Elsevier

Journal of Differential Equations

Volume 270, 5 January 2021, Pages 1196-1217
Journal of Differential Equations

Interaction of Dirac δ-waves in the nonlinear Klein-Gordon equation

https://doi.org/10.1016/j.jde.2020.09.012Get rights and content

Highlights

  • A review on solving nonlinear hyperbolic systems of PDEs containing Dirac δ measures.

  • A review on α-product applications to some nonlinear models arising in mathematical physics.

  • Under certain conditions, δ waves in the Klein-Gordon equation behave like classical solitons in the sine-Gordon equation.

Abstract

The present paper studies the interaction of Dirac δ-waves in models ruled by the nonlinear Klein-Gordon equation uttc2uxx=ϕ(u), where c>0 is a real number and ϕ is an entire function taking real values on the real axis. Such study is made using a product of distributions that both extends the meaning of ϕ(u) for certain distributions u and allows the definition of a solution concept consistent with the classical solution concept. From such study, it emerges that in several nonlinear Klein-Gordon equations Dirac δ-waves behave like classical solitons in the sine-Gordon equation. As particular cases, this work examines the phi-four equation and the sine-Gordon equation.

Section snippets

Introduction and contents

The Klein-Gordon equation has over 90 years of history. Proposed in 1926 and 1927 by Klein, Fock, and Gordon [1], [2], [3], [4], it was considered the relativistic counterpart of the Schrödinger wave equation. Nowadays, it is accepted that the (nonlinear) Klein-Gordon equation describes the evolution of a spin-zero scalar field in the presence of a potential (see [5] pp. 20, 21). This equation plays a prominent role in a wide range of fields such as physics, engineering, chemistry, and biology

Multiplication of distributions

For readers' convenience and paper's self-containedness, this section summarizes Sarrico's distributional products theory to provide all that is essential to understand the present work. To get an up to date overview of his distributional products, see [72] Sec. 2. For a thorough presentation, see [41], [73].

Let D be the space of indefinitely differentiable complex-valued functions defined on R with compact support, and let D be the space of Schwartz distributions. In his theory, each function

Powers of distributions

This section defines α-powers for some distributions with the aim of later composing entire functions with certain distributions relevant for this work.

The α-product (6) allows to define α-powers Tαn with TCp(DpDμ) and nN0 by the recurrence relationTαn=(Tαn1)α˙T for n1, with Tα0=1 for T0. See [51] for details. Naturally, 0αn=0 for all n1. As an example, if mC{0}, by (7) it follows that (mδ)α0=1, (mδ)α1=mδ, and for n2, δαn=[α(0)]n1δ. Since Sarrico's distributional products are

Composition of an entire function with a distribution

This section defines the composition of entire functions with certain distributions. Such definition affords a rigorous meaning to the right-hand side ϕ(u) of the Klein-Gordon equation (1) for the distributions u studied in this work.

Let zC and ϕ:CC be an entire function given by the MacLaurin seriesϕ(z)=a0+a1z+a2z2+ with coefficients on C and ϕ˘:CC the entire function given by the seriesϕ˘(z)=a1+a2z+a3z2+ The sum of the above series can easily be computed by the following formulaϕ˘(z)={ϕ(z

The α-solution concept

This section presents the α-solution concept and relates it with the classical solution concept.

Let F(R) be the space of continuously differentiable maps u˜:RD in the sense of the usual topology of D. For tR, the notation [u˜(t)](x) is sometimes used to stress that the distribution u˜(t) acts on functions ξD depending on x.

Let Σ(R) be the space of functions u:R×RR such that:

  • (a)

    for each tR, u(x,t)Lloc1(R);

  • (b)

    u˜:RD, defined by [u˜(t)](x)=u(x,t) is in F(R).

The natural injection uu˜ from Σ(R)

Interaction of δ-waves

This section solves the Cauchy problem (1), (2), (3) in solution space (4) by considering ϕ as an entire function taking real values on the real axis.

Having in mind the identification uu˜, equation (1) must be substituted by equation (33) and the solution (4) byu˜(t)=f(t)τγ1(t)δ+g(t)τγ2(t)δ, where γ1,γ2,f,g:RR are C2-functions. For short, U˜F(R) will denote the set of functions u˜:RD of the form (35) and Q denotes the space of entire functions taking real values on the real axis.

Let us

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    Supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020.

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