c Pleiades Publishing, Ltd., 2016. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2016, Vol. 295, pp. 148–157. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 163–173.
Spectral Stability Theory of Heteroclinic Solutions to the Korteweg de Vries Burgers Equation with an Arbitrary Potential A. T. Il’ichev a and A. P. Chugainova a Received June 10, 2016
Abstract—The analysis of stability of heteroclinic solutions to the Korteweg–de Vries–Burgers equation is generalized to the case of an arbitrary potential that gives rise to heteroclinic states. An example of a specific nonconvex potential is given for which there exists a wide set of heteroclinic solutions of different types. Stability of the corresponding solutions in the context of uniqueness of a solution to the problem of decay of an arbitrary discontinuity is discussed. DOI: 10.1134/S0081543816080083
1. INTRODUCTION In order that a discontinuity could be thought of as an admissible one, i.e., physically realizable, a number of conditions should hold. One of these conditions consists in the existence of a structure of a discontinuity. At the same time, under detailed consideration, a discontinuity is assumed to represent a narrow domain with sharp but continuous variation of quantities describing the solution. This narrow domain is called a structure of a discontinuity. Inside a structure the variation of quantities is described by a more complicated system of equations than outside the structure. The terms that distinguish the internal system of equations from the external one form a singular perturbation of the latter system. Tendency of this perturbation to zero does not mean in general that the corresponding solutions of the internal system approach solutions of the external one. Those solutions of the internal system of equations that turn into solutions of the limiting external system when the singular perturbation tends to zero represent solutions with structure. A structure of a discontinuity can be represented either by a stationary solution (of traveling wave type) or by a nonstationary solution (see, e.g., the discussion of this issue in [17, Ch. 7]). In the studies [2, 4, 14–17] dealing with the Hopf equation with complicated nonlinearity, the selection of solutions used for constructing a solution to the problem of decay of an arbitrary discontinuity included the requirement of existence of a stationary structure of discontinuities. The structure of discontinuities was described by the generalized Korteweg–de Vries–Burgers (KdVB) equation. It was noted in those studies that not all discontinuities with structure are suitable for this purpose [4]. The appearance of the recent studies [5, 12], in which the stability of solutions describing a structure of discontinuities is examined, makes it possible to effectively include the requirement of stability of a structure in the notion of admissibility of a discontinuity, and to revise from this point of view the results obtained earlier. The indicated studies analyzed the spectral (linear) stability of stationary structures described by the generalized KdVB equation with a nonconvex potential of a specific form. In [12] the stability of stationary structures of special discontinuities (heteroclinic structures doubly asymptotic to different saddles) was studied. It was shown that among a variety a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.
E-mail addresses:
[email protected] (A.T. Il’ichev),
[email protected] (A.P. Chugainova).
