Symmetries and conservation laws of the classical Boussinesq equation

A recursion operator for the classical Boussinesq equation is given, which yields infinitely many symmetries and conservation laws. It is also shown that these symmetries define a hierarchy of the classical Boussinesq equation each of which is a hamiltonian system.     

Formulas to Solve the （2＋1）—Dimensional System：The （2＋1）—Extension of Classical Boussinesq System

We deal with the (2 1)-extension of classical Boussinesq system,which can reduce to several meaningful (1 1)-dimensional systems.By studying its lax paire,we put forward invariances of Lax pair at first,then a recursion formula depending on an arbitrary function is derived,At last,some solutions of the (2 1)-extension of classical Boussinesq system are digged out by using the formula.     

Dispersionless noncommutative Kadomtsev–Petviashvili hierarchy: Multi-Hamiltonian structure and large N-like V-algebras

We propose a dispersionless limit of the NC Kadomtsev–Petviashvili hierarchy. Multi-Hamiltonian formulation, l-reductions and relevant W-algebras are investigated. As examples, Gelfand–Dickey (GD) Poisson brackets for dispersionless NCKdV and NC Boussinesq hierarchies are constructed explicitly. The associated GD Poisson algebras are shown large N-like analogs of classical V-algebras.     

General explicit solutions of a classical Boussinesq system

Seeking a travelling wave solution of the classical Boussinesq system and making an ansatz for the solution,we obtain a nonlinear system of algebraic equations.We solve the system using an effective algorithm and then two general explicit solutions are obtained which are of physical interest.     

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlev\'{e test and we also calculate its one- and two-soliton solutions.     

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

Abstract

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlev\'{e test and we also calculate its one- and two-soliton solutions.     

New Approach to Find Exact Solutions to Classical Boussinesq System

In this paper, based on a new system of three Riccati equations, we give a new method to construct more new exact solutions of nonlinear differential equations in mathematical physics. The classical Boussinesq system is chosen to illustrate our method. As a consequence, more families of new exact solutions are obtained, which include solitary wave solutions and periodic solutions.     

New Approach to Find Exact Solutions to Classical Boussinesq System

In this paper, based on a new system of three Riccati equations, we give a new method to construct more new exact solutions of nonlinear differential equations in mathematical physics. The classical Boussinesq system is chosen to illustrate our method. As a consequence, more families of new exact solutions are obtained, which include solitary wave solutions and periodic solutions.     

Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law

We find that classical similarity transformations in the coherent state representation projects onto the similarity transformation operators (STO), these operators constitute a loyal representation of symplectic group. Remarkably, the multiplication rule of the STOs naturally leads to the quantum optical generalized ABCD law, which is the quantum mechanical correspondence of the classical optical ABCD law. Throughout the whole derivation, the technique of integration within an ordered product (IWOP) of operators is employed.     

Symmetry Analysis of Nonlinear Incompressible Non-Hydrostatic Boussinesq Equations

The symmetries of the （2＋1）-dimensional nonlinear incompressible non-hydrostatic Boussinesq （INHB） equations, which describe the atmospheric gravity waves （GWs）, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. Calculation shows the INHB equations are invariant under some Galilean transformations, scaling transformations, and space-time translations. The symmetry reduction equations and similar solutions of the INHB equations are proposed.     

Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K_m−1 of arithmetic genus m−1, from which we establish the associated Baker-Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.     

Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl(2)⊗sl(2)-valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes-Cummings-Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes-Cummings-Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.     

Boundary Value Problems for Boussinesq Type Systems

We characterise the boundary conditions that yield a linearly well posed problem for the so-called KdV–KdV system and for the classical Boussinesq system. Each of them is a system of two evolution PDEs modelling two-way propagation of water waves. We study these problems with the spatial variable in either the half-line or in a finite interval. The results are obtained by extending a spectral transform approach, recently developed for the analysis of scalar evolution PDEs, to the case of systems of PDEs. The knowledge of the boundary conditions that should be imposed in order for the problem to be linearly well posed can be used to obtain an integral representation of the solution. This knowledge is also necessary in order to conduct numerical simulations for the fully nonlinear systems. Mathematics Subject Classifications (2000) 34A30, 34A34, 35F10.     

Runge-Lenz vector, accidental SU(2) symmetry, and unusual multiplets for motion on a cone

We consider a particle moving on a cone and bound to its tip by 1/r or harmonic oscillator potentials. When the deficit angle of the cone divided by 2π is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some of the corresponding multiplets have fractional “spin” and unusual degeneracies.     

Integral equations satisfied by the H- X- and Y-functions and the albedo problem

We come back to a nonlinear integral equation satisfied by the function H, which is distinct from the classical H-equation. Established for the first time by Busbridge (1955), it appeared occasionally in the literature since then. First of all, this equation is generalized over the whole complex plane using the method of residues. Then its counterpart in a finite slab is derived; it consists in two series of integral equations for the X- and Y-functions. These integral equations are finally applied to the solution of the albedo problem in a slab.     

Quantum-classical correspondence for motion on a plane with deficit angle

A particle constrained to move on a cone and bound to its tip by harmonic oscillator and Coulomb-Kepler potentials is considered both in the classical as well as in the quantum formulations. The SU(2) coherent states are formally derived for the former model and used for showing some relations between closed classical orbits and quantum probability densities. Similar relations are shown for the Coulomb-Kepler problem. In both cases a perfect localization of the densities on the classical solutions is obtained even for low values of quantum numbers.     

Scalable preparation of three-atom and four-atom W states via atom-cavity-laser interaction

We propose the optical generation of W states for three atomic and four atomic qubits, with each qubit trapped in a separate cavity and coupled to the cavity laser. A single-photon source and two classical fields are employed in the present scheme. By encoding the quantum information of each qubit on the degenerate ground states of the atom, we obtain the atomic entanglement that is relatively stable against spontaneous emission. It is demonstrated that the three- and four-atomic W states can be produced deterministically via a proper manipulation of the atom-cavity interaction sequence and time. Generalization of the present scheme to prepare multi-atomic W states is also discussed.