Abstract
In this article, the Neumann problem on the semiline 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 problem1 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 Kortewegde Vries equation (KdV), and the SineGordon equation [18]. Other studies have been devoted to IBV problems for nonlinear PDEs which are Cintegrable, namely, which are exactly linearizable via a change of variables: wellknown examples in this class are the Burgers equation and the Eckhaus equation [915].
It is the aim of this article to analyze the Neumann problem for the Burgers equation:
on the semiinfinite domain x ∈ [0, ∞) characterized by the following set of initial and boundary data:
with
where F(t) is a continuous, bounded function of its argument:
and the initial datum u_{0}(x) is assumed to be integrable on the semiline:
We point out that such problem was previously considered in [12], where it was shown to be equivalent to a nonlinear integrodifferential 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 contractionmapping 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 (2a2e) 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 L^{2 }(0,1) was proven in [16] by means of the Galerkin method, the Neumann problem on the semiline 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, 2a2e) in a onetoone 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) ,u_{x}(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, 2a2e).
2 Reduction to a nonlinear integral equation
We begin our analysis by introducing the following ("generalized" HopfCole) linearizing transformation [11,12]:
with
The above transformation maps the Burgers equation (1) into the linear heat equation
with the "compatibility" condition for C(t) given by
where (hereafter) the dot indicates differentiation with respect to time. Through transformation (3a3c), from the Neumann IBV data for the u(x,t), (2a2e), we obtain the IBV data for the υ(x,t) that characterize (4):
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:
According to (6c), the boundary datum υ_{x}(0,t) for the heat equation (4) is a nonlinear combination of known (u_{x}(0,t)) and unknown (u(0,t)) boundary data for the Burgers equation (1).
The Neumann problem on the semiline for υ(x,t) is then in principle solved through the following prescription:
1. Solve the Neumann problem on the semiline 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 (2a2e) to the determination of the couple {υ(x,t), C(t)}, where υ(x,t) solves the heat equation (4) with the Neumann condition (6a6c) 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 cosineFourier transform:
Using (8b) and (8c) with (8a), via (7), we equivalently get
In the righthand side of (9), υ(0,t) is unknown. It is then convenient to set z(t) = υ(0,t) and to calculate left and righthand 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
b) from the convolution properties of the cosineFourier transform (8), we have
From this, at x = 0, we can restate (9) as follows:
or, via (7),
Making use of (10a), we can put in a more explicit form the previously given prescription to solve the Neumann problem on the semiline, (1) and (2a2e), for u(x, t):
1. Given the Neumann data on the semiline, (2a2e), compute C(t) by substituting (10b) into (10c), namely, from the following nonlinear integrodifferential equation:
with C(0) = 1 as in (3c);
2. Evaluate the solution to the heat equation (4) with IBV data (6a6c), 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 u_{0}(x) and F(t), there is no general technique for solving a nonlinear integrodifferential 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 (2a2e) 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 S_{M}(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 S_{M}(σ), we introduce the transformation:
where coincides with the righthand 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 is a contraction mapping in S_{M}(σ) for t ∈ [0,σ).
In order to prove this Lemma, we need to prove that, for t ∈ [0, σ), is closed and contractive in S_{M}(σ).
3.1 Closure of in S_{M}(σ) for t ∈ [0,σ)
We need to prove that if z(t) ∈ S_{M}(σ) then w(t) ∈ S_{M}(σ) as well, namely that z(t) ≤ M for t ∈ [0,σ) entails for t ∈ [0,σ).
The first step is to obtain an upper and lower bounds for C(t). Integrating (10c), we obtain
From the fact that e^{x} = e^{x }≤ e^{x }for any x ∈ ℝ, applying the triangular inequality (x  y ≤ x + y < x + y) on (13) we get
imposing the last righthand side of (14b) to be strictly greater than zero, we have the following condition on σ:
where W is the LambertW function, implicitly defined as the inverse function of f(W) = W e^{w}.
The second step, is to obtain an upper bound for . Applying the triangular inequality on (10b), via (16), we get
From (6a) and (2e) we can write
then, for the first term on the righthand side of (15), we get
For the second and the third terms in the righthand side of (15), inequalities (14a14c) and (2d) entail
Defining M = αA, with α > 1, and combining (15) with (17a17c), we get
where
with σ satisfying condition (14c). On the other hand, in the interval , for B > 0 and M > 0, β(σ) is a monotonic, increasing bijective function on the positive Reals, and so there exists a value such that
Taking σ ≤ σ*, from (18a), we have w(t) ≤ M. Thus the mapping is closed. □
3.2 Contractivity of in S_{M}(σ) for t ∈ [0,σ)
We need to prove that, given two solutions of (12), z(t) and ẑ(t), with z(t)ẑ(t) = δ < 2M; it then follows that with 0 <θ < 1.
We now write
Let us recall
Notice that, if , then, for 0 ≤ t ≤ σ, via (14a) and (14b), we have that C(t) is a nonzero bounded function of t:
thus X(t) and are bounded functions of t as well.
The identity
entails
where
Let us recall
existence and welldefinedness 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 , from (22), we can write
or, integrating once with initial condition C(0)Ĉ(0) = 0,
Formula (25b), via (19), implies
Choosing
we get θ < 1 and, for what we saw in the previous Subsection 3.1, the mapping remains closed. Thus, Lemma 1 is proved, namely, is a contraction operator on S_{M}(σ). □
Lemma 1 means that there exists a unique fixed point of in S_{M}(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 (2a2e) for the Burgers equation (1), and derive the corresponding expression for z(t).
A solution to the Burgers equation is given by
where A and t_{0 }> 0 are two real constants and
From (28), we obtain for the squared modulus
Where
and
Thus the solution (28) on the whole line is a single hump with (negative) peak value given by
where y_{p }is the solution of the equation
the peak value is attained at
and moves to the left with velocity
The corresponding initial datum and boundary condition, which associate the given solution (28) to the Burgers equation (1) are
Notice that, if t_{0 }= 0, then, from (31a), it turns out that u_{0}(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 semiline x ∈ [0, +∞) is a particular solution of the Neumann problem (2a2e) for the Burgers equation (1). To this end, we start noting that from the solution (28), we get
so that
this last relation, via (5) and (3c), implies
which in turn, from (6b) and (32), implies
On the other hand, it is now immediate to see that the integral equation (10b), when (34) and (35) are used, reduces to
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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