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of such discontinuities only one type of discontinuities with monotonous structure is stable. Special discontinuities of other types (with nonmonotonous structure) are unstable. Numerical simulation of the process of development of perturbations with nonmonotonous structure shows the decay of a special discontinuity and the impossibility of its realization [4, 15]. In [5] the stability of nonspecial admissible discontinuities with structure (heteroclinic structures doubly asymptotic to points one of which is a saddle and the other is either a focus or a node) was studied and various stable and unstable structures were revealed. In problems related to the decay of an arbitrary discontinuity, among other states, singular limits of heteroclinic states are formed to which the data that determine the configuration of the initial discontinuity evolve. These states necessarily correspond to solutions of the model equation (or system of equations) in question. They determine the configuration of stationary (corresponding to a traveling wave) or nonstationary discontinuous solutions. For a given solution to be asymptotically realizable in the problem of decay of an arbitrary discontinuity, it is necessary that this solution have a structure arising in small-scale examination (in particular, in the presence of dissipation or dispersion, or both together; they are essential on small scales and in a number of cases provide a mechanism that yields the existence of a structure). Discontinuities with a stationary structure were called admissible in the literature [6–8, 14, 17]. Nevertheless, the indicated necessary condition is not sufficient to select the structures that arise during the evolution of an arbitrary discontinuity: an illustrative example is given by the multiple nonuniqueness in the problem of decay of an arbitrary discontinuity for the generalized Hopf equation [14, 15]. The nonuniqueness means that the Cauchy data in the form of an arbitrary discontinuity at which, in general, shock conditions are not fulfilled can evolve in various ways, and the solution of the Cauchy problem is constructed from various combinations of admissible discontinuities and nonbreaking simple waves. There may be several such combinations, despite the imposed requirement of the existence of a structure of discontinuities included in these combinations. In [3] a sufficient condition for selecting solutions is proposed that allows one to guarantee the uniqueness of the asymptotics formed in the problem in question. This is made on the basis of an example of the simplest model KdVB equation with nonmonotonous potential of a specific form that has various solutions of discontinuity-with-structure type. This sufficient condition is the spectral stability of solutions under consideration. To investigate this stability, we use the well-known approach of [18]. By admissible discontinuities (i.e., realizable in practice in the problem of decay of an arbitrary discontinuity) we now mean discontinuities with structure that have the stability property. Introducing the stability requirement in the notion of an admissible discontinuity leads to substantial reduction of the set of admissible discontinuities described in [14, 15] (where all discontinuities with structure were considered admissible), and this eliminates the nonuniqueness of a solution to the problem of decay of an arbitrary discontinuity that was discovered in the previous studies [14]. We note here that considerable attention has been recently paid to problems of continuum mechanics that are based on nonlinear elasticity models (see, e.g., [9–11, 13], where nonlinear waves in hydrodynamic media occupying domains with elastic boundaries are treated). Wave processes in elastic media are as a rule described by a hyperbolic system of equations, and hence discontinuities appear inevitably. In contrast to gas dynamics, such discontinuities turn out to be nonunique, thereby giving a fundamental example of nonuniqueness of a shock wave in continuum mechanics [16]. Therefore, the problem of selecting admissible discontinuous solutions (shock waves) is extremely topical in this field. Thus, the investigation of stability of heteroclinic states (including those of the KdVB equation) plays a key role in the correct solution of the problem of decay of an arbitrary discontinuity. In the present paper, the analysis of stability of heteroclinic solutions of the KdVB equation with a convex potential that was performed in [18] is generalized to the case of an arbitrary potential that gives rise to heteroclinic states. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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The paper is organized as follows. In Section 2 we formulate the problem whose solutions are to be analyzed for stability. In Section 3 we briefly present the stability theory of heteroclinic solutions of the KdVB equation. Section 4 contains the proof of a lemma on continuous asymptotic behavior of the Evans function, which is analytic in an unbounded domain Ω+ on the complex plane of the spectral parameter λ, as |λ| → ∞. In Section 5 we give examples of stable and unstable heteroclinic solutions of the KdVB equation for a nonconvex potential of special form. 2. FORMULATION OF THE PROBLEM Let us consider the generalized KdVB equation ∂ϕ(v) ∂2v ∂3v ∂v + = μ 2 −m 3, ∂t ∂x ∂x ∂x
m, μ = const,
(2.1)
v = v(x, t).
On the right-hand side of (2.1), the term containing the coefficient m describes dispersive effects (the coefficient m is the dispersion parameter). The term containing the coefficient μ allows for viscous effects and causes dissipation (the coefficient μ is the dissipation parameter). In this paper a general potential ϕ(v) will be treated. We make the following assumption about this potential. Assumption. There exists a heteroclinic solution with stationary structure that is described by the following equations: ξ = x − W t,
v = v(ξ), m
dv d2 v = W v − ϕ(v), −μ dξ 2 dξ
lim v(ξ) = vl ,
lim v(ξ) = vr .
