Kaup-Boussinesq Equation Research Articles

Formation of the undular bores in shallow water generalized Kaup–Boussinesq model

The formation of undular bores in Riemann problems of the good generalized Kaup–Boussinesq equation is investigated by Whitham modulation theory. Firstly, the Whitham equations associated with the one-phase and two-phase periodic wave solutions are given by the finite-gap averaging method. Secondly, the basic structures of rarefaction wave and dispersive shock wave are discussed by considering the self-similar solutions of the Whitham equations. Then we propose a complete classification of all possible wave patterns for the initial discontinuity of the good generalized Kaup–Boussinesq equation, in which it has been shown that the theoretical results from Whitham modulation theory are in good agreement with the full numerical simulations. The correspondences between the soliton frontiers of the dispersive shock waves and the exact soliton solution of the good generalized Kaup–Boussinesq equation are considered, and exotic undular bores are found. Finally, the dam break problem and piston problem are explored to show the important physical applications of the theoretical results, from which certain inspiring phenomena of wave breaking are discovered for the first time.

Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions

<p style='text-indent:20px;'>The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.</p>

Soliton gas in bidirectional dispersive hydrodynamics.

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs. We construct kinetic equations for both types of bidirectional soliton gases and solve the respective shock-tube problems for the collision of two "monochromatic" soliton beams consisting of solitons of approximately the same amplitude and velocity. The corresponding weak solutions of the kinetic equations consisting of differing uniform states separated by contact discontinuities for the mean flow are constructed. Concrete examples of bidirectional Eulerian soliton gases for the defocusing nonlinear Schrödinger (NLS) equation and the resonant NLS equation are considered. The kinetic equation of the resonant NLS soliton gas is shown to be equivalent to that of the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq equations. The analytical results for shock-tube Riemann problems for bidirectional soliton gases are shown to be in excellent agreement with direct numerical simulations.

Explicit Solutions of Coupled Time Fractional Kaup-Boussinesq Equation with Weak Dispersion

In the current article, we consider a system of time fractional Kaup-Boussinesq shallow water equations. Using Lie group analysis, we obtain the symmetry group of transformations which reduces the system of fractional partial differential equations (FPDEs) to system of fractional ordinary differential equations (FODEs). Further, we investigate the exact explicit group invariant solution as well as power series solution of the given system of equations. Next the physical significance of the group invariant solution under the influence of fractional order is studied graphically. Lastly, conserved vectors for the FPDEs are obtained using the conservation theorem.

On invariant analysis and conservation laws for degenerate coupled multi-KdV equations for multiplicity $$l = 3$$ l = 3

The degenerate coupled multi-KdV equations for coupled multiplicity l=3 are studied. The equations also known as three fields Kaup-Boussinesq equations are considered for invariant analysis and conservation laws. The classical Lie's symmetry method is used to analyze the symmetries of equations. Based on Killing's form which is invariant of adjoint action, the full classification for Lie algebra is presented. Further, one-dimensional optimal group classification is used to obtain invariant solutions. Besides this, using general theorem proved by Ibragimov we find several nonlocal conservation laws for these equations. The conserved currents obtained in this work can be useful for the better understanding of some physical phenomena modeled by the underlying equations.

Solitons of shallow-water models from energy-dependent spectral problems

The current work investigates the soliton solutions of the Kaup-Boussinesq equation using the inverse scattering transform method. We outline the construction of the Riemann-Hilbert problem for a pair of energy-dependent spectral problems for the system, which we then use to construct the solution of this hydrodynamic system.

Invariant solutions and conservation laws of the generalized Kaup–Boussinesq equation

The generalized Kaup–Boussinesq equation is a model which is used to describe the water wave. In this paper, Lie group analysis method is used to perform detailed analysis on the generalized Kaup–Boussinesq equation. Some invariant solutions are obtained under the transformation groups. The conservation laws of the generalized Kaup–Boussinesq equation are constructed using two methods: multiplier method and Ibragimov theorem.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion.

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

Modified homogeneous balance method: Applications and new solutions

This work is devoted to application of the modified homogeneous balance method to obtain generalized bilinear forms of some well-known soliton equations: the Korteweg de Vries equation, the scalar Boussinesq equation and the Kaup–Boussinesq equations. These bilinear forms are solved for new solutions using the perturbation method and the principle of superposition.

INTEGRABLE MODELS FOR SHALLOW WATER WITH ENERGY DEPENDENT SPECTRAL PROBLEMS

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse problem as a Riemann–Hilbert problem with a ℤ2 reduction group. The soliton solutions are explicitly obtained.

N = 2 supersymmetric a=4-Korteweg–de Vries hierarchy derived via Gardner’s deformation of Kaup–Boussinesq equation

We consider the problem of constructing Gardner’s deformations for the N=2 supersymmetric a=4-Korteweg–de Vries (SKdV) equation; such deformations yield recurrence relations between the super-Hamiltonians of the hierarchy. We prove the nonexistence of supersymmetry-invariant deformations that retract to Gardner’s formulas for the Korteweg-de Vries (KdV) with equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner’s deformation of the Kaup–Boussinesq equation, which is contained in the bosonic limit of the superhierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full N=2, a=4-SKdV hierarchy. Our method is applicable toward the solution of Gardner’s deformation problems for other supersymmetric KdV-type systems.

Gardner's deformations of the Boussinesq equations

Using the algebraic method of Gardner's deformations for completely integrable systems, we construct the recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these systems, we obtain new integrable equations adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.

Asymptotic soliton train solutions of Kaup–Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h 0( x) and u 0( x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u 0( x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.

On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization

New transformations between the nonlinear Schrodinger, Kaup and non-local Boussinesq equations as well as their modified counterparts are found and analysed. The bilinear representations of these equations, including an alternative bilinear form of the Chen - Lee - Liu equation, are obtained by a direct method based on the Bell's exponential polynomials. Explicit Wronskian solutions to these equations are also presented.

Real finite-gap regular solutions of the Kaup-Boussinesq equation

Real finite-gap regular solutions of the Kaup-Boussinesq equation