The long-time behavior of solitary waves for the weakly damped KdV equation

In this paper, we first introduce the long-time behavior stability of solitary waves for the weakly damped Korteweg–de Vries equation. More concretely, solutions of the dissipative system with the initial values near a c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}$\end{document}-speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed |c(t)−c0|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|c(t)-c_{0}|$\end{document} being small, in the long-time period (i.e., 0⩽t⩽O(1ϵτ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leqslant t\leqslant O(\frac{1}{\epsilon ^{\tau}})$\end{document}). Meanwhile, the approximation difference in a suitably weighted space Ha1(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}_{a}(\mathbb{R})$\end{document} is of the order of the damping coefficient and of some kind of exponential weight form. As a comparison, we also study the long-time behavior stability, i.e., for 0⩽t<+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leqslant t<+\infty $\end{document}, the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c(t)=c0e−βt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c(t)= c_{0}e^{-\beta t}$\end{document} (0<β⩽1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\beta \leqslant 1$\end{document}), when the initial values are near a c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}$\end{document}-speed solitary wave. However, here, the approximation difference merely defined in H1(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}(\mathbb{R})$\end{document} depends on the damping coefficient ϵ and the exponential decay coefficient β.

To give a more explicit picture, we describe the undamped and damped KdV equations separately.
Undamped Case: If = 0 in equation (1.1), one can define the Hamiltonian (1.2) and the impulse functional (see [9,10]) Obviously, the profiles of traveling-wave solutions of the KdV equation are critical points of the Hamiltonian H for fixed values of I, namely, relative equilibria (see [4]). The family of all traveling-wave profiles is called the manifold of relative equilibrium (MRE), which is the two-dimensional manifold of the form u(x, t) = u c (xct + γ ) for all c > 0, γ ∈ R. In addition, the profile of the solitary wave conforms to u c (y) → 0 as |y| → ∞, i.e., u c (y) = αsech 2 ςy with α = 3c, ς = 1 2 √ c, (1.4) which uniquely (up to the space translations) satisfies the equation (see [11]) -∂ 2 y u c + cu c - A solitary wave has a permanent phase shift or a different speed when a solitary wave acquires a small perturbation. Therefore, the orbital stability of solitary waves was introduced in [12][13][14]. Weinstein in [15,16] and Bona, Souganidis, and Strauss in [17,18] asserted that a solution that is initially close to a solitary wave u c (xct) in the Sobolev space H 1 (R), will forever remain close to the set of translates u c (xct + γ ) of the wave. More precisely, for sufficiently small δ > 0, one has inf γ u(·, t)u c (· + γ ) H 1 ≤ δ, ∀t > 0, (1.6) if the same quantity is small at the initial time t = 0. In particular, Pego and Weistein showed the asymptotic stability of the traveling wave in [9] that if u(x, t) is initially a small perturbation in the weighted norms space H 2 (R)∩H 1 a (R) of a given solitary wave u c (xct + γ ), then u(x, t)u c + (xc + t + γ + ) H 2 (R)∩H 1 a (R) → 0 as t → +∞, (1.7) for some c + near c and γ + near γ . Here, the exponential weights are of the form e ay (a > 0) as follows: where ∂ x H(u) = -∂ x [u xx + 1 2 u 2 ]. Clearly, I(u(t)) = I(u(0))e -2 t . This implies that lim t→+∞ I(u(t)) = 0 and lim t→+∞ u(t, x) = 0 almost everywhere in R.