ξ→−∞
ξ→+∞
We investigate the stability of solutions of traveling wave type that represent discontinuity structures of solutions of the equation ∂ϕ(v) ∂v + = 0. ∂t ∂x
(2.2)
Equation (2.1) transforms into this equation when we consider solutions characterized by a large scale L of variation of quantities with respect to the variable x (in (2.1) both terms on the right-hand side become small in comparison with the terms on the left-hand side). Equation (2.2) can be called the generalized Hopf equation, and it coincides with the classical Hopf equation when ϕ(v) is a quadratic function of v. Equation (2.2) (as well as (2.1)) expresses a conservation law; therefore, the corresponding relation at a discontinuity can be written as W =
[ϕ(v)] . [v]
(2.3)
Here W is the speed of the discontinuity, and the square brackets denote the difference of values of a function in front of and behind the discontinuity. For μ = 0 we change the variables in (2.1) as follows: √ t → t m,
√ x → x m,
√ γ=
m , μ
ϕ(v) = f (u + vr ),
v = u + vr .
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Equation (2.1) in the new variables takes the form 1 ∂2u ∂f (u) ∂3u ∂u = . + + ∂t ∂x ∂x3 γ ∂x2
(2.5)
The stationary structure of discontinuities is described by the equations u = u(ξ),
ξ = x − W t,
d2 u 1 du = W u − f (u), − 2 dξ γ dξ lim u(ξ) = ul ,
ξ→−∞
(2.6)
lim u(ξ) = ur ≡ 0.
ξ→+∞
If a solution of problem (2.6) exists, then from the point of view of large length scale it must represent a discontinuity with stationary structure in which u = ur ≡ 0 and u = ul are the states in front of (ξ > 0) and behind (ξ < 0) the discontinuity. The states ul and ur satisfy (2.3) and represent, therefore, the states behind and in front of the discontinuity corresponding to the conservation law. 3. SPECTRAL STABILITY OF HETEROCLINIC SOLUTIONS Below we present the stability theory developed for heteroclinic solutions of the KdVB equation in the case of a monotonous potential in [18]. This theory is also applicable to a potential f (u) of general form under the assumption made above. To investigate the spectral (linear) stability of a heteroclinic stationary solution to equation (2.5), we look for a solution of the form u(x, t) = U (ξ) + w(ξ, t). The function w(ξ, t) satisfies the equation ∂ ∂2w 1 ∂w ∂w = − 2 + + W − f (U (ξ)) w , ∂t ∂ξ ∂ξ γ ∂ξ
f (U ) =
df (U ) . dU
(3.1)
The function U (ξ) obeys equation (2.6) and the conditions lim U (ξ) = ul ,
lim U (ξ) = 0.
ξ→−∞
ξ→+∞
Definition 1. A solution U (ξ) of equation (2.6) is said to be spectrally (linearly) unstable if there exists a solution to equation (3.1) of the form w(ξ, t) = eλt Y (ξ) with Re λ > 0. Thus, it follows from (2.6) that the function Y (ξ) must satisfy the linear inhomogeneous ordinary differential equation with coefficients depending on W and λ. This equation represents the spectral problem Aγ Y = λY (ξ), ∂ ∂ 2 Y (ξ) 1 ∂Y (ξ) − + W − f (U (ξ)) Y (ξ) , + Aγ Y = (3.2) ∂ξ ∂ξ 2 γ ∂ξ Y (ξ) → 0
as
ξ → ±∞.
