Open Access Research

Neumann problem on the semi-line for the Burgers equation

Silvana De Lillo12 and Matteo Sommacal3*

Author affiliations

1 Dipartimento di Matematica ed Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy

2 INFN, Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Italy

3 Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France

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Citation and License

Boundary Value Problems 2011, 2011:34  doi:10.1186/1687-2770-2011-34


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/34


Received:28 April 2011
Accepted:14 October 2011
Published:14 October 2011

© 2011 De Lillo and Sommacal; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, the Neumann problem on the semi-line for the Burgers equation is considered. The problem is reduced to a nonlinear integral equation in one independent variable, whose unique solution is proven to exist for small time. An explicit solution is discussed as well.

Keywords:
Burgers equation; Neumann problem

1 Introduction

Initial/boundary value (IBV) problems for integrable nonlinear PDEs frequently appear in physical applications and have originated several important studies in the past few decades. Much interest has been devoted to IBV problems for nonlinear PDEs which are treatable by the inverse scattering transform method, such as the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the Sine-Gordon equation [1-8]. Other studies have been devoted to IBV problems for nonlinear PDEs which are C-integrable, namely, which are exactly linearizable via a change of variables: well-known examples in this class are the Burgers equation and the Eckhaus equation [9-15].

It is the aim of this article to analyze the Neumann problem for the Burgers equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M1">View MathML</a>

(1)

on the semi-infinite domain x ∈ [0, ∞) characterized by the following set of initial and boundary data:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M2">View MathML</a>

(2a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M3">View MathML</a>

(2b)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M4">View MathML</a>

(2c)

where F(t) is a continuous, bounded function of its argument:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M5">View MathML</a>

(2d)

and the initial datum u0(x) is assumed to be integrable on the semi-line:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M6">View MathML</a>

(2e)

We point out that such problem was previously considered in [12], where it was shown to be equivalent to a nonlinear integro-differential equation (in one independent variable), which however cannot generally be solved. In this article, our analysis is based on the method developed in [10] for the solution of Dirichlet problem and on the use of the contraction-mapping technique, analogously to what was done for the Eckhaus equation in [15]. In particular, the main result of the present study is to prove the following.

Theorem 1 There exists a finite constant σ ∈ ℝ, 0 <σ < ∞, such that the solution to the Neumann problem (2a-2e) for (1) exists and is unique for 0 ≤ t <σ.

Unlike the Neumann problem on the finite interval (0,1), for which the existence of a solution in L2 (0,1) was proven in [16] by means of the Galerkin method, the Neumann problem on the semi-line for the Burgers equation has not received much attention in the literature in the past. To the best of the authors' knowledge, Theorem 1 (as well as Lemma 1 in Section 3) are new.

In Section 2, we put problem (1, 2a-2e) in a one-to-one correspondence with a Neumann problem for the heat equation, characterized by a boundary datum which is a nonlinear combination of the boundary data {u(x,0) ,ux(0,t)} of the Burgers equation. We reduce such a problem to a nonlinear integral equation of Volterra type in one independent variable (t). In Section 3, we prove the existence and uniqueness of the solution for small time. In Section 4, we discuss a special solution of the problem (1, 2a-2e).

2 Reduction to a nonlinear integral equation

We begin our analysis by introducing the following ("generalized" Hopf-Cole) linearizing transformation [11,12]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M7">View MathML</a>

(3a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M8">View MathML</a>

(3b)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M9">View MathML</a>

(3c)

The above transformation maps the Burgers equation (1) into the linear heat equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M10">View MathML</a>

(4)

with the "compatibility" condition for C(t) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M11">View MathML</a>

(5)

where (hereafter) the dot indicates differentiation with respect to time. Through transformation (3a-3c), from the Neumann IBV data for the u(x,t), (2a-2e), we obtain the IBV data for the υ(x,t) that characterize (4):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M12">View MathML</a>

(6a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M13">View MathML</a>

(6b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M14">View MathML</a>

(6c)

Comparing (6c) with (5) and making use of (6b) with (2b), we can restate the compatibility condition for C(t), (5), in the following shape:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M15">View MathML</a>

(7)

According to (6c), the boundary datum υx(0,t) for the heat equation (4) is a nonlinear combination of known (ux(0,t)) and unknown (u(0,t)) boundary data for the Burgers equation (1).