The authors in [19,20] used the symmetry group to reduce the energy momentum and then obtained the stability of relative equilibria. In [21,22], the authors analyzed the spectrum property of the self-adjoint operator generated by an energy functional, and then they found sharp conditions for the stability and instability of solitary waves or multisolitons. Specifically, Derks and Groesen in [23] considered the damped KdV equation in the bounded periodic domain x ∈ [0, 2π]. By applying the implicit theorem, they constructed an energy-decaying manifold M ∼ O(e -2εt ), which is related to the damping coefficient , and then they obtained the long-time behavior stability of solutions near the constructing manifold M , where the approximation difference is O( e -2εt ).
Here, inspired by the ideas about the spectral analysis given in [9] and the construction of the energy-decaying manifold given in [23], we study the long-time and long-time behavior, respectively, for the weakly damped equation (1.1) in the whole space x ∈ R. Our first result is about the long-time behavior: where 0 < a < c 3 , τ < 1 2 , C are constants.
Remark 1.1 1. The restriction 0 < a < c 3 is imposed in Theorem 1.1 since the expression a(ca 2 ) is maximized at a = c 3 (see Proposition 2.5 in Ref. [9]). 2. It is natural to expect the solution to approximate the initial solitary wave as long as possible if the initial value has a slight perturbation. However, (1.10) implies that all solutions will vanish as t → +∞. Hence, it is valid to consider the stability near the initial solitary wave in the long-time period where T = O( 1 τ ) means the same order T ≈ 1 τ and the restraint on the quantity 1 τ follows from (2.35).
3. To analyze the property of the damping condition and solitary wave, the solution to equation (1.1) will be formally expressed in the form where θ (t) = γ (t) -t 0 c(s) ds and the leading (dominant) term u c(t) (x + θ (t)) is an exact solitary-wave solution of (1.1) with = 0, when c(t), γ (t) are just near the initial c 0 , γ 0 . 4. Substitution of (1.13) into (1.1) yields an equation of the form where (u c(t) , v) will be given in (2.6) and Meanwhile, differentiating (1.5) with respect to y and c, we know that the operator ∂ y L c in L 2 is degenerate, i.e., These give rise to solutions ∂ y u c and ∂ c u c -t∂ y u c to the linearized problem (1.17) 5. As in References [16,24], to obtain more exponential decay, it is appropriate to require that the right-hand side of (1.14) is orthogonal to the 2-dimensional generalized kernel of the adjoint of ∂ y L c . These constraints yield two coupled first-order differential equations for c(t) and γ (t) (called modulation equations), which are coupled to the infinitedimensional dispersive evolution equation for v(·, t).
Next, we discuss the long-time behavior stability of solutions. In contrast to the restriction c(t) near c 0 given in Theorem 1.1, we need that c(t) decays exponentially to zero as t → +∞. This is presented as follows:

