We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations. We then discuss when the associated linearized equation admits solutions exponentiall...

In this paper, using the Lie group analysis method, we study the invariance properties of the general time fractional fifth-order Korteweg-de Vries (KdV) equation. A systematic research to derive Lie point symmetries of the equation is performed. In the sense of point symmetry, all of the geometric vector fields and the symmetry reductions of the equation are obtained, the exact power series so...

1. Continuous Transformation Groups 1.1. Local Transformation Groups 1.2. Lie Equations 1.3. Invariants 1.4. Invariant Manifolds 2. Invariant Differential Equations 2.1. The Continuation of Point Transformations 2.2. Defining Equation 2.3. Invariant and Partly Invariant Solutions 3. Tangential Transformations 3.1. Contact Transformations 3.2. Tangential Transformations of Finite and Infinite Or...

In this paper, we consider multi-component generalizations of the Hirota–Satsuma coupled Korteweg–de Vries (KdV) equation. By introducing a Lax pair, we present a matrix generalization of the Hirota–Satsuma coupled KdV equation, which is shown to be reduced to a vector Hirota–Satsuma coupled KdV equation. By using Hirota's bilinear method, we find a few soliton solutions to the vector Hirota–Sa...

In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient condit...

In this paper we study several aspects of solitary wave solutions of the Ostrovsky equation. Using variational methods, we show that as the rotation parameter goes to zero, ground state solitary waves of the Ostrovsky equation converge to solitary waves of the Korteweg-deVries equation. We also investigate the properties of the function d(c) which determines the stability of the ground states. ...

mentioned in [1] as a model for the propagation of one-dimensional, unidirectional small amplitude long waves in nonlinear media. In GKdVB equation, μ, γ > 0 and α is a positive integer is considered, the independent variable x represents the medium of propagation, t is proportional to elapsed time, and u(x, t) is a velocity at the point x at time t. And If ν = 0, α = 1, the GKdVB equation (1) ...

The behavior of linear and nonlinear dust acoustic waves (DAWs) in an unmagnetized plasma including inertialess electrons and positrons, ions, and mobile positive/negative dust grains are studied. Reductive perturbation method is employed for small and finite amplitude DAWs. To investigate the solitary waves, the Korteweg–de Vries (KdV) equation is derived and the solution is presented. B...

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still pr...

We study numerical approximations of systems of partial differential equations modeling the interaction of short and long waves. The short waves are modeled by a nonlinear Schrödinger equation which is coupled to another equation modeling the long waves. Here, we consider the case where the long wave equation is either a hyperbolic conservation law or a Korteweg–de Vries equation. In the former...