The Neumann problem on the semi-line for υ(x,t) is then in principle solved through the following prescription:

1. Solve the Neumann problem on the semi-line for υ(x, t), with initial datum (6a) and (6c);

2. Determine the unknown function C(t) by means of the transformation (3a) and (3c);

3. Recover u(x,t) via the inverse transformation (3b).

We reduced the problem of solving Burgers equation (1) with the Neumann condition (2a-2e) to the determination of the couple {υ(x,t), C(t)}, where υ(x,t) solves the heat equation (4) with the Neumann condition (6a-6c) and C(t) simultaneously satisfying (7).

In order to explicitly evaluate the solution υ(x,t) of (4), as in [12], it is convenient to introduce the cosine-Fourier transform:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M16">View MathML</a>

(8a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M17">View MathML</a>

(8b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M18">View MathML</a>

(8c)

Using (8b) and (8c) with (8a), via (7), we equivalently get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M19">View MathML</a>

(9)

In the right-hand side of (9), υ(0,t) is unknown. It is then convenient to set z(t) = υ(0,t) and to calculate left- and right-hand sides of (9) at x = 0. To do so, let us start by recalling that

(a) for an arbitrary function of t, g(t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M20">View MathML</a>

b) from the convolution properties of the cosine-Fourier transform (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M21">View MathML</a>

From this, at x = 0, we can restate (9) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M22">View MathML</a>

(10a)

or, via (7),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M23">View MathML</a>

(10b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M24">View MathML</a>

(10c)

Making use of (10a), we can put in a more explicit form the previously given prescription to solve the Neumann problem on the semi-line, (1) and (2a-2e), for u(x, t):

1. Given the Neumann data on the semi-line, (2a-2e), compute C(t) by substituting (10b) into (10c), namely, from the following nonlinear integro-differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M25">View MathML</a>

(11)

with C(0) = 1 as in (3c);

2. Evaluate the solution to the heat equation (4) with IBV data (6a-6c), υ(x,t), by means of (8a) making use of (8b) and (8c);

3. Recover u(x,t) from υ(x,t) via (3b).

For arbitrary u0(x) and F(t), there is no general technique for solving a nonlinear integro-differential equation like (11). On the other hand, the determination of the solution u(x,t) has been reduced to the solution of the nonlinear integral equation (10b)--with C(t) satisfying (10c)--for only one independent variable (t). In the next section, we prove the existence and uniqueness of the function z(t) for 0 ≤ t <σ, with 0 <σ < ∞ (Lemma 1). Once the existence and uniqueness of z(t) are established, the existence and uniqueness of υ(x,t) for 0 ≤ t <σ then follow, via (9), with C(t) being obtained via (10c). Then, via the inverse transformation (3b), Theorem 1 immediately follows, namely the solution of the original Neumann problem (2a-2e) for the Burgers equation (1) exists and is unique (for 0 ≤ t <σ).

3 Contraction mapping

In order to analyze the existence properties of z(t) for 0 ≤ t <σ < ∞, we denote by SM(σ) the closed sphere ||z|| ≤ M in the Banach space of continuous functions z(t) for t ∈ [0,σ), with the uniform norm ||z|| = l.u.b.|z(t)|. On the sphere SM(σ), we introduce the transformation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M26">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M27">View MathML</a> coincides with the right-hand side of (10b). To prove the existence and the uniqueness of the solution of the integral equation (10b) for a finite interval of time, we will prove the following:

Lemma 1 The mapping operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a>is a contraction mapping in SM(σ) for t ∈ [0,σ).

In order to prove this Lemma, we need to prove that, for t ∈ [0, σ), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> is closed and contractive in SM(σ).

3.1 Closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> in SM(σ) for t ∈ [0,σ)

We need to prove that if z(t) ∈ SM(σ) then w(t) ∈ SM(σ) as well, namely that ||z(t)|| ≤ M for t ∈ [0,σ) entails <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M29">View MathML</a> for t ∈ [0,σ).

The first step is to obtain an upper and lower bounds for |C(t)|. Integrating (10c), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M30">View MathML</a>

(13)

From the fact that |ex| = ex ≤ e|x| for any x ∈ ℝ, applying the triangular inequality (|x| - |y| |x + y| < |x| + |y|) on (13) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M33">View MathML</a>

(14a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M34">View MathML</a>

(14b)

imposing the last right-hand side of (14b) to be strictly greater than zero, we have the following condition on σ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M35">View MathML</a>

(14c)

where W is the Lambert-W function, implicitly defined as the inverse function of f(W) = W ew.

