Solitary waves of the rotation-generalized Benjamin-Ono equation

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.


Introduction
In the present paper we are concerned with studying the rotation-generalized Benjamin-Ono (RGBO) equation which can be written as (u t + βH u xx + (f (u)) x ) x = γu, x ∈ R, t > 0, (1.1) where γ > 0 and β = 0 are real constants, f is a C 2 function which is homogeneous of degree p > 1 such that sf (s) = pf (s), and H denotes the Hilbert transform defined by where p.v. denotes the Cauchy principal value. When f (u) = 1 2 u 2 , equation (1.1), which is so-called the rotation-modified Benjamin-Ono (RMBO) equation, models the propagation of long internal waves in a deep rotating fluid [15,18,22,30]. In the context of shallow water the propagation of long waves in rotating fluid is described by the Ostrovsky equation [10,16,27,28] (u t + βu xxx + (u 2 ) x ) x = γu, x ∈ R, t > 0, (1.2) which is also called the rotation-modified Korteweg-de Vries (RMKdV) equation. See also [13,14] for the two-dimensional long internal waves in a rotating fluid. The parameter γ is a measure of the effect of rotation. Setting γ = 0 in (1.1), integrating the result with respect to x and setting the constant of integration to zero, one obtains the generalized Benjamin-Ono (GBO) equation u t + βH u xx + (f (u)) x = 0. (1.3) Most attention in this work is paid to the existence, the stability and the properties of localized traveling waves (commonly referred to as solitary waves) of (1.1). Using variational methods and the Pohozaev-type identities, we prove the existence and nonexistence of solitary waves for a range of the parameters of (1.1). We also show that our solitary waves (of (1.1)) are the ground states, i.e. they have minimal action. We also consider the effect of letting the rotation parameter γ approach zero. Actually we show that the ground state solitary waves converge to solitary waves of the GBO equation.
It was shown by Linares and Milanes [22] that the RMBO equation (1.1) is well-posed in the space for s > 3/2, where the operator ∂ −1 x is defined via the Fourier transform as ∂ −1 x g(ξ) = (iξ) −1 g(ξ). The methods therein also imply the same result for the RGBO equation (1.1).
These conserved quantities play an important role in our stability analysis.
We show in Theorems 6.1 and 7.2 that the function d(c) defined by (6.1) determines the stability of the solitary waves in the sense that if d (c) > 0, then G (β, c, γ) is X -stable, while if d (c) < 0, then O ϕ is X -unstable, where the space X is defined in (1.7). In Theorem 8.1, we use the ideas of [17], and provides sufficient conditions for instability directly in terms of the parameters β, γ and p.
We also investigate the properties of the function d(c) which determines the stability of the ground states. Using an important scaling identity, together with numerical approximations of the solitary waves, we are able to numerically approximate d(c). Remark 1.1 It is noteworthy that despite our regularity assumption on f , one can observe that all our results are valid for the nonlinearity f (u) = −|u|u.
For s ∈ R, we denote by H s (R), the nonhomogeneous Sobolev space defined by the closure ϕ ∈ S (R) : ϕ H s (R) < ∞ , with respect to the norm ϕ H s (R) = (1 + |ζ|) s 2 ϕ(ζ) where S (R) is the space of tempered distributions. Let X be the space defined by with the norm f X = f H 1/2 (R) + ∂ −1 x f L 2 (R) .

Solitary Waves
By a solitary wave solution of the RGBO equation, we mean a traveling-wave solution of equation (1.1) of the form ϕ(x − ct), where ϕ ∈ X and c ∈ R is the speed of wave propagation. Alternatively, it is a solution ϕ(x) in X of the stationary equation We will prove existence of solitary waves in the space X by considering the following variational problem. Define the functionals and K(u) = −(p + 1) R F (u)dx; (2.3) and consider the following minimization problem for some λ > 0. First we observe that M λ > 0 for any λ > 0. In fact, for c ≤ 0 max{β, c, γ} u 2 where c 1 = 1 − 1 2 + 2c 3 27γβ 2 and c 2 = 27β 2 −4c 3 27γβ 2 +4c 3 . On the other hand, for u ∈ X , we have Indeed, by using the Sobolev embedding and an interpolation, we find Then the Cauchy-Schwarz inequality implies that Now inequality (2.7) is obtained from (2.8) and (2.9). We also note that u 2 X is equivalent to I(u). Indeed, (2.5), (2.6) and the inequality lead to the coercivity condition I(u) ∼ u 2 X . Thus, it can be deduced from (2.7) that Then if ψ ∈ X achieves the minimum of problem (2.4), for some λ > 0, then there exists a Lagrange multiplier µ ∈ R such that βH ψ x − cψ − γ∂ −2 x ψ = −µf (ψ). Hence ϕ = µ 1/(p−1) ψ satisfies (2.1). We denote the set of such solutions by G(β, c, γ). By the homogeneity of I and K, u ∈ G(β, c, γ) also achieve the minimum m = m(β, c, γ) = inf I(u) ; u ∈ X , K(u) > 0 and it follows that M λ = mλ 2 p+1 . We note that if we multiply (2.1) by ϕ and integrate, we find that I(ϕ; β, c, γ) = K(ϕ). Thus we may characterize the set G(β, c, γ) as G(β, c, γ) = ϕ ∈ X ; K(ϕ) = I(ϕ; β, c, γ) = (m(β, c, γ)) We now seek to prove that this set is nonempty.
We say that a sequence ψ n is a minimizing sequence if for some λ > 0, lim n→∞ K(ψ n ) = λ and lim n→∞ I(ψ n ) = M λ . Theorem 2.1 Let p ≥ 2, β > 0, γ > 0 and c < c * = 3(β 2 γ/4) 1/3 . Let {ψ n } n be a minimizing sequence for some λ > 0. Then there exist a subsequence (renamed ψ n ) and scalars y n ∈ R and ψ ∈ X such that ψ n (· + y n ) → ψ in X . The function ψ achieves the minimum I(ψ) = M λ subject to the constraint Proof. The result is an application of the concentration compactness lemma of Lions [23], similar to [1,6,25]. We give the sketch of the proof here.
Let {ψ n } be a minimizing sequence, then we deduce from the coercivity of I that the sequence {ψ n } is bounded in X , so if we define then after extracting a subsequence, we may assume lim n→∞ ρ n L 1 (R) = L > 0. We may assume further after normalizing that ρ n L 1 (R) = L for all n. By the concentration compactness lemma, a further subsequence ρ n satisfies one of vanishing, dichotomy or compactness conditions. We can easily see that M λ = λ 2/(p+1) M 1 , so that the strict subadditivity condition M α + M λ−α > M λ holds for any α ∈ (0, λ). In the same manner as in [19,20,25], it follows from the coercivity of I, inequality (2.7), and the subadditivity condition that both vanishing and dichotomy may be ruled out, and therefore the sequence ρ n is compact. Now set ϕ n (x) = ψ(x + y n ). Since ϕ n is bounded in X , a subsequence ϕ n converges weakly to some ψ ∈ X , and by the weak lower semicontinuity of I over X , we have I(ψ) ≤ lim n→∞ I(ϕ n ) = M λ . Moreover, weak convergence in X , compactness of ρ n , and inequality (2.7) imply strong convergence of ϕ n to ψ in L p+1 (R). Therefore K(ψ) = lim K(ϕ n ) = λ, so I(ψ) ≥ M λ . Together with the inequality above, this implies I(ψ) = M λ , so ψ is a minimizer of I subject to the constraint K(·) = λ. Finally, since I is equivalent to the norm on X , ϕ n ψ, and I(ϕ n ) → I(ψ), it follows that ϕ n converges strongly to ψ in X .
Proposition 2.1 Let β > 0, γ > 0 and c < c * be as in Theorem (2.1), then there exists C ∈ R, C = 0, such that any solution of (2.1) satisfies Since K is an even function, hence Then by using the residue theorem, there holds that where b + ia is the complex root of β|ξ| 3 − cξ 2 + γ, with a, b > 0. Therefore K ∈ C ∞ (R \ {0}). It is therefore concluded for c < c * that (2.20) Now the change of variable η = xy in the first term of the right-hand side of (2.20) reveals that Applying Theorem 3.1.5 in [12], it transpires that there exists C = 0 such that (2.17) holds.

Nonexistence
In this section we present conditions on the parameters β, c, γ and the nonlinearity f (u) that guarantee equation (1.1) has no solitary wave solutions in the space X . These conditions follow from the following functional identities.
Proof. These relations follow by multiplying equation (2.1) by ϕ and xϕ x , respectively and integrating over R. To see that the β term vanishes in the second relation, first observe that since ϕ x has zero mass, it follows that H (xϕ x ) = xH (ϕ x ). Then, using the anti-commutative property of H we have This completes the proof.
Theorem 3.1 Equation (2.1) has no solution in X provided any of the following conditions hold.
Proof. Eliminating the terms on the right hand sides of (3.1), we find that − 2β Now suppose β < 0 and γ > 0. Then since the expression so if c satisfies the inequality in (i), the left hand side of (3.2) will be negative, a contradiction. This proves statement (i). Statement (ii) follows similarly. Next, subtracting the two relations in (3.2), we have The right and left hand sides of this equation have opposite signs when either condition (iii) or condition (iv) holds.

Ground States and Variational Characterizations
A ground state of (2.1) is a solitary wave of (1.1) which minimizes the action S(u) = E(u) − cQ(u) among all nonzero solutions of (2.1), where E(u) and Q(u) are defined in (1.4) and (1.5), respectively. Recall that a solitary wave of (1.1) corresponds to a critical point of S(u), that is, S (u) = 0. Thus, the set of ground states may be characterized as The theorem below finds a ground state of (2.1) as a minimizer for S(ϕ) under a new constraint. Our result is related to that in [24]. where Proof. First, we prove that there is a nontrivial minimizer for (4.2) which is a solution of (2.1). By (2.7), one can easily observe that there exist ε 1 , ε 2 > 0 such that for every nontrivial function Now, let {ϕ n } ⊂ N be a minimizing sequence of (4.2). Then ϕ n X ≥ ε 1 and so that {ϕ n } n is bounded in X . To show that there is a convergent subsequence, with a limit ϕ ∈ X , similar to [1,6], we use again the concentration-compactness lemma [23], applied to the sequence First similar to Theorem 2.1, the evanescence case is excluded. To rule out the dichotomy case, one shows that for all σ < 0. Now if the dichotomy would occur, i.e. ϕ n splits into a sum ϕ 1 n + ϕ 2 n and the distance of the supports of these functions tends to +∞, then one shows that P ϕ 1 n → σ, P ϕ 2 n → −σ, σ ∈ R and J ≥ J σ + J −σ > J which is a contradiction. Therefore the sequence ϕ n concentrates and the limit ϕ satisfies P (ϕ) ≤ 0. The case P (ϕ) < 0 can be excluded by the same reason as above, and we see that ϕ ∈ X is a minimizer for (4.2). Now since ϕ is a minimizer for (4.2), there exists a Lagrange multiplier θ such that S (ϕ) = θP (ϕ). Since S (ϕ), ϕ = 0 and we see that θ = 0, i.e. ϕ is a solution of (2.1).
Finally we show that a ground state of (2.1) achieves the minimum J in (4.2). Let u ∈ X satisfy This completes the proof.
The following proposition proves that minima for M λ in (2.4) are exactly the ground states of (2.1).

