The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles.
The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles.
The Hirota transformation for the soliton solutions of the classical Sine-Gordon equation is suggestive of an extremely simple way for the construction of a nonlinear quantum-dynamical system of spin 1/2 particles that is equivalent to the classical system over the soliton sector. The soliton solution of the classical equation is mapped onto an operator, U, a nonlinear functional of the particle-number operators, that solves the classical equation. Multi-particle states in the Fock space are the eigenstates of U; the eigenvalues are the soliton solutions of the Sine-Gordon equation. The fact that solitons can have positive as well negative velocities is reflected by the characterization of particles in the Fock space by two quantum numbers: a wave number k, and a spin projection, σ (= +-1). Thanks to the simplicity of the construction, incorporation of particle interactions, which induce soliton effects that do not have a classical analog, is simple.
Despite the fact that it is not integrable, the (1 + 2)-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N ≥ 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in (1 + 2) dimensions, is constructed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to “annihilate” and “create” solitons – an effect that does not have an analog in perturbed classical nonlinear evolution equations.
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.
Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schr\"odinger equation containing a complex nonlinearity $W(\ensuremath{\rho},\mathbf{j})+i\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect $W(\ensuremath{\rho},\mathbf{j})$ and determines $\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ univocally. In the above reference $W(\ensuremath{\rho},\mathbf{j})$ was deduced by means of a stochastic quantization approach, in this way obtaining a noncanonical quantum system. In this work we introduce a family of nonlinearities $W(\ensuremath{\rho},\mathbf{j})$ generating a family of nonlinear canonical quantum systems, and derive their Lagrangian and the Hamiltonian functions and the evolution equations of the fields. We derive also the Ehrenfest relations and study the soliton properties. The shape of the soliton, propagating in the system obeying the EIP, can be obtained by solving a first-order ordinary differential equation. We show that, in the case of soliton solutions, by means of a unitary transformation, the EIP potential is equivalent to a real algebraic nonlinear potential proportional to $\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).$
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational GL(n)-invariant R-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.
The investigation of the dynamics of nonlinear systems becomes a hot topic in modern science and engineering. One of the most challenging problem is the quantum evolution of nonlinear excitations which has many applications in condensed matter physics, quantum optics, quantum information and other areas of physics. Many different methods were suggested and implemented in this direction, both analytical and numerical. Among the most fruitful approaches are the so called phase space representations of quantum dynamics. In this method the evolution equation for quantum density operator is converted to the partial differential equation for some classical quasi-distribution function, which in some cases can be reduced to the system of ordinary stochastic differential equations. In this paper we are presenting the results of the investigation of quantum Ablowitz-Ladik model by the phase space representation methods. The classical Ablowitz-Ladik equation is the integrable discretization of Nonlinear Schrodinger Equation. The quantum dynamics of solitons and their interaction will be discussed,
based on the results of the numerical simulation. The possibility of the quantum entanglement and squeezing in quantum Ablowitz-Ladik model will be investigated.