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From (3.2) for ξ → ±∞ one has Y (ξ) ∼ y± (ξ), λy+ (ξ) = −
∂ 3 y+ (ξ) 1 ∂ 2 y+ (ξ) ∂y+ (ξ) , + + (W − f (0)) 3 2 ∂ξ γ ∂ξ ∂ξ
(3.3)
∂ 3 y− (ξ) 1 ∂ 2 y− (ξ) ∂y− (ξ) . + + (W − f (ul )) λy− (ξ) = − 3 2 ∂ξ γ ∂ξ ∂ξ Equations (3.3) have solutions of the form y± (ξ) = eν P+ (ν) = 0 and P− (ν) = 0, respectively, where P+ (ν) = ν 3 −
1 2 ν − (W − f (0))ν + λ, γ
±ξ
. The quantities ν ± satisfy the equations
P− (ν) = ν 3 −
1 2 ν − (W − f (ul ))ν + λ. γ
(3.4)
For pure imaginary ν = iτ , from (3.4) we have 1 λ = − τ 2 + iτ (τ 2 + W − f (0)), γ
1 λ = − τ 2 + iτ (τ 2 + W − f (ul )). γ
(3.5)
Following [18], we denote by Ω+ the component containing the right λ-half-plane bounded from the left by the rightmost curve (3.5). Equations (3.5) imply Lemma 1. The continuous spectrum of the operator Aγ in (3.2) lies entirely in the left halfplane {λ, Re λ < 0}, while the right half-plane {λ, Re λ > 0} can only contain isolated eigenvalues. We note that because of the translational invariance of (2.5), λ = 0 is always an eigenvalue of (3.2) with the eigenfunction ∂ξ U (ξ). Let the polynomials (3.4) for λ ∈ Ω+ have roots νj± (λ), j = 1, 2, 3, respectively. Then from (3.4) we also obtain Lemma 2. Re ν1± (λ) < 0 < Re νk± (λ), k = 2, 3, for λ ∈ Ω+ . Proof. It is easily seen that (3.4) have pure imaginary roots if and only if λ obeys equation (3.5) for the continuous spectrum. It then follows immediately that the roots νi , i = 1, 2, 3, of (3.4) may cross the imaginary axis only when λ crosses the rightmost curve (3.5), and for λ ∈ Ω+ their number in the right and left half-planes remains constant. At the same time, for |λ| large it is evident that ν ± = (−λ)1/3 +
1 + O(λ−1/3 ). 3γ
(3.6)
It follows immediately from (3.6) that for λ ∈ Ω+ one root lies in the left half-plane and the other two lie in the right half-plane. Equation (3.2) has a solution Y + (ξ, λ) (corresponding to the exponentially decaying solution exp(ν1+ (λ)ξ) of the first equation in (3.3)): Y + (ξ, λ) ∼ exp(ν1+ (λ)ξ)
as ξ → ∞.
We define the Evans function with the help of the solution Y + (ξ, λ). Definition 2. The scattering coefficient D(λ) in Y + (ξ, λ) ∼ D(λ) exp(ν1− (λ)ξ)
as ξ → −∞
is called the Evans function. Theorem 1 [18]. A number λ = λ0 ∈ Ω+ is an eigenvalue of Aγ if and only if D(λ0 ) = 0. Let us treat equation (3.2) as the system of equations dy = A(ξ, λ)y, dξ PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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⎛
y = Y (ξ, λ), Y (ξ, λ), Y (ξ, λ) ,
⎜ A(ξ, λ) = ⎝ −λ −
0 0
df (U (ξ)) dξ
153
⎞ 1 0 0 1 ⎟ ⎠. −1 W − f (U (ξ)) γ
The adjoint equation to (3.7) has the form dz = −zA(ξ, λ), dξ
(3.8)
where z is a three-component row vector. The quantities ν ± (λ) are the eigenvalues of the asymptotic matrices A± (λ) = lim A(ξ, λ). ξ→±∞
Lemma 2 provides the existence of solutions ζ + (ξ, λ) of (3.7) and η − (ξ, λ) of (3.8) that are analytic with respect to λ in Ω+ (see [19]) and such that +
lim e−ν1 ξ ζ + (ξ, λ) = r(λ),
ξ→∞
−
lim eν1 ξ η − (ξ, λ) = l(λ).