The second step, is to obtain an upper bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M27">View MathML</a>. Applying the triangular inequality on (10b), via (16), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M36">View MathML</a>

(15)

From (6a) and (2e) we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M37">View MathML</a>

(16)

then, for the first term on the right-hand side of (15), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M38">View MathML</a>

(17a)

For the second and the third terms in the right-hand side of (15), inequalities (14a-14c) and (2d) entail

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M39">View MathML</a>

(17b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M40">View MathML</a>

(17c)

Defining M = αA, with α > 1, and combining (15) with (17a-17c), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M41">View MathML</a>

(18a)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M42">View MathML</a>

(18b)

with σ satisfying condition (14c). On the other hand, in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M43">View MathML</a>, for B > 0 and M > 0, β(σ) is a monotonic, increasing bijective function on the positive Reals, and so there exists a value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M45">View MathML</a>

(18c)

Taking σ σ*, from (18a), we have ||w(t)|| ≤ M. Thus the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> is closed.    □

3.2 Contractivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> in SM(σ) for t ∈ [0,σ)

We need to prove that, given two solutions of (12), z(t) and (t), with ||z(t)-(t)|| = δ < 2M; it then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M46">View MathML</a> with 0 <θ < 1.

We now write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M47">View MathML</a>

(19)

Let us recall

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M48">View MathML</a>

(20)

Notice that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M49">View MathML</a>, then, for 0 ≤ t σ, via (14a) and (14b), we have that C(t) is a nonzero bounded function of t:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M50">View MathML</a>

thus X(t) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M51">View MathML</a> are bounded functions of t as well.

The identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M52">View MathML</a>

(21)

entails

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M53">View MathML</a>

(22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M54">View MathML</a>

(23a)

Let us recall

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M55">View MathML</a>

(23b)

existence and well-definedness of which are implied by our previous considerations about the boundedness of C(t).

Since, for an arbitrary function of t, g(t), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M56">View MathML</a>, from (22), we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M57">View MathML</a>

(24)

or, integrating once with initial condition |C(0)-Ĉ(0)| = 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M58">View MathML</a>

(25a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M59">View MathML</a>

(25b)

Formula (25b), via (19), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M60">View MathML</a>

(26)

Choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M61">View MathML</a>

(27)

we get θ < 1 and, for what we saw in the previous Subsection 3.1, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> remains closed. Thus, Lemma 1 is proved, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> is a contraction operator on SM(σ).   □

Lemma 1 means that there exists a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M62">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M28">View MathML</a> in SM(σ), for 0 ≤ t σ. We have thereby proven the existence and uniqueness of the solution of the integral equation (10b) for 0 ≤ t <σ. Then, as explained at the end of Section 2, from the existence and uniqueness of z(t) in the interval 0 ≤ t <σ, we get, via (9) and (10c), the existence and uniqueness of υ(x,t) in the same interval, and via the inverse transformation (3b), we immediately get Theorem 1.

4 A special solution

In this section, we consider a particular solution of the Neumann problem (2a-2e) for the Burgers equation (1), and derive the corresponding expression for z(t).

A solution to the Burgers equation is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M63">View MathML</a>

(28)

where A and t0 > 0 are two real constants and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M64">View MathML</a>

From (28), we obtain for the squared modulus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M65">View MathML</a>

(29a)

Where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M66">View MathML</a>

(29b)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M67">View MathML</a>

(29c)

Thus the solution (28) on the whole line is a single hump with (negative) peak value given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M68">View MathML</a>

(30a)

where yp is the solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M69">View MathML</a>

(30b)

the peak value is attained at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M70">View MathML</a>

(30c)

and moves to the left with velocity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M71">View MathML</a>

(30d)

The corresponding initial datum and boundary condition, which associate the given solution (28) to the Burgers equation (1) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M72">View MathML</a>

(31a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M73">View MathML</a>

(31b)

Notice that, if t0 = 0, then, from (31a), it turns out that u0(x) = Aδ(x), where δ(x) is the Dirac delta function; in this case, all the following calculations can still be performed.

Next, we prove that (28) considered on the semi-line x ∈ [0, +∞) is a particular solution of the Neumann problem (2a-2e) for the Burgers equation (1). To this end, we start noting that from the solution (28), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M74">View MathML</a>

(32)

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M75">View MathML</a>

(33)

this last relation, via (5) and (3c), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M76">View MathML</a>

(34)

which in turn, from (6b) and (32), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M77">View MathML</a>

(35)

On the other hand, it is now immediate to see that the integral equation (10b), when (34) and (35) are used, reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M78">View MathML</a>

(36a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/34/mathml/M79">View MathML</a>

(36b)

An explicit computation of the integral (36a) with (36b) yields immediately the same expression (35) for z(t).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors wish to thank M. J. Ablowitz for enlightening discussions on this topic. MS wishes to thank T. Horikis and M. Hoefer, particularly the latter for bringing to his attention a different way to linearize (1).

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