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Laurent Delisle

Publications and source records attributed to Laurent Delisle.

16 recordsLinked to original sources
arXiv2026

A Classification of Hirota-Integrable Supersymmetric Bilinear KdV-Type Equations

We present a classification of supersymmetric bilinear KdV-type equations admitting unconstrained three-super-soliton solutions. Extending Hirota's classical three-soliton criterion to the supersymmetric setting, we derive the complete bosonic and fermionic compatibility conditions governing the existence of three-super-soliton solutions. We prove that every supersymmetric bilinear KdV-type equation possesses unconstrained one- and two-super-soliton solutions, whereas three-super-soliton solutions exist only when eight integrability conditions are satisfied. These conditions provide a supersymmetric analogue of Hirota's classical integrability criterion and naturally recover the fermionic relations previously introduced by Carstea, revealing their structural origin. As a consequence, we classify the supersymmetric extensions of Hirota bilinear KdV-type equations and show that only a subset of Hietarinta's classical classification remains valid in the unrestricted supersymmetric framework.

nlin.SI↗
arXiv2026

Vector Representation of Exact Soliton Dynamics in Multi-component Nonlinear Schrödinger Systems

Multicomponent nonlinear Schrödinger equations constitute fundamental models for coherent matter waves in multicomponent Bose--Einstein condensates, spinor quantum fluids, and vector nonlinear optical systems. We develop a vector formulation of the Hirota bilinear formalism for the completely integrable Manakov system that treats the coupled nonlinear Schrödinger equations directly at the vector level rather than through the conventional component-wise decomposition. This framework naturally retains the intrinsic multicomponent representation of the model while providing compact analytical expressions for exact vector soliton solutions. Within this approach, we systematically construct bright, dark, and mixed one-, two-, and three-soliton solutions and show how the underlying vector structure provides a unified description of their nonlinear interactions. In particular, the proposed formalism makes the coupling between the different components explicit while preserving the geometric organization of the vector system throughout the bilinearization procedure. Beyond its analytical simplicity, the framework offers a natural perspective for the study of coherent multicomponent nonlinear excitations and provides a foundation for extending vector Hirota methods to other classes of exact solutions, including rogue waves, periodic waves, and rational solutions.

quant-ph↗
arXiv2026

A Vector Bilinear Framework for Soliton Dynamics in Coupled Modified KdV Systems

We investigate the integrable structure and soliton dynamics of a coupled modified Korteweg-de Vries (cmKdV) system with a real symmetric coupling matrix. We introduce a vector reformulation of Hirota's bilinear formalism in which both the bilinear equations and their solutions are expressed directly at the vector level, rather than through a component-wise construction. This formulation preserves the intrinsic structure of the coupled system and provides a compact framework for multi-component nonlinear wave dynamics. Within this approach, we construct explicit one-, two-, and three-soliton solutions in closed vector form and recover the three-soliton condition directly at the vector level, confirming consistency with integrability. The method enables a unified treatment of focusing, defocusing, and mixed-sign regimes. In particular, for indefinite coupling, it reveals the existence of nontrivial vector ground states, leading to soliton solutions on non-zero backgrounds. These results highlight the structural advantages of the vector bilinear approach and open perspectives for the study of more general nonlinear excitations in multi-component integrable systems.

nlin.SI↗
arXiv2024

Optically tuned soliton dynamics in Bose-Einstein condensates within dark traps

This study investigates the formation and dynamics of solitons in Bose-Einstein condensates (BECs) within dark traps generated by two crossed Laguerre-Gaussian (LG) beams with varying azimuthal indices $\ell$. As the index $\ell$ increases, the potential transitions from a harmonic trap when $\ell = 1$ to a square-well potential for larger values of $\ell$. This transition allows us to study a range of soliton dynamics under different confinement conditions while maintaining the same BEC volume. Through the derivation of the Gross-Pitaevskii equation (GPE) and under these specific conditions in both one-dimensional (1D) and two-dimensional (2D) configurations, we explore the dynamics of solitons across multiple scenarios. The study examines two primary methods for solitons generation: the temporal modulation of the scattering length and the implementation of an initial potential barrier that is subsequently removed. The results indicate that the trap shape plays a critical role in the generation and interaction dynamics of solitons. In harmonic traps, solitons exhibit a behavior different from those observed in anharmonic traps, where the dynamics is significantly influenced by the azimuthal index of the trap. The ability to control soliton dynamics in BECs holds significant promise for applications in quantum technologies, precision sensing, and the exploration of fundamental quantum phenomena.