(3.9)
ξ→−∞
Here (A+ − ν1+ (λ)I)r(λ) = 0,
l(λ)(A− − ν1− (λ)I) = 0,
or (recall that the prime denotes differentiation)
l = P− (ν1− )−1 ν1− (ν1− − γ −1 ) − W + f (ul ), ν1− − γ −1 , 1 . r = 1, ν1+ , (ν1+ )2 , If y(ξ, λ) and z(ξ, λ) are arbitrary solutions to (3.7) and (3.8), respectively, then z · y (ordinary product in R3 ) is independent of ξ. Lemma 3 [18, 19]. D(λ) = η − · ζ + . For λ ∈ Ω+ , the roots ν1± (λ) are simple, which follows from Lemma 2. Then analyticity of r(λ) and l(λ) in this domain follows, which in turn implies analyticity of the solutions ζ + and η − (see, e.g., [1, (4.3)] for analogous constructions). Then Lemma 3 implies analyticity of the Evans function D(λ). Therefore, the following lemma holds. Lemma 4. The Evans function D(λ) is analytic for λ ∈ Ω+ . Moreover, the Evans function can be analytically continued through the imaginary axis while the inequality Re ν1± (λ) < Re νk± (λ),
(3.10)
k = 2, 3,
holds and, consequently, the roots ν1± involved in the construction of the Evans function are simple. Indeed, for λ = 0 we have (without loss of generality assume that 1/γ 2 + W − f (0) > 0 and 1/γ 2 + W − f (ul ) > 0) 1 1 1 1 + + + + W − f (0) < ν2 = 0 < ν3 = + + W − f (0), ν1 = − γ γ2 γ γ2 1 1 1 1 − − − + W − f (ul ) < ν2 = 0 < ν3 = + + W − f (ul ). ν1 = − γ γ2 γ γ2 Inequality (3.10) obviously remains valid in some neighborhood of zero. Thus, the following lemma is true. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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Lemma 5. The Evans function D(λ) can be analytically continued to the domain − 0 < Re λ for some 0 > 0. The following lemma also holds for the asymptotic behavior of the Evans function when λ tends to the infinitely remote point of the complex plane. Lemma 6. D(λ) → 1 as |λ| → ∞. According to Theorem 1, the problem of finding the unstable eigenvalues of the operator Aγ (eigenvalues located in Ω+ ) is analogous to determining the zeroes of the Evans function D(λ) that are also located in Ω+ . The number of zeroes of D(λ) can be computed using the argument principle. According to Lemma 6, D(λ) is continuous at infinity, and from Lemma 5 it follows that the imaginary axis belongs to the domain of continuity of D(λ); therefore, the number of zeroes of D(λ) in Ω+ is determined by the number of revolutions of the image of the imaginary axis {λ : λ = iτ, τ ∈ R} under the mapping D(·). Thus, it is necessary to construct the function D(λ) on the imaginary axis, i.e., to solve numerically the ordinary differential equations (3.7) and (3.8) under conditions (3.9) for λ lying in a sufficiently large symmetric segment of the imaginary axis [18]. The stability of special discontinuities (heteroclinic states doubly asymptotic to singular points of a saddle type) and nonspecial discontinuities (heteroclinic states connecting a saddle with either a focus or a node) for the nonconvex potential of the particular form ϕ(v) = v 4 − v 2