19)
where C is a constant and m( , β, t) depends on , β, and t such that (1.20) Remark 1.2 Note that here it is impossible to consider the long-time stability of solutions in weight space H 1 a as in Theorem 1.1, since a → 0 as t → +∞ follows from a < c(t) 3 and c(t) = c 0 e -βt (0 < β ≤ 1). Remark 1.3 The approximation exponent given in (1.12) of Theorem 1.1 and (1.19) of Theorem 1.2 strictly depends on the damping coefficient . This is in sharp contrast to the asymptotic stability (1.7) with the exponent weight e -a(c-a 2 )t of decay given in Reference [9]. In other words, the weakly damped term will dominate the exponential decay rate.
The rest of this paper is organized as follows: In Sect. 2, we justify the representation (1.13) of the solution for nonlinear equations, and derive the equation of motion of the new variables (c(t), γ (t), w(y, t)). Moreover, we study the long-time behavior to finish the proof of Theorem 1.1. In Sect. 3, we also justify the new representation (3.1) and prove Theorem 1.2 for the long-time behavior stability. In the Appendix, we review the spectral analysis and certain smoothing and exponential decay estimates of the linearized operator ∂ y L c in (1.14).

The long-time behavior stability 2.1 Decomposition of the solution
Due to the weak damping term, we use time-dependent tubular coordinates in a neighborhood of solitary waves and skillfully represent solutions of the initial value problem (1.1) in the form (see also (1.13)) and u c(t) (y) belongs to the family of traveling waves. In order to achieve exponential decay for the perturbation v(y, t) in the weighted space H 1 a , we wish to impose the constraint that w(y, t) = e ay v(y, t) ∈ range(Q) = ker(P), where the projections P, Q are given in Proposition A.2 (see the Appendix). This requirement corresponds to the two scalar constraints w, η k = 0, k = 1, 2, cf. (A.14), which follows the modulation equations, namely, two coupled first-order differential equations for c(t), γ (t) as t > 0. As this point, let us begin the proof of Theorem 1.1.
The solution u(x, t) of the initial problem (1.1) satisfies, for ∀t > 0, Moreover, u is a classical solution of (1.1) for t > 0. Given the initial data in (1.11), if the We now derive evolution equations for γ (t), c(t), and v(y, t) that are valid pointwise for Hence, where we write Meanwhile, (2.4) implies that this equation is initially justified in C([0, t], H -3 ), but also holds in C([0, t], L 2 ) and moreover is pointwise. The constraint w ∈ range(Q) in (2.3) now yields the following system of evolution equations for (w, γ , c): Written as an integral equation, the initial value problem for (2.9) becomes: The equation PF = 0 yields equations forγ ,ċ as follows. Introduce the notation 11) and note that e ay ∂ y e -ay w, η k =v, ∂ yηk for k = 1, 2, by integration by parts. Then, by (A.14), the condition PF = 0 is equivalent to Using the biorthogonality relation ξ j , η k = δ jk , we obtain a system of equations for γ (t) and c(t): and (2.14) The matrix A(t) satisfies In order to obtain reversibility of the matrix A(t), in some sense, we need the term et I ≈ I. In other words, it is possible to consider stability in the long-time period 0 ≤ t ≤ T = O( 1 τ ) (given in (2.35)) instead of the long time "t → +∞". Otherwise, et I → 0 as t → +∞.

The long-time behavior
In order to complete the proof of Theorem 1.1. It remains to establish the priori estimates from the evolution equations in (2.10)-(2.13). We have Proposition 2.1 There exist δ * > 0, 0 > 0, C > 0 such that, if the decomposition (2.10), (2.11), and (2.12) exists for Proof The proof follows the two stages as given in Proposition 4.1 in Ref. [9] but with different detailed estimates. If δ * is sufficiently small and 0 ≤ t ≤ T = O( 1 τ ), then A(t) defined in (2.13) has a bounded inverse, so we may estimate (2.13) to find (2.18) From (2.8), using that e ay ∂ y e -ay = ∂ ya and the expression (1.4) (or the following estimate (3.19)), we obtain the estimates , satisfies the condition of Lemma A.2. We may then estimate (2.10) as follows, for t > 0: where the variable b is constrained in Remark A.1 (see the Appendix). Then, multiplying (2.20) by e (b+ )t , we find, for t > 0, Due to lim →0 + e -b τ -1 2 = 0, integrating (2.25) by parts, we have where the last inequality follows from the monotone theorem and the fact that Taking the supremum over 0 ≤ t ≤ T 0 (= O( τ )), we find that if δ * is sufficient small, then Next, we estimate |c(t)c 0 |. Using (2.18) and (2.31), we find that However, if we consider (2.33) on the long-time point T 0 (= O( τ )), we know that To obtain a small estimate |c(t)c 0 |, we need to consider the more appropriate long-time Obviously, it is sufficient to choose τ < 1 2 such that lim →0 + 1 τ 0 (e -2 ses ) ds = 0. Similarly, the fourth estimate of (2.17) holds.
Taking the supremum over 0 ≤ t ≤ T(= O( 1 τ )), we find that if δ * is sufficient small, then Remark 2.1 In some sense, there is a balance between the long-time point T = O( 1 τ ) and the exponent weight b = τ . In other words, if the long-time point is smaller, then the exponent weight of decay is larger. Here, we cannot obtain the exponent weight of decay e -a(c-a 2 )t as in Ref. [9] due to perturbation estimates (2.23) and (2.36) caused by the weakly damped term.
Proof (ii) H 1 estimate: We make use of the damping quantity Since u c 0 is a critical point of the functional E, we have for any z ∈ H 1 , Now, we take z = u(x, t)u c 0 (y) = et u c(t) (y) + v(y, t)u c 0 (y) above, and observe that δE At the same time, we estimate (2.43) as follows. Note that, for δ * sufficiently small, Then, for some k 1 > 0, Since e -ay u c 0 (y) is bounded in y, we may estimate where we have used the estimate ab ≤ δa 2 + C(δ)b 2 for a suitably small δ. Finally, since This completes the proof of Proposition 2.1, which implies the conclusions of Theorem 1.1.

