, when the instability becomes large enough and localized extreme events are formed. Monte Carlo investigation shows that the qualitative behavior of the inhomogeneity depends only on the background spectrum, and not on the initial inhomogeneity.

Periodic waves in the Jaulent--Miodek equation: modulational stability and algebraic solitons

Jinbing Chen Southeast University Peoples Rep of China Co-Author(s):    Zhihang Gu

Abstract: We study the finite-dimensional integrable reduction for the Jaulent--Miodek (JM) equation associated with the energy-dependent Schr\{o}dinger operator, in which a traveling periodic wave expressed by Jacobi elliptic functions is obtained for the JM equation. Solutions of the linearized JM equation are represented as squared eigenfunctions of the Lax system, so that the stability spectrum are connected with the Lax spectrum via a characteristic polynomial. The Lax spectrum are numerically computed by using the Floquet--Bloch decomposition of periodic solutions of Lax system, while the stability spectrum are traced out via the characteristic polynomial. Since the band of stability spectrum lies on the imaginary axis, the traveling periodic wave of JM equation is proved to be modulationally stable. Followed by a gauge transformation, the Darboux transformation is retrieved in a different way, from which a new algebraic soliton is obtained at the endpoint of continuous spectral band, and three new periodic waves are derived with three discrete eigenvalues.

Smooth soliton solutions of the Camasssa-Holm type equations

Nianhua Li Huaqiao University Peoples Rep of China Co-Author(s):

Abstract: We will present some new results on integrable systems of Camassa-Holm (CH) type. First, we will provide a nearly complete list of (1+1)-dimensional CH-type equations and propose several new extensions, including super and (2+1)-dimensional generalizations. Next, we will discuss their integrability properties and exact solutions. Finally, taking the modified CH equation as an example, we shall show how to construct smooth soliton solutions of the CH type equations by an approach of combining the Darboux transformation with the reciprocal transformation.

Soliton interactions in the sharp-line Maxwell-Bloch system

Sitai Li Xiamen University Peoples Rep of China Co-Author(s):

Abstract: We consider $N$-soliton solutions of the sharp-line Maxwell-Bloch system that may contain multiple degenerate soliton groups (DSGs). A DSG is a localized coherent nonlinear traveling-wave structure, consisting of inseparable solitons with identical velocities. Thus, DSGs are generalizations of single solitons (considered as $1$-DSGs), and constitute fundamental building blocks for the solutions of many integrable systems. We provide explicit formulae for an $N$-DSG and its center, and calculate the long-time asymptotics for such solutions. We also briefly discuss high-order soliton solutions, which can be regarded as generalizations of the $N$-soliton solutions.

Dispersive regularization of Talanov focusing in the AKNS hierarchy

Peter D Miller University of Michigan USA Co-Author(s):

Abstract: It was shown in the 1960s by Talanov that there exist solutions of the dispersionless focusing nonlinear Schr\odinger equation with a parabolic amplitude profile that collapse and blow up in finite time. Accounting for the effect of small dispersion, Suleimanov conjectured in 2017 that the corresponding solution of the nonlinear Schr\odinger equation should be regularized with a particular wave profile that was later characterized as a certain infinite-order limit of fundamental rogue waves. We describe a recent proof of a version of Suleimanov's conjecture and explain how it generalizes to the full AKNS hierarchy.

Stability of Elliptic Function Solutions and Dynamics of Elliptic-Localized Waves

Xuan Sun Donghua University Peoples Rep of China Co-Author(s):    Liming Ling

Abstract: This report investigates the stability of elliptic function solutions and elliptic localized wave solutions to integrable nonlinear soliton equations. We construct elliptic localized wave solutions in rational form and establish a correspondence between elliptic rogue waves and modulational instability. Furthermore, we analyze the spectral and orbital stability of the elliptic function solutions.

The Kadomtsev-Petviashvili reduction method and its applications to the (coupled) Sasa-Satsuma equation

Chengfa Wu Shenzhen University Peoples Rep of China Co-Author(s):    Feng Bao-Feng, Shi Changyan, and Zhang Guangxiong.

Abstract: In this talk, we will provide an overview of the Kadomtsev-Petviashvili (KP) reduction method, a powerful technique for deriving exact solutions to nonlinear partial differential equations. In particular, we will focus on the Sasa-Satsuma equation, a higher-order nonlinear Schr\{o}dinger-type equation that plays a crucial role in nonlinear optics and fluid dynamics. Using the KP reduction method, we construct explicit rogue wave solutions. This talk is based on joint work with Feng Bao-Feng, Shi Changyan, and Zhang Guangxiong.

Equivalence between Wronskian- and Grammian-type solutions and asymptotic analysis of N-soliton solutions for the Gerdjikov-Ivanov equation

Tao Xu China University of Petroleum-Beijing Peoples Rep of China Co-Author(s):    Yuxin Yang, Chuanxin Xu, Min Li

Abstract: For the Gerdjikov-Ivanov (GI) equation, we rigorously prove the equivalence between the Wronskian- and Grammian-type solutions derived from the elementary and binary Darboux transformations, respectively. The proof is finished by making complete Wronskian expansions and establishing the relations between the corresponding numerators and denominators of two determinant solutions. Meanwhile, some determinant identities are obtained as a byproduct upon comparing the coefficients of the same terms in the expansions. Furthermore, we conduct asymptotic analysis for N-soliton solutions on the zero and plane-wave backgrounds. Explicit asymptotic expressions are obtained as t goes to infinity, yielding the physical information of interacting solitons, such as amplitudes, velocities, and phase shifts before and after collisions. In particular, we derive the general parametric conditions for synchronous N-soliton collisions at arbitrary space-time points on both backgrounds. This scenario may be useful for understanding complex behavior for a large number of solitons, e.g., the generation of rogue waves via soliton collisions.

Modeling and Mathematical Analysis of a Higher-Order CH-KP Model for Shallow Water Waves

Shouming Zhou Chongqing normal University Peoples Rep of China Co-Author(s):

Abstract: This study employs a systematic asymptotic expansion method to derive a two-dimensional nonlocal model for shallow water wave propagation from the full hydrodynamic governing equations. The local well-posedness of the resulting higher-order two-dimensional extension of the Camassa-Holm type equation is established in a suitably constructed Sobolev-type space, and the blow-up criterion as well as peaked solitary-wave solutions are also considered.