Weak Rotation Limit
In this section, we show that the solitary waves of the RGBO equation (1.1) converge to those of the generalized Benjamin-ono equation (1.3). We remark that such a relationship is somewhat surprising since the solitary waves of (1.1) have zero mass, as can be seen by integrating (1.1) with respect to x, while it is well-known (see [1,2,3,4,5] and references therein) that the solitary waves of (1.3) are strictly negative functions and do not have zero mass.
In order to precisely state the convergence result, it is worth noting that for each c < 0 and β > 0 the GBO equation (1.3) possesses a nontrivial solitary wave ϕ and it satisfies The uniqueness of solitary waves of the GBO equation for p > 1 is unknown; however Amick and Toland [7] showed that the solitary wave solutions of the classical Benjamin-Ono (p = 1) are unique (up to translation). The explicit solution was found by Benjamin [11]: One can see that contrary to the unique solitary wave of the KdV equation, the solitary wave of the Benjamin-Ono equation does not decay exponentially.
Theorem 5.1 For β > 0 and c < 0 fixed, let a sequence γ n → 0 + as n → ∞, and let ψ n any element of G(β, c, γ n ). Then there exists a subsequence (still denoted as γ n ), translations y n and a solitary wave ψ ∈ H 1/2 (R) of (5.1) so that ψ n (· + y n ) → ψ in H 1/2 (R), as γ n → 0 + . That is, the solitary waves of the GBO equation are the limits in H 1/2 (R) of solitary waves of the RGBO equation.
To prove this result, we first note for β > 0 and c < 0 that solutions of (5.1) satisfies in a variational problem of the type of Theorem 2.1. More precisely, ground states of (5.1) achieve the minimum Analogous to Theorem 2.1, one can show that for a given sequence there exists a subsequence, renamed ψ n , scalars y n ∈ R and ϕ 0 ∈ H 1/2 (R) such that ψ n (· + y n ) → ϕ 0 in H 1/2 (R).
The proof of Theorem 5.1 is approached via the following lemmas. where c * is defined in Theorem 2.1.

Lemma 5.2
The space X is dense in H 1/2 (R).
Proof. For any u ∈ H 1/2 (R) and δ > 0, let us define u δ as u δ (ξ) = u(ξ)χ |ξ|>δ (ξ). By Parseval's identity follow that x u L 2 (R) < +∞, it follows that u δ ∈ X . In view of the definition of u δ and u ∈ H 1/2 (R) then the inequality holds true. Hence from continuity we may choose δ > 0 sufficiently small so that which completes the proof.

Stability
In this section we investigate the stability of the set G (β, c, γ) of ground state solitary waves. We first state precisely our definition of stability.
then the solution u(t) of (1.1) with initial value u(0) = u 0 can be extended to a solution in the space C([0, +∞), X ∩ X s ) and satisfies sup Otherwise we say that Ω is X -unstable.
Since ground state solitary waves minimize the action S(u) = E(u)−cQ(u), it is natural to consider the function where ϕ is any element of G (β, c, γ). The fact that d is well-defined follows from the relation Together with Lemma 5.1, this relation also implies that d is continuous on the domain β > 0, γ > 0, c < c * , strictly increasing in γ and β and strictly decreasing in c. It can also be shown as in [20] and [21] that d has the following differentiability properties.
Lemma 6.1 For each fixed β > 0 and γ > 0, the partial derivative d c (β, c, γ) exists for all but countably many c. For fixed c and γ, d β (β, c, γ) exists for all but countably many β and for fixed β and c, d γ (β, c, γ) exists for all but countably many γ. At points of differentiability, we have For the remainder of this section we shall regard β > 0 and γ > 0 as fixed and denote d(c) = d(β, c, γ), d (c) = d c (β, c, γ) and d (c) = d cc (β, c, γ). The main stability result is that the stability of the set of ground states is determined by the sign of d (c). Theorem 6.1 Let β > 0, γ > 0, c < c * and ϕ ∈ G (β, c, γ). If d (c) > 0, then the set of ground states G (β, c, γ) is X -stable.