A new decomposition of the solution
Note that, in the long-time stability case, the expression (2.15): may not be reversible as t → +∞, which is derived by setting the form of solution (2.1). Hence, we subtly analyze the following new form of the solution and u c(t) (y) belongs to the family of traveling waves with c(t) = c 0 e -βt (0 < β ≤ 1). Substituting (3.1) into (1.1), we similarly derive evolution equations for γ (t), c(t), and v(y, t) as follows: Hence, Therefore, Since here the speeds c(t) of the traveling wave will decay to zero, the exponential weight a(< c(t) 3 ) will also decay. On the other hand, due to the fifth item in Remark 1.1, we cannot initially set the exponential weight a = 0 in H 1 a . Otherwise, it may follow more than a 2-dimensional generalized kernel. Hence, in contrast to the long-time stability case by setting (2.3) to prove Theorem 1.1, we need to set w(y, t) = e a(t)y v(y, t), A a (t) = e a(t)y ∂ y L c(t) e -a(t)y and L c(t) = -∂ 2 y + c(t)u c(t) . Then, we deduce that where, for simplicity, writing a = a(t) if there is no risk of confusion, F = -e ay (ċ∂ c +γ ∂ y )u c(t) -γ e ay ∂ y e -ay w + F, Meanwhile, (2.4) implies that this equation is initially justified in C([0, t], H -3 ), but also holds in C([0, t], L 2 ) and moreover is pointwise. As in the long-time behavior case above, we wish to impose the similar projections P, Q given in Proposition A.2 w(y, t) = e ay v(y, t) ∈ range(Q) = ker(P). (3.8) However, here, we should denote them by P(t), Q(t). Indeed, the current assumption of Proposition A.2: 0 < a(t) < c(t) 3 depends on t, from which it follows that the ξ j = ξ j (t) and η k = η k (t) depend on t for j, k = 1, 2. This requirement corresponds to the two scalar constraints w, η k (t) = 0, k = 1, 2, cf. (A.14), which also generates the modulation equations, namely, two coupled first-order differential equations for c(t), γ (t) as t > 0. Hence, the constraint w ∈ range(Q) in (3.8) now yields the following system of evolution equations for (w, γ , c): Written as an integral equation, the initial value problem for (3.9) becomes: Then, similarly by (A.14), the condition PF = 0 is equivalent to Using the biorthogonality relation ξ j , η k = δ jk , we obtain a system of equations for γ (t) and c(t): and A(t) = e ay ∂ y u c(t) , η 1 + (∂ ya)w,η 1 , e ay ∂ c u c(t) , η 1 e ay ∂ y u c(t) , η 2 + (∂ ya)w,η 2 , e ay ∂ c u c(t) , η 2 = e ay ∂ y u c(t) , η 1 + e ay ∂ y e -ay w,η 1 , e ay ∂ c u c(t) , η 1 e ay ∂ y u c(t) , η 2 + e ay ∂ y e -ay w,η 2 , e ay ∂ c u c(t) , η 2 = e ay ∂ y u c(t) , η 1 , e ay ∂ c u c(t) , η 1 e ay ∂ y u c(t) , η 2 , e ay ∂ c u c(t) , η 2 + e ay ∂ y e -ay w,η 1 , e ay ∂ c u c(t) , η 1 e ay ∂ y e -ay w,η 2 , e ay ∂ c u c(t) , η 2 . (3.13) The matrix A(t) satisfies (3.14)

