Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation

On the variational structure of breather solutions

In this paper we give a systematic and simple account that put in evidence that many breather solutions of integrable equations satisfy suitable variational elliptic equations, which also implies that the stability problem reduces in some sense to $(i)$ the study of the spectrum of explicit linear systems (\emph{spectral stability}), and $(ii)$ the understanding of how bad directions (if any) can be controlled using low regularity conservation laws. We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV), Gardner, and sine-Gordon (SG) equations. Then we perform numerical simulations that confirm, at the level of the spectral problem, our previous rigorous results, where we showed that mKdV breathers are $H^2$ and $H^1$ stable, respectively. In a second step, we also discuss the Gardner and the Sine-Gordon cases, where the spectral study of a fourth-order linear matrix system is the key element to show stability. Using numerical methods, we confirm that all spectral assumptions leading to the $H^2\times H^1$ stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime. In a second part, we study the periodic mKdV case, where a periodic breather is known from the work of Kevrekidis et al. We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability. However, we also identify the source of nonlinear instability in the case described in Kevrekidis et al. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.Comment: 55 pages; This paper is an improved version of our previous paper 1309.0625 and hence we replace i

ANALYTICAL SOLUTION OF THE RELATIVISTIC KLEIN-GORDON WAVE EQUATION

In this study, the solution to Klein-Gordon equations with focus on analytical methods is discussed. The analytical methods used in this research are the Variational Iteration Method (VIM) developed by Ji-Huan He, Adomian Decomposition Method (ADM) by Adomian and New Iterative Method (NIM) developed by Daftardar Gejji and Jafari. The modified Adomian Decomposition method by Wazwaz was used to solve the linear inhomogeneous and nonlinear Klein-Gordon equations to accelerate the convergence of the solution and minimizes the size of calculation while still maintaining high accuracy of the analytical solution. All the problems considered yield the exact solutions with few iterations. The solutions obtained were compared with the exact solution and the solutions obtained by other existing methods. The solutions obtained by the three methods yield the same results and all the problems considered show that the Variational Iteration Method, Adomian Decomposition Method and New Iterative Method are very powerful and potent in solving Klein-Gordon equations and can be used to obtain closed form solutions of linear and nonlinear differential equations (ordinary and partial)

Structure of the broken phase of the sine-Gordon model using functional renormalization

We study in this paper the sine-Gordon model using functional Renormalization Group (fRG) at Local Potential Approximation (LPA) using different RG schemes. In $d=2$, using Wegner-Houghton RG we demonstrate that the location of the phase boundary is entirely driven by the relative position to the Coleman fixed point even for strongly coupled bare theories. We show the existence of a set of IR fixed points in the broken phase that are reached independently of the bare coupling. The bad convergence of the Fourier series in the broken phase is discussed and we demonstrate that these fixed-points can be found only using a global resolution of the effective potential. We then introduce the methodology for the use of Average action method where the regulator breaks periodicity and show that it provides the same conclusions for various regulators. The behavior of the model is then discussed in $d\ne 2$ and the absence of the previous fixed points is interpreted.Comment: 43 pages, 32 figures, accepted versio