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arXiv:2606.27569

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

Maxime Foucher, Laurent Delisle, Amine Jaouadi

quant-phSubmitted Jun 25, 2026• Announced Jun 29, 2026
arXivHTMLPDF

Abstract

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.

Comments
32 pages, 9 figures

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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.
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.
We analyze the dynamics of two-component vector solitons, namely, dark-bright solitons, via the variational approximation in Bose-Einstein condensates. The system is described by a vector nonlinear Schr\"odinger equation appropriate to multicomponent Bose-Einstein condensates. The variational approximation is based on a hyperbolic tangent (hyperbolic secant) for the dark (bright) component, which leads to a system of coupled ordinary differential equations for the evolution of the ansatz parameters. We obtain the oscillation dynamics of two-component dark-bright solitons. Analytical calculations are performed for same-width components in the vector soliton, and numerical calculations extend the results to arbitrary widths. We calculate the binding energy of the system and find it to be proportional to the intercomponent coupling interaction and numerically demonstrate the breakup or unbinding of a dark-bright soliton. Our calculations explore observable eigenmodes, namely, the internal oscillation eigenmode and the Goldstone eigenmode. We find analytically that the number of atoms in the bright component is required to be less than the number of atoms displaced by the dark soliton in the other component in order to find the internal oscillation eigenmode of the vector soliton and support the existence of the dark-bright soliton. This outcome is confirmed by numerical results. Numerically, we find that the oscillation frequency is amplitude independent. For dark-bright solitons in 87Rb^{87}\mathrm{Rb}87Rb we find that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments.
We obtain soliton and plane wave solutions for the coupled nonlinear Schrödinger equations, which describe the dynamics of the three-component Bose–Einstein condensates by using the Hirota method. Meanwhile we find that the system which has attractive atomic interaction will only possess a shape changing (inelastic) collision property due to intensity redistribution in the absence of the spin-exchange interaction. As a discussed example, we investigate the one-soliton, two-soliton solutions and collisional effects between bright two-soliotn solution, which lead to the intensity redistribution.
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important. In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants g, respectively, and indeed have been observed in a quasione-dimensional (q1D) optically constructed waveguide (Strecker et al.; 2002) (Khaykovich et al.; 2002). Such matter-wave solitons are expected in atom optics for applications in atom laser, atom interferometry, and coherent atom transport (Meystre; 2002). In this chapter, we extend the analysis of the matter-wave solitons to a multicomponent case by considering the so-called spinor condensate (Stenger et al.; 1998) whose spin degrees of freedom are liberated under optical traps. Based on theoretical and experimental results, we introduce a new integrable model which describes the dynamical properties of the matter-wave soliton of spinor condensates (Ieda et al.; 2004a). We employ the inverse scattering method to solve this model exactly. As a result, we predict the occurrence of undiscovered physical phenomena such as macroscopic spin precession and spin switching. The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we
We study the dynamics of non-autonomous bright-dark matter-wave solitons in two- and three- component Bose-Einstein condensates. Our setting includes a time-dependent parabolic potential and scattering length, as well as Rabi coupling of the separate hyperfine states. By means of a similarity transformation, we transform the non-autonomous coupled Gross-Pitaevskii equations into the completely integrable Manakov model with defocusing nonlinearity, and construct the explicit form of the non-autonomous soliton solutions. The propagation characteristics for the one-soliton state, and collision scenarios for multiple soliton states are discussed in detail for two types of time-dependent nonlinearities: a kink-like one and a periodically modulated one, with appropriate time-dependence of the trapping potential. We find that in the two-component condensates the nature of soliton propagation is determined predominantly by the nature of the nonlinearity, as well as the temporal modulation of the harmonic potential; switching in this setting is essentially due to Rabi coupling. We also perform direct numerical simulation of the non-autonomous two-component coupled Gross-Pitaevskii equations to corroborate our analytical predictions. More interestingly, in the case of the three-component condensates, we find that the solitons can lead to collision-induced energy switching (energy-sharing collision), that can be profitably used to control Rabi switching or vice-versa. An interesting possibility of reversal of the nature of the constituent soliton, i.e., bright (dark) into dark (bright) due to Rabi coupling is demonstrated in the three-component setting.