(3.11)
was studied in [12] and [3], respectively, in the context of the problem of decay of an arbitrary discontinuity. At the same time, the proof of Lemma 6 was omitted. This proof is given in the next section. 4. PROOF OF LEMMA 6 Let us define
V± (λ)
as the matrices of right (column) eigenvectors of A± (λ), ⎛ ⎞ 1 1 1 ⎜ ± ν2± ν3± ⎟ V± (λ) = ⎝ ν1 ⎠, (ν1± )2 (ν2± )2 (ν3± )2
and define W± (λ) as the corresponding matrices of left (row) eigenvectors, ⎛ + −1 + + ⎞ P+ (ν1 ) {ν1 (ν1 − γ −1 ) − W + f (0)} P+ (ν1+ )−1 (ν1+ − γ −1 ) P+ (ν1+ )−1 ⎜ ⎟ W+ (λ) = ⎝P+ (ν2+ )−1 {ν2+ (ν2+ − γ −1 ) − W + f (0)} P+ (ν2+ )−1 (ν2+ − γ −1 ) P+ (ν2+ )−1 ⎠ , ⎛
P+ (ν3+ )−1 {ν3+ (ν3+ − γ −1 ) − W + f (0)} P+ (ν3+ )−1 (ν3+ − γ −1 ) P+ (ν3+ )−1 P− (ν1− )−1 {ν1− (ν1− − γ −1 ) − W + f (ul )} P− (ν1− )−1 (ν1− − γ −1 ) P− (ν1− )−1
⎞
⎜ ⎟ W− (λ) = ⎝P− (ν2− )−1 {ν2− (ν2− − γ −1 ) − W + f (ul )} P− (ν2− )−1 (ν2− − γ −1 ) P− (ν2− )−1 ⎠ . P− (ν3− )−1 {ν3− (ν3− − γ −1 ) − W + f (ul )} P− (ν3− )−1 (ν3− − γ −1 ) P− (ν3− )−1
Owing to the normalization of the eigenvectors, we have W± (λ)V± (λ) = I,
W± (λ)A± (λ)V± (λ) = diag(ν1± , ν2± , ν3± ).
Further let us consider matrices F± (ξ, λ): F+ (ξ, λ) = W+ (λ) A(ξ, λ) − A+ (λ) V+ (λ), F− (ξ, λ) = W− (λ) A(ξ, λ) − A− (λ) V− (λ),
ξ > 0, ξ < 0.
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As a result we have
df (U (ξ)) + − νj f (U (ξ)) − f (0) , − [F ]ij (ξ, λ) = P dξ df (U (ξ)) − − −1 − − νj f (U (ξ)) − f (ul ) , − [F ]ij (ξ, λ) = P (νi ) dξ
+
(νi+ )−1
Define a matrix F(ξ, λ) as
F(ξ, λ) =
F+ (ξ, λ),
ξ > 0,
F− (ξ, λ),
ξ < 0.
ξ > 0, (4.1) ξ < 0.
(4.2)
From (4.1) and (4.2) we have ∞ [F(ξ, λ)]ij dξ < ∞
(4.3)
−∞
for fixed λ, and from (3.4), (3.6), (4.1), and (4.2) we have [F(ξ, λ)]ij → 0
uniformly as |λ| → ∞.
(4.4)
Conditions (4.3) and (4.4) imply the validity of Proposition 1.17 in [19], which yields W+ (λ)ζ + = e1 + o(1),
η − V− (λ)e1 = 1 + o(1),
λ → ∞,
(4.5)
where e1 = (1, 0, 0) . Further note that (3.6) implies V+ (λ) = V− (λ) · (I + o(1)).
(4.6)
Then from (4.5) and (4.6) one has D(λ) = η − · ζ + = η − V+ (λ) · W+ (λ)ζ + = 1 + o(1),
|λ| → 1.
5. DISCUSSION As an example of application of the above theory we consider equation (2.5) with the nonconvex potential f (u) obtained from (3.11) by the substitution (2.4) with vr = −1: f (u) = (u − 1)4 − (u − 1)2 .
