Existence and stability of traveling waves for an integro-differential equation for slow erosion

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order nonlinear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at ±∞. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approach the traveling waves asymptotically as t→+∞$t\to +\infty$.     

The IVP for the Benjamin-Ono equation in weighted Sobolev spaces

We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces , s∈R, s?1 and s?r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.     

On the stability of the solitary waves to the (generalized) Kawahara equation

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space $H^2(\mathbb{R})$ of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [4] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [9]. The second family consists of even traveling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3-19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.     

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.     

Well-posedness for the Cauchy problem associated to the Hirota-Satsuma equation: Periodic case

We consider a system of Korteweg-de Vries (KdV) equations coupled through nonlinear terms, called the Hirota-Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restriction norm method, we prove local well-posedness whenever s>−1/2. Also, with some restriction on the parameters of the system, we use the recent technique introduced by Colliander et al., called I-method and almost conserved quantities, to prove global well-posedness for s>−3/14.     

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.     

On the instability of elliptic traveling wave solutions of the modified Camassa–Holm equation

This paper is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period related to the modified Camassa–Holm equation. These solutions, called snoidal waves, are written in terms of the Jacobi elliptic functions. To prove our result we use the abstract method of Grillakis, Shatah and Strauss [23], the Floquet theory for periodic eigenvalue problems and the n-gaps potentials theory of Dubrovin, Matveev and Novikov [19].     

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.     

The stability of traveling waves with non-critical speeds for some competition systems

This paper is concerned with some quasilinear cross-diffusion systems which model competing species in mathematical ecology.By detailed spectral analysis,each traveling wave solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted spaces.     

Well-posedness and blow-up phenomena for the generalized Degasperis-Procesi equation

In this paper, we mainly study the Cauchy problem of the generalized Degasperis-Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time.     

Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case

The following nonlinear Schrödinger equation is studied

     

The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability

Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin-Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices.     

Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity

ABSTRACT

In this paper, we consider a viscoelastic plate equation with a velocity-dependent material density and a logarithmic nonlinearity. Using the Faedo-Galaerkin approximations and the multiplier method, we establish the existence of the solutions of the problem and we prove an explicit and general decay rate result. These results extend and improve many results in the literature.     

The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion

This paper is concerned with the existence and stability of traveling waves with transition layers for a quasi-linear competition system with cross diffusion,which was first proposed by Shegesada,Kawasaki and Teramoto.When one of the random diffusion rates is small and the cross-diffusion rate is not small,by the geometric singular perturbation method,the existence of traveling waves with transition layers is obtained.Further,by the detailed spectral analysis and topological index method,the traveling waves...     

On the stability of the linear differential equation of higher order with constant coefficients

We obtain a result on stability of the linear differential equation of higher order with constant coefficients in Aoki-Rassias sense. As a consequence we obtain the Hyers-Ulam stability of the above mentioned equation. A connection with dynamical sytems perturbation is established.     

Isoinertial family of operators and convergence of KdV cnoidal waves to solitons

In this paper we show the convergence of Korteweg-de Vries cnoidal waves to the limit soliton. It is proved that the convergence is uniform and in H²-norm, as the period of the solutions tends to infinity. Families of Hill operators are also studied. We obtain a condition under which families of operators are isoinertial. This condition is satisfied for classes of Hill operators that are obtained by linearization. Our application is to the family of linearized operators at the KdV cnoidal waves. It is proved that this family is isoinertial and also the value of the inertial index is calculated.     

Fronts,traveling fronts,and their stability in the generalized Swift-Hohenberg equation

The generalized Swift-Hohenberg equation with an additional quadratic term is studied. Time-stable localized stationary solutions of the pulse and front types are found. It is shown that stationary fronts give rise to traveling fronts, whose branches are also obtained. This study combines theoretical methods for dynamical systems (in particular, the theory of homo-and heteroclinic orbits) and numerical simulation.     

The initial-boundary value problem for the “good” Boussinesq equation on the bounded domain

The initial-boundary value problem for the “good” Boussinesq equation on the bounded domain is studied in this article. The local and global well-posedness of this initial-boundary value problem is given.