Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Interaction of solitary waves for the generalized KdV equation

We consider a class of generalized KdV equations with a small parameter and nonlinearities of the type u^m. We create a finite differences scheme to simulate the solution of the Cauchy problem and present some numerical results for the problem of the solitary waves interaction. In particular, we consider sufficient condition under which pairs of solitary waves interact, in the asymptotic sense, in accordance with the soliton scenario.     

On the Cauchy problem for the generalized Kawahara equation

We consider global weak solutions of the Cauchy problem for the generalized Kawahara equation. We prove theorems on the existence, uniqueness, and interior regularity as well as the exponential decay for large time.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.     

Solution of Hyers-Ulam stability problem for generalized Pappus' equation

The purpose of this paper is to solve the stability problem of Ulam for an approximate mapping of the following generalized Pappus' equation:

n²Q(x+my)+mnQ(x−ny)=(m+n)[nQ(x)+mQ(ny)]     

On the stability of a cubic functional equation

In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation
2f（x ＋ 2y） ＋ f（2x - y） = 5f（x ＋ y） ＋ 5f（x - y）＋ 15f（y）
in the spirit of Hyers, Ulam, Rassias and Gavruta.     

On the stability of the generalized sine functional equations

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：
g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2
Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.     

On the stability of a quartic functional equation

In this paper, we prove the generalized Hyers-Ulam stability for the following quartic functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y).     

Ill-posedness of Kawahara equation and Kaup-Kupershmidt equation

In this paper we consider the Cauchy problems for the Kawahara equation and the Kaup-Kupershmidt equation. By using the general well-posedness principle introduced by I. Bejenaru and T. Tao (2006) [1], we prove that the Kawahara equation is ill-posed for the initial data in Hs(R) with and the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with .     

Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ‐expansion method

In this paper, variable coefficients Kawahara equation (VCKE) and variable coefficients modified Kawahara equation (VCMKE), which arise in modeling of various physical phenomena, are studied by Lie group analysis. The similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations. Moreover, some exact analytic solutions are considered by the power series method. Further, a generalized ‐expansion method is applied to VCKE and VCMKE for constructing some new exact solutions. As a result, hyperbolic function solutions, trigonometric function solutions and some rational function solutions with parameters are furnished. Copyright © 2012 John Wiley & Sons, Ltd.     

On the stability of the solitary waves to the rotation Benjamin‐Ono equation

In this paper, we study several aspects of solitary wave solutions of the rotation Benjamin‐Ono equation. By solving a minimization problem on the line, we construct a family of even travelling waves ψ_c,γ. We then prove the uniqueness of even ground states associated with large speed and their orbital stability. Note that this improves the results in Esfahani and Levandosky, where only the stability of the set of ground states is proven.     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.     

On the Hyers-Ulam stability of the linear differential equation

We obtain some results on generalized Hyers-Ulam stability of the linear differential equation in a Banach space. As a consequence we improve some known estimates of the difference between the perturbed and the exact solutions.