(5.1)
As already mentioned, the problem of decay of an arbitrary discontinuity as described by (2.5) with the potential (5.1) was discussed in [3]. This potential f (u) was selected, in particular, because (2.5) in this case admits a rich variety of heteroclinic solutions, including special discontinuities that are absent in the case of standardly used convex potentials. For a sufficiently large value of γ, the set of those states ul (with ur = 0 fixed) for which there exist stationary heteroclinic solutions of (2.5) consists of segments and separate points of the u-axis. In the figure this set is indicated on the graph of f (u) by the bold segments (curves AB0 , B1 B2 , B3 B4 , D0 D1 , D2 D3 , and the part of the graph above the point D4 ) and the separate points Ci , i = 0, . . . , 4, corresponding to special discontinuities. This set was constructed numerically [15] for the parameters ur = 0, m = 1.3, and μ = 0.05 (γ ≈ 22.8). Moreover, the following theorem holds. Theorem 2 [14]. Equation (2.6) either has no solutions or has a unique solution with specified asymptotic values. For a sufficiently large value of γ, the set of states ul for given ur consists of PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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A.T. IL’ICHEV, A.P. CHUGAINOVA M f (u) A
B2 B3 B4
u D4
C4 C3 B1 B0
D3
C2
D2
C1 C0 P
D1 D0
Set of discontinuities with stationary structure for γ ≈ 22.8.
segments and separate points of the u-axis. This set is shown in the figure on the graph of f (u) by bold segments and a finite number of separate points Ci , i = 0, 1, . . . . The structure of the discontinuities A → Ci represents an integral curve connecting singular points of saddle type. Definition 3. For every i the discontinuity A → Ci is said to be special. The number of special discontinuities grows when the parameter γ grows, and for γ < γ0 special discontinuities are absent. In problems related to an arbitrary discontinuity decay, in a number of cases singular limits of heteroclinic states are formed. The data that determine the configuration of an initial discontinuity evolve to these singular limits, which must necessarily be solutions of the corresponding model equation (or system of equations). These limits correspond, in particular, to a stationary discontinuous solution (corresponding, in turn, to a traveling wave). A necessary condition for this solution to be realizable in the asymptotics of the problem of decay of an arbitrary discontinuity consists in the existence of its structure, i.e., a smooth domain of the solution front. This smooth domain arises under small-scale consideration. Nevertheless, this necessary condition is not sufficient to select structures that arise when an arbitrary discontinuity evolves. To make a selection of solutions that yields uniqueness of the asymptotics of the problem in question, it suffices to require, along with the existence of a structure, the stability of the solutions under consideration. Further, for simplicity, we restrict ourselves to the consideration of heteroclinic solutions emanating from the point A in the figure. Let the point F on the graph of f (u) (see the figure) correspond to the state behind a discontinuity. The results of [3] imply the following: • if F belongs to the segment AB0 , then the stationary structure A → F is stable and its speed W is determined from (2.3); • if F belongs to the segment B2i−1 B2i , i = 1, . . . (the segments of existence of a stationary structure), then the stationary structure A → F is unstable; • for any value of γ greater than γ0 , starting with which the first structure A → C0 of special discontinuity exists, the monotonous structure A → C0 is stable and all other structures A → Cj with integer j ≥ 1 (if any) are unstable; • if F belongs to the segment P D0 , then the stationary structure A → F is stable and its speed W is determined from (2.3); • if F belongs to the segments D0 D1 , D2 D3 , and D4 M (the segments of existence of a stationary structure), then the stationary structure A → F