Define the -neighborhood of the set of ground states defined by Since d is strictly decreasing in c and K is continuous on X , we may define for u ∈ U for sufficiently small > 0.
Proof of Theorem 6.1. Assume that G (β, c, γ) is X -unstable with regard to the flow of the RGBO equation. Then there exists a sequence of the initial data u k (0) such that inf ϕ∈G (β,c,γ) Let u k (t) be the solution of (1.1) with initial data u k (0). We can also choose δ > 0 and a sequence of times t k such that inf Moreover we can find ϕ ∈ G (β, c, γ) such that lim k→∞ u k (0) − ϕ k X = 0.
Since E and V are conserved by the flow of (1.1), By using Lemma 6.2, we have for δ sufficiently small that By (6.4) there is some ψ k ∈ G (β, c, γ) such that u k (t k )) X < 2δ, and by using the fact I(u) = I(u; β, c, γ) ≥ C u 2 X , we obtain u k (t k )) X ≤ ψ k X + 2δ ≤ C −1 I(ψ k ; β, c, γ) + 2δ = 2(p + 1) Thus since K is Lipschitz continuous on X and d −1 is continuous, it follows that c(u k (t k )) is uniformly bounded in k. Thus by (6.5)-(6.7) it follows that lim k→∞ c(u k (t k )) = c; and therefore lim k→∞ K(u k (t k )) = lim This implies that Hence it follows from (6.5), (6.6) and (6.8) that lim k→∞ I(u k (t k )) = 2(p+1)d(c)/(p−1). Thus u k (t k ) is a minimizing sequence and therefore has a subsequence which converges in X to some ϕ ∈ G (β, c, γ). This contradicts (6.4), so the proof of the theorem is complete.
Let ψ be as in Theorem 7.1. Define another vector field B ψ by for u ∈ Ω ϕ, . Geometrically, B ψ can be interpreted as the derivative of the orthogonal component of τ α(·) ψ with regard to τ α(·) ϕ .
Lemma 7.2 Let ψ be as in Theorem 7.1. Then the map B ψ : Ω ϕ, 0 → X is C 1 with bounded derivative. Moreover, Proof. The proof follows the same lines from the proof of Lemma 3.5 in [8], Lemma 3.3 in [9] or Lemma 4.7 in [20].
Proof. It suffices to show that there exists a function ψ that satisfies the conditions of Theorem 7.1. Define Then since ϕ c ∈ X and d dc ϕ c ∈ X it follows that ψ ∈ X , and thus ψ ∈ L 2 . Since w = d dc ϕ c satisfies the linear equation it follows as in the proof of Theorem 2.2 that w ∈ H ∞ and ∂ −1 Next using again (3.1) an the facts F = f and pf (ϕ) = f (ϕ)ϕ, it yields that Finally we show that S (xϕ ), xϕ = 3γ R (∂ −1 x ϕ) 2 dx. First we observe from (2.1) and (3.1) that and by using (3.1) again, we obtain Therefore we deduce from (8.1),(8.2) and (8.3) that Theorem 8.1 Let β > 0, γ > 0, c < c * = 3(β 2 γ/4) 1/3 and ϕ ∈ G (β, c, γ). Then the orbit O ϕ is X -unstable if one the following cases occurs: (i) c < 0, p > 3 and γ is sufficiently small, (ii) p > 5 and c < p−5 p−1 c * Proof. By Theorem 7.1 and Lemma 8.1, we only need to check condition (7.1) for ψ defined in Lemma 8.1. First we note that lim γ→0 γ R (∂ −1 x ϕ) 2 dx = 0. Indeed, we already know from (3.1) that Applying Theorem 5.1, it transpires that where φ is a ground state of (1.3) with c < 0. Applying Theorem 5.1 once more we see that . Therefore by Lemma 8.1 one has S (ϕ)ψ , ψ < 0 for γ > 0 sufficiently small. This which proves (i).
