Global and singular solutions to the generalized Proudman-Johnson equation

We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters a having the form for n∈N, thereby extending a result of Bressan and Constantin (2005) [2]. Furthermore, we present new proofs of existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates.     

Instability of solitary wave solutions for the generalized BO-ZK equation

In this paper we study the generalized BO-ZK equation in two space dimensions

ut+upux+αHuxx+εuxyy=0.     

Global well-posedness of the critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t_∂u+ux_∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where .     

Global conservative solutions of the generalized hyperelastic-rod wave equation

We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa-Holm equation (f(u)=u²/2 and g(u)=κu+u²) as well as the hyperelastic-rod wave equation (f(u)=γu²/2 and g(u)=(3−γ)u²/2) as special cases. It is shown that the problem is well-posed for initial data in H¹(R) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

Global weak solutions to a generalized Benjamin-Bona-Mahony-Burgers equation

The existence of global weak solutions for a generalized Benjamin-Bona-MahonyBurgers equation is established in the space C([0, ∞) × R) ∩ L~∞([0, ∞); H1(R)) under the condition that its initial value belongs to the space H1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first order derivatives of the solution with respect to the space variable are derived to prove the existence.     

The application of the auxiliary equation technique to a generalized mKdV equation with variable coefficients

The auxiliary differential equation technique is employed to investigate a generalized mKdV equation with variable coefficients. The Jacobi elliptic function wave-like solutions of the equation are expressed under several circumstances. The degenerated soliton-like and trigonometric function solutions are discussed in detail as the modulus of the Jacobi elliptic wave-like solutions tends to 1 and 0, respectively.     

Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Sufficient and necessary conditions for the embeddings between Besov spaces and modulation spaces are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces is shown.     

On the long time behaviour of a generalized KdV equation

We consider the Cauchy problem for the generalized Korteweg-de Vries equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkGi2oaaBaaaleaacaaIXaaabeaakiaadwhacqGHRaWkcqGH% ciITdaWgaaWcbaGaamiEaaqabaGccaGGOaGaeyOeI0IaeyOaIy7aa0% baaSqaaiaadIhaaeaacaaIYaaaaOGaaiykamaaCaaaleqabaGaeqyS% degaaOGaamyDaiabgUcaRiabgkGi2oaaBaaaleaacaWG4baabeaakm% aabmGabaWaaSaaaeaacaWG1bWaaWbaaSqabeaacqaH7oaBaaaakeaa% cqaH7oaBaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa!56D5!\[\partial _1 u + \partial _x ( - \partial _x^2 )^\alpha u + \partial _x \left( {\frac{{u^\lambda }}{\lambda }} \right) = 0\]where is a positive real and and integer larger than 1. We obtain the detailed large distance behaviour of the fundamental solution of the linear problem and show that for 1/2 and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeU7aSjabg6da+iabeg7aHjabgUcaRmaalaaabaGaaG4maaqa% aiaaikdaaaGaey4kaSYaaeWaceaacqaHXoqydaahaaWcbeqaaiaaik% daaaGccqGHRaWkcaaIZaGaeqySdeMaey4kaSYaaSaaaeaacaaI1aaa% baGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVa% GaaGOmaaaaaaa!4FF7!\[\lambda > \alpha + \frac{3}{2} + \left( {\alpha ^2 + 3\alpha + \frac{5}{4}} \right)^{1/2} \], solutions of the nonlinear equation with small initial conditions are smooth in the large and asymptotic when t± to solutions of the linear problem.     

Global well-posedness of the Cauchy problem of the fifth-order shallow water equation

By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation is globally well-posed in the Sobolev space Hs(R) provided .     

Global well-posedness for the Benjamin equation in low regularity

In this paper we consider the initial value problem of the Benjamin equation

     

On the fifth-order KdV equation: Local well-posedness and lack of uniform continuity of the solution map

In this paper we prove that the following fifth-order equation arising from the KdV hierarchy

     

Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing

The forced Korteweg-de Vries equation with Burgers’ damping (fKdVB) on a periodic domain, which arises as a model for water waves in a shallow tank with forcing near resonance, is considered. A method for construction of asymptotic solutions is presented, valid in cases where dispersion and damping are small. Through variation of a detuning parameter, families of resonant solutions are obtained providing detailed insight into the resonant response character of the system and allowing for direct comparison with the experimental results of Chester and Bones (1968).     

Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation

     

A KdV equation in 2+1 dimensions: Painlevé analysis,solutions and similarity reductions

The nonlinear equationm _ty =(m _yxx +m _x m _y ) _x is throughly analyzed. The Painlevé test yields a positive result. The Bäckhand transformations are found and the Darboux-MoutardMatveev formalism arises in the context of this analysis. Some solutions and their interactions are also analyzed. The singular manifold equations are also used to determine symmetry reductions. This procedure can be related with the direct method of Clarkson and Kruskal.     

Global existence for the critical generalized KdV equation

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,

The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.

     

Global existence and blow-up for the generalized sixth-order Boussinesq equation

In this paper we prove local well-posedness in L²(R)$L^2\left(\mathbb{R}\right)$ and H¹(R)$H^1\left(\mathbb{R}\right)$ for the generalized sixth-order Boussinesq equation _utt=_uxx+β_uxxxx+_uxxxxxx+_(|u|αu)xx$u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+{\left({\left|u\right|}^{\alpha }u\right)}_{xx}$. Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) [14]. We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

On the support of solutions to the Zakharov-Kuznetsov equation

In this article we prove that sufficiently smooth solutions of the Zakharov-Kuznetsov equation: