In this paper, we study the Cauchy problem for Benjamin-Ono-Burgers equation $${\partial _t}u - \epsilon \partial _x^2u + {\cal H}\partial u{u_x} = 0$$ , where $${\cal H}$$ denotes Hilbert transform operator. We obtain that it is uniformly locally well-posed small data in refined Sobolev space $${\tilde H^\sigma }(\mathbb{R})\,\,(\sigma \geqslant 0)$$ which a subspace of L2(ℝ). It worth noting ...

In this work we consider the problem on group classification and conservation laws of general first order evolution equations. We obtain subclasses these equations which are quasi-self-adjoint self-adjoint. By using recent Ibragimov's Theorem laws, establish admiting self-adjoint illustrate our results applying them to inviscid Burgers' equation. particular an infinite number new symmetries fou...

In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially...

We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation ∂tu− ∂ txxu + 4u∂xu = 3∂xu∂ xxu + u∂ xxxu. In a recent paper [3], we proved for this equation the existence and uniqueness of L1∩BV weak solutions satisfying an infinite family of Kružkov-type entropy inequalities. The purpose of this paper is to replace the Kruž...

A two-step strategy is proposed for the computation of singularities in nonlinear PDEs. The first step is the numerical solution of the PDE using a Fourier spectral method; the second step involves numerical analytical continuation into the complex plane using the epsilon algorithm to sum the Fourier series. Test examples include the inviscid Burgers and nonlinear heat equations, as well as a t...

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We...

The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: ut + uux = 0. Often referred to as the inviscid Burgers equation, many are familiar with this equation in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascad...

We consider the inviscid Burgers' equation enlarged with a control function and its adjoint equation. For several discretization schemes of the inviscid Burgers' equation the \adjoint" nite diier-ence methods are derived. Applying these discretization methods and the checkpointing routine treeverse, approximations of the solution of both diierential equations are calculated and compared. 1 The ...

We study the vanishing viscosity limit of one-dimensional Burgers equation near nondegenerate shock formation. develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up moment first forms. The inner part this has novel structure based on fractional spacetime Taylor series for inviscid solution. obtain sharp rates in variety norms, including $$L^\infty...

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV–Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method. 2010 Elsevier Inc. All rights reserved.