The long-time behavior
Now, we will estimate the weighted perturbation, w(y, t) = e ay v(y, t), in H 1 , via the integral equation (3.10), the modulation equation (3.12), and the linear semigroup estimates of Lemma A.2.
Since v L 2 decays to zero with respect to t, in the expression (3.14), A(t) has a bounded inverse as 0 ≤ t < +∞. We may estimate (3.12) to find (3.15) From (3.7), using that e ay ∂ y e -ay = ∂ ya, we obtain the estimates For simplicity, also writing b = b(t) if there is no risk of confusion, and then multiplying (3.17) by e bt , we find from (3.19) that, for t > 0, We first deal with the term I. For 0 ≤ t ≤ 1 and 0 < β ≤ 1, one can easily deduce that On the other hand, for 1 ≤ t < ∞, one has Since b(t) → 0 as t → +∞, we can choose any β > 0 satisfying βb > 0. Hence, one can (3.23)

Appendix
For the reader's convenience, we list out the spectral property and developing analysis of the operator A 0 = ∂ y L c given in (1.17) in the space L 2 and L 2 a . The interested reader is referred to References [9,25,26].

A.1 Spectral theory in L 2 and L 2 a
The spectrum of the operator A 0 = ∂ y L c on L 2 consists of a discrete spectrum (isolated eigenvalues of finite multiplicity) and an essential spectrum (everything else in the spectrum).
Lemma A.1 (Theorem 2.1 Ref. [9]) A 0 has no isolated eigenvalues whose spectrum coincides with the imaginary axis.
In fact, if λ is an eigenvalue of A 0 with L 2 -eigenfunction Y (y), then Since the solitary wave u c (y) → 0 at an exponential rate as |y| → ∞ (see (1.4)), it follows that the constant coefficient equation is Hence, the essential spectrum of A 0 = ∂ y L c is the imaginary axis and the corresponding eigenvalue function Y (y) exponentially decays to zero as y → ∞.
The following functions are also described in [9], in relation to the isolated eigenvalue Thus, the spectral theory of A 0 = ∂ y L c in L 2 a is equivalent to the spectral theory of A a in L 2 . Since u c (y) and ∂ y u c (y) decay to zero at an exponential rate as |y| → ∞, the essential spectrum of A a also agrees with the spectrum of the constant coefficient operator For the generalized eigenspaces of A a and its adjoint A * a = -e -ay L c ∂ y e ay , one has: Proposition A.2 (Proposition 2.8 Ref. [9]) Assume dI[u c ] dc = 0 and 0 < a < √ c/3. Then, λ = 0 is the only eigenvalue for A a with algebraic multiplicity two, and ker g (A a ) = ker A 2 a = span{ξ 1 , ξ 2 }, ker g A * a = ker A * 2 a = span{η 1 , η 2 }, (A. 13) where ξ j = e ayξ j and η j = e -ayη j for j = 1, 2, i.e., where θ 1 , θ 2 , and θ 3 are as in (A.5). In addition, the ξ j and η k are biorthogonal, with ξ j , η k = δ jk for j, k = 1, 2. Thus, the spectral projection P for A a , associated with the eigenvalue λ = 0, and the complementary spectral projection Q, are given by w, η k ξ k , (A. 16) for w ∈ L 2 . These projections satisfy PA a w = A a Pw, QA a w = A a Qw, for w ∈ dom A a . Since u c exponentially decays to zero as |y| → ∞, the coefficients in (A.18) converge to those of the free evolution equation

A.2 Decay of smoothing estimates
Using the Fourier transform, one obtains: Proposition A.3 (Proposition 4.1 Ref. [9]) For any integer n ≥ 0, and 0 < a < √ c/3, there exists C = C(n, a) such that, for any w ∈ L 2 and for all t > 0, ∂ n y e A 0 a t w L 2 ≤ Ct -n/2 e -a(c-a 2 )t w L 2 . (A.21)