is unstable; PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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• if F lies on the graph of f (u) above the point M , then the stationary structure A → F is stable and its speed W is determined from (2.3). Note that stable stationary structures form the asymptotics of the decay of corresponding arbitrary discontinuities (with the final state corresponding to the point F located on “stable” parts of the graph of f (u)). ACKNOWLEDGMENTS This work is supported by the Russian Science Foundation under grant 14-50-00005. REFERENCES 1. J. C. Alexander and R. Sachs, “Linear instability of solitary waves of a Boussinesq type equation: A computer assisted computation,” Nonlinear World 2 (4), 471–507 (1995). 2. A. P. Chugainova, “Nonstationary Solutions of a Generalized Korteweg–de Vries–Burgers Equation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 281, 215–223 (2013) [Proc. Steklov Inst. Math. 281, 204–212 (2013)]. 3. A. P. Chugainova, A. T. Il’ichev, A. G. Kulikovskii, and V. A. Shargatov, “Problem of an arbitrary discontinuity disintegration for the generalized Hopf equation: Selection conditions for the unique solution,” IMA J. Appl. Math. (in press). 4. A. P. Chugainova and V. A. Shargatov, “Stability of nonstationary solutions of the generalized KdV–Burgers equation,” Zh. Vychisl. Mat. Mat. Fiz. 55 (2), 253–266 (2015) [Comput. Math. Math. Phys. 55, 251–263 (2015)]. 5. A. P. Chugainova and V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV– Burgers equation,” Zh. Vychisl. Mat. Mat. Fiz. 56 (2), 259–274 (2016) [Comput. Math. Math. Phys. 56, 263–277 (2016)]. 6. I. M. Gel’fand, “Some problems in the theory of quasilinear equations,” Usp. Mat. Nauk 14 (2), 87–158 (1959) [Am. Math. Soc. Transl., Ser. 2, 29, 295–381 (1963)]. 7. S. K. Godunov, “On nonunique ‘blurring’ of discontinuities in solutions of quasilinear systems,” Dokl. Akad. Nauk SSSR 136 (2), 272–273 (1961) [Sov. Math., Dokl. 2, 43–44 (1961)]. 8. S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Kluwer, New York, 2003). 9. A. T. Il’ichev, “Envelope solitary waves and dark solitons at a water–ice interface,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 289, 163–177 (2015) [Proc. Steklov Inst. Math. 289, 152–166 (2015)]. 10. A. T. Il’ichev, “Soliton-like structures on a water–ice interface,” Usp. Mat. Nauk 70 (6), 85–138 (2015) [Russ. Math. Surv. 70, 1051–1103 (2015)]. 11. A. T. Il’ichev, “Solitary wave packets beneath a compressed ice cover,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 32–42 (2016) [Fluid Dyn. 51 (3), 327–337 (2016)]. 12. A. T. Il’ichev, A. P. Chugainova, and V. A. Shargatov, “Spectral stability of special discontinuities,” Dokl. Akad. Nauk 462 (5), 512–516 (2015) [Dokl. Math. 91 (3), 347–351 (2015)]. 13. A. T. Il’ichev and Y. B. Fu, “Stability of an inflated hyperelastic membrane tube with localized wall thinning,” Int. J. Eng. Sci. 80, 53–61 (2014). 14. A. G. Kulikovskii, “A possible effect of oscillations in the structure of a discontinuity on the set of admissible discontinuities,” Dokl. Akad. Nauk SSSR 275 (6), 1349–1352 (1984) [Sov. Phys., Dokl. 29 (4), 283–285 (1984)]. 15. A. G. Kulikovskii and A. P. Chugainova, “Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena,” Zh. Vychisl. Mat. Mat. Fiz. 44 (6), 1119–1126 (2004) [Comput. Math. Math. Phys. 44, 1062–1068 (2004)]. 16. A. G. Kulikovskii and A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory,” Usp. Mat. Nauk 63 (2), 85–152 (2008) [Russ. Math. Surv. 63, 283–350 (2008)]. 17. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman & Hall/CRC, Boca Raton, FL, 2001), Monogr. Surv. Pure Appl. Math. 118. 18. R. L. Pego, P. Smereka, and M. I. Weinstein, “Oscillatory instability of traveling waves for a KdV–Burgers equation,” Physica D 67 (1–3), 45–65 (1993). 19. R. L. Pego and M. I. Weinstein, “Eigenvalues, and instabilities of solitary waves,” Philos. Trans. R. Soc. London A 340 (1656), 47–94 (1992).
Translated by the authors PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
Vol. 295
2016