cond-mat.quant-gas↗
arXiv2023

A novel Hirota bilinear approach to $N=2$ supersymmetric equations

This article presents a novel application of the Hirota bilinear formalism to the $N=2$ supersymmetric KdV and Burgers equations. This new approach avoids splitting N=2 equations into two $N=1$ equations. We use the super Bell polynomials to obtain bilinear representations and present multi-soliton solutions.

nlin.SI↗
arXiv2015

A N=2 extension of the Hirota bilinear formalism and the supersymmetric KdV equation

We present a bilinear Hirota representation of the N=2 supersymmetric extension of the Korteweg-de Vries equation. This representation is deduced using binary Bell polynomials, hierarchies and fermionic limits. We, also, propose a new approach for the generalisation of the Hirota bilinear formalism in the N=2 supersymmetric context.

math-ph↗
arXiv2015

Superforms and the $\mathbb{C}P^{N-1}$ supersymmetric sigma model

We present a characterisation of Maurer-Cartan 1-superforms associated to the two-dimensional supersymmetric $\mathbb{C}P^{N-1}$ sigma model. We, then, solve the associated linear spectral problem and use its solutions to describe an integrable system for a $su(N)$-valued map.

math-ph↗
arXiv2015

General solutions of the supersymmetric $\mathbb{C}P^2$ sigma model and its generalisation to $\mathbb{C}P^{N-1}$

A new approach for the construction of finite action solutions of the supersymmetric $\mathbb{C}P^{N-1}$ sigma model is presented. We show that this approach produces more non-holomorphic solutions than those obtained in previous approaches. We study the $\mathbb{C}P^2$ model in detail and present its solutions in an explicit form. We also show how to generalise this construction to $N>3$.

math-ph↗
arXiv2014

Geometry of surfaces associated to grassmannian sigma models

We investigate the geometric characteristics of constant gaussian curvature surfaces obtained from solutions of the $G(m,n)$ sigma model. Most of these solutions are related to the Veronese sequence. We show that we can distinguish surfaces with the same gaussian curvature using additional quantities like the topological charge and the mean curvature. The cases of $G(1,n)=\mathbb{C}P^{n-1}$ and $G(2,n)$ are used to illustrate these characteristics.

math-ph↗
arXiv2014

Constant curvature surfaces of the supersymmetric $\mathbb{C}P^{N-1}$ sigma model

Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric $\mathbb{C}P^{N-1}$ sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic solutions of the model. We extend our analysis to general supersymmetric Grassmannian models.

math-ph↗
arXiv2013

Constant curvature solutions of Grassmannian sigma models: (2) Non-holomorphic solutions

We generalize here our general procedure for constructing constant curvature maps of 2-spheres into Grassmannian manifolds G(m,n) this time concentrating our attention on maps which are non-holomorphic. We present some expressions describing these solutions in the general case and discuss how to use these results to construct solutions of constant curvature. We also discuss possible values of this constant curvature.

math-ph↗
arXiv2013

Classical and SUSY solutions of the Boiti-Leon-Manna-Pempinelli equation

In this paper, we propose the study of the Boiti-Leon-Manna-Pempinelli equation from two point of views: the classical and supersymmetric cases. In the classical case, we construct new solutions of this equation from Wronskian formalism and Hirota method. We, then, introduce a N = 1 supersymmetric extension of the Boiti-Leon-Manna-Pempinelli equation. We thus produce a bilinear form and give multisolitons and superpartner solutions. As an application, we produce a pair of Bäcklund transformations.

math-ph↗
arXiv2012

Soliton and similarity solutions of N=2,4 supersymmetric equations

We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg-de Vries and modified KdV equations. We give new representations of the $τ$-functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts.

math-ph↗
arXiv2012

Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

We present a general formula for the Gaussian curvature of curved holomorphic 2-spheres in Grassmannian manifolds G(m, n). We then show how to construct such solutions with constant curvature. We also make some relevant conjectures for the admissible constant curvatures in G(m, n) and give some explicit expressions, in particular, for G(2, 4) and G(2, 5).

math-ph↗
arXiv2011

Links between symmetry reduction and Hirota methods of the N=2 susy KdV equation

We consider the resolution of the N=2 supersymmetric KdV equation with a=-2 (SKdV_{a=-2}) from two approaches, the group invariant method (or symmetry reduction) and the Hirota formalism. A bilinear form of the SKdV_{a=-2} equation is constructed. Links between the two methods are established and new solutions are obtained from both approaches.

math-ph↗