Attention is now given to the proof of (ii). Suppose p > 5. By Lemma 8.1 and equation (3.1) we have This is clearly negative when c ≤ 0. Now for c > 0, a straightforward calculation reveals that and thus The term on the right hand side is negative when c < p−5 p−1 c * . This completes the proof.
Remark 8.1 Notice that as p → ∞, p−5 p−1 c * → c * , so the region of instability approaches the entire domain of existence.
We now investigate what conclusions may be drawn from Theorems 6.1 and 7.2, which state that stability is determined by the sign of d (c). Although no explicit formula for d is available, it is possible to determine the behavior of d (c) for small γ > 0. The following scaling property is the main ingredient in this analysis. Let u ∈ X with K(u) > 0. For any r > 0 we have I(u; rβ, rc, rγ) = rI(u; β, c, γ), so m(rβ, rc, rγ) = rm(β, c, γ). Next let v(x) = u(sx) for s > 0. Then and consequently m(β, cs −1 , s −3 γ) = s Next we set q = 2 p−1 and assume that d is twice differentiable. Then differentiating with respect to c gives and .

(8.6)
Theorem 8.2 Assume d is twice differentiable on the domain c < c * .
Proof. First observe that This is positive when 1 < p < 3 and negative when p > 3. As shown in the proof of Theorem 8.1, the term vanishes as γ approaches zero. It therefore remains to show that the term γ 2 d γγ vanishes as well. To do so, define Then since γd γ → 0 as γ → 0 + , g defines a continuous function for γ ≥ 0. Furthermore by the assumption that d is differentiable, it follows that g is differentiable for γ > 0. By the Mean Value Theorem, for each integer k there exists γ k ∈ (0, 1/k) such that g(1/k) − g(0) = 1 k g (γ k ), and thus We next consider the behavior of d for c near c * = 3(β 2 γ/4) 1/3 . Using appropriately chosen trial functions, we obtain upper bounds on d as c approaches c * for the nonlinearities f (u) = |u| p and f (u) = −|u| p−1 u. In both cases, for any p ≥ 2, these bounds imply that d(c) → 0 as c → c * . In the case of the odd nonlinearity f (u) = −|u| p−1 u, the bound implies that d is convex (and hence G (β, c, γ) is stable) for c near c * .
Our choice of trial function is u = w x , where w(x) = e −a|x| sin(bx) for appropriately chosen a > 0 and b = 0. It is clear that u ∈ X , and we have This integral may be evaluated explicitly using Maple to obtain For c < c * , the cubic βr 3 − cr 2 + γ has one real root and two complex roots. Let b ± ai denote the complex roots. Then by the cubic formula, we have As c → c * = 3(β 2 γ/4) 1/3 , we have D → −2c * and thus lim c→c * a = 0 Moreover, as c → c * .

Lemma 8.3 Suppose that d(c)
is differentiable for c < c * . Then for c < c * it holds that Proof. By (5.3), (6.2) and Lemma 6.1, it follows that Hence, we obtain that and therefore (8.8) follows.
Lemma 8.4 For u, a and b as chosen above, we have as c → c * , it suffices to show that the term in parentheses in expression (8.7) is O(c * − c). Using the expansion which holds for small x > 0, we have Combining this with the other two terms in equation (8.7) we are left with .
This bound on I(u), together with a lower bound on K(u), leads to an upper bound on m(β, c, γ). The lower bound on K(u) depends on the nonlinear term f (u). For even nonlinearities we have the following bound.
as c approaches c * .
Proof. It suffices to prove that K(u) ≥ C √ c * − c for some constant C independent of c. For then m(β, c, γ) ≤ I(u) and it follows from (6.2) that To obtain the lower bound on K(u), first write and after the change of variable y = bx + φ this becomes ∞ φ e −a(p+1)y/b | cos(y)| p cos(y) dy.
While the bound in the previous lemma shows that d → 0 as c → c * , unfortunately it does not provide any information about the sign of d (c). For the odd nonlinearity f (u) = −|u| p−1 u, however, the integrand of the functional K is nonnegative, and we have the following stronger bound. as c approaches c * .
Proof. It suffices to prove that K(u) ≥ C(c * − c) −1/2 for some constant C independent of c. For then m(β, c, γ) ≤ I(u) and the lemma follows from (6.2). Now, using the calculations from the previous lemma, we have e −a(p+1)y/b | cos(y)| p+1 dy.
Writing the integral as e −a(p+1)z/b cos(z) p+1 dz = e a(p+1)π/b e a(p+1)π/b − 1 For small a this is approximately b a(p + 1)π Proof. For 1 < p < 5 the function (c * − c) p+3 2(p−1) is convex and vanishes at c = c * . Since d is positive and is bounded above by a multiple of this convex function, its second derivative must be positive at points c arbitrarily close to c * .

Numerical Studies
In this section we present numerical results which illustrate the behavior of the solitary waves as the parameters c and γ are varied, and provide insight into the nature of the function d(c) whose concavity determines the stability of the solitary waves. To obtain the numerical approximations we use a spectral method due to Petviashvili. First observe that the solitary wave equation (2.1) may be written βH ϕ xxx − cϕ xx + f (ϕ) xx = γϕ.
Writing ψ xx = ϕ this becomes −βH ψ xxx + cψ xx + γψ = f (ϕ) so taking the Fourier transform yields Thus a natural iterative scheme is the following: Unfortunately, the algorithm has poor convergence properties. However, the algorithm with stabilizing factor M n defined by has much better convergence properties. It was shown in [29] that this algorithm converges for 1 < α < (p + 1)/(p − 1) and the rate of converges is fastest when α = α * = p/(p − 1). This algorithm was implemented in MATLAB using a large spatial domain to compute the solitary waves for a range of parameter values (β, c, γ). Figures 1 and 2 show several numerically computed solitary waves for the nonlinearity f (u) = u 2 . Figure 1 illustrates the oscillatory tails that develop as c approaches c * , while   Table 1. For p = 2 and p = 2.2 we have d cc > 0 for all c < c * . However, when p = 2.4 there is a small interval of speeds for which d cc < 0. As p increases this interval grows, and when p = 4 we have d cc < 0 for all c < c * . The behavior for small γ > 0 agrees with the results of Theorems 8.1 and 8.2 in that when p < 3 we have d cc > 0 for small γ and when p > 3 we have d cc < 0 for small γ. We note that in the case p = 3, to which these theorems do not apply, we have d cc > 0 for small γ. The behavior for c near c * is rather interesting. It appears that, for p < 3, d cc → +∞ as c → c * , while for p > 3, d cc → −∞ as c → c * . When p = 3, d cc appears to approach some finite negative value. Regions where d cc > 0. 2 c < c * 2.2 c < c * 2.4 c < 0.980c * and c > 0.991c * 2.6 c < 0.976c * and c > 0.994c * 2.8 c < 0.972c * and c > 0.996c * 3 c < 0.968c * 3.2 −1.287c * < c < 0.962c * 3.4 −0.023c * < c < 0.954c * 3.6 0.465c * < c < 0.942c * 3.8 0.738c * < c < 0.915c * 4 empty The results for the odd nonlinearity f (u) = −|u| p−1 u are shown in Figures 6 and 7 and summarized in Table 2. When p ≤ 3 we have d cc > 0 for all c < c * . On the other hand, when p ≥ 5 we have d cc < 0 for all c < c * . When 3 < p < 5 it appears that there exists some speed c p such that d cc < 0 for c < c p and d cc > 0 for c p < c < c * . Once again, the behavior for small γ > 0 agrees with the results of Theorems 8.1 and 8.2. The behavior for c near c * is similar to that of the even nonlinearity, only the critical exponent appears to be p = 5 in this case, in agreement with Theorem 8.3.      The second plot is a blowup of the first, and better illustrates the plots for 3 ≤ p ≤ 4.