Wave breaking and solitary wave solutions for a generalized Novikov equation

Applied Mathematics Letters 100 (2020) 106014

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Applied Mathematics Letters www.elsevier.com/locate/aml

Wave breaking and solitary wave solutions for a generalized Novikov equation Rudong Zheng a , Zhaoyang Yin a,b ,∗ a b

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China Faculty of Information Technology, Macau University of Science and Technology, Macau, China

article

info

Article history: Received 31 July 2019 Received in revised form 15 August 2019 Accepted 15 August 2019 Available online 23 August 2019 Keywords: A generalized Novikov equation Blow-up Solitary wave solutions Weak solutions

abstract In the paper, we mainly study wave-breaking and solitary wave solutions for a generalized Novikov equation. We first obtain a new blow-up result for strong solutions of a generalized Novikov equation in H s (R) for any s > 25 . We then study the solitary wave solutions and find a single solitary wave solution which blows up in H 1 (R). This implies that the weak solutions of the equation may not be unique. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we consider the following generalized Novikov equation: { mt + (2u2x − uux − u2 )mx = (4ux − u)m2 + (2ux − u)(ux − u)m, m = u − uxx ,

(1.1)

which can be rewritten as an equation of u, 1 1 2 1 1 1 ut = − u3 + u2 ux + uu2x − u3x + (p + px ) ∗ ( u3 + uu2x + u3x ), 3 2 3 3 2 6

p(x) =

1 −|x| e . 2

(1.2)

Eq. (1.1) was proposed by Novikov in [1]. It possesses an infinite hierarchy of quasi-local higher symmetries and has the form (1 − ∂x2 )ut = F (u, ux , uxx , uxxx ), (1.3) where F is a homogeneous polynomial. The most famous member of (1.3) is Camassa–Holm (CH) equation: mt + umx + 2ux m = 0, m = u − uxx . ∗ Corresponding author at: Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China. E-mail addresses: [email protected] (R. Zheng), [email protected] (Z. Yin).

https://doi.org/10.1016/j.aml.2019.106014 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

R. Zheng and Z. Yin / Applied Mathematics Letters 100 (2020) 106014

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The CH equation can be regarded as a shallow water wave equation [2,3]. It is completely integrable [4–6]. It has a bi-Hamiltonian structure [7], and has peakon solutions of the form ce−|x−ct| with c > 0, which are orbitally stable [8]. It is worth mentioning that the peakons are suggested by the form of the Stokes water wave of greatest height, see discussions in [9–14]. The local well-posedness and ill-posedness for the Cauchy problem of the CH equation were studied in [15–20]. Blow-up phenomena and global existence of strong solutions were presented in [15,21–23]. The global weak solutions [24,25], conservative and dissipative solutions were discussed in [26,27] respectively. The second integrable example of (1.3) is the Novikov equation: mt + u2 mx + 3uux m = 0, m = u − uxx .

(1.4)

Different from the CH equation, it has cubic nonlinearity. It also possesses a bi-Hamiltonian structure and √ admits exact peakon solutions u(t, x) = ± ce|x−ct| with c > 0 in [28]. The local well-posedness was studied in [29–32]. It also has global strong solutions [29] and solutions which blow up in finite time [32]. The global weak solutions were studied in [33]. Recently, we establish the local well-posedness for the Cauchy problem of (1.1) in Besov spaces with high regularity in [34]. There we find that the strong solutions of (1.1) keep the quantity ∥u∥H 1 (R) invariant. We also obtain a blow-up criteria and a blow up result. When the initial data m0 ∈ M+ (R) (a positive Radon measure), by vanishing viscosity method, we prove in [35] that there exists a global weak solution + + m ∈ L∞ loc (R ; M (R)) satisfying ∥u∥L∞ (R+ ;H 1 (R)) ≤ ∥u0 ∥H 1 (R) . Moreover, the weak solution is unique when ∞ + 1 m ∈ Lloc (R ; L (R)). In this paper, by analyzing the evolution of uux along the particle trajectories, we can get a new blowup result which improves considerably the recent blow-up result in [34]. Then we study the solitary wave solutions and find that a single solitary wave solution may blow up in H s (R) for any s < 32 . As a consequence, the weak solution m(t, x) is not unique in L∞ (0, T ; M+ (R)) for any T < ∞. To the best of our knowledge, it is the first time for one to prove that Eq. (1.1) within Camassa–Holm type equations has the solitary wave solution blowing-up in H 1 (R). Our paper is organized as follows. In Section 2, we present a new blow-up result. Section 3 is devoted to the study of the solitary wave solution of Eq. (1.1). 2. Blow up First we recall some results for strong solutions of the initial problem (1.1). Lemma 2.1. Let m0 ∈ H s (R), s > 21 . Then there exists a time T > 0 such that the initial problem (1.1) is locally well-posed in C([0, T ]; H s (R)) ∩ C 1 ([0, T ]; H s−1 (R)). If T ∗ is the maximal existence time of the solution, then T ∗ < ∞ if and only if lim supt→T ∗ ∥m(t)∥L∞ (R) = ∞. In addition, the equation admits the conservation law: ∥u(t)∥H 1 (R) = ∥u0 ∥H 1 (R) , ∀t ∈ [0, T ∗ ). Next, by a straightforward calculation, we can get some useful equalities. Lemma 2.2. The evolution equations for u, ux , w ≜

ux u

(when u ̸= 0) are as follows.

1 1 4 1 ut + (2u2x − uux − u2 )ux = − u3 − uu2x + u3x + (p + px ) ∗ (u + ux )3 6 2 3 6 1 3 3 2 1 3 1 2 2 2 uxt + (2ux − uux − u )uxx = − u − u ux + uux + ux + (p + px ) ∗ (u + ux )3 , 6 2 3 6 (u − ux )t + (2u2x − uux − u2 )(u − ux )x = u2 ux − 2uu2x + u3x = ux (u − ux )2 , u2 u − ux wt + (2u2x − uux − u2 )wx = (w − 1)(w + 1)(−8w2 + 5w + 1) + (p + px ) ∗ (u + ux )3 . 6 6u2

(2.1)

(2.2) (2.3)

R. Zheng and Z. Yin / Applied Mathematics Letters 100 (2020) 106014

Define the flow map q(t, x), which satisfies { qt (t, x) = (2u2x − uux − u2 )(t, q(t, x)), t ∈ [0, T ∗ ), q(0, x) = x, x ∈ R.

3

(2.4)

Since u ∈ C([0, T ∗ ); H s (R)) with s > 25 , then q ∈ C 1 ([0, T ∗ ) × R; R) and qx (t, x) > 0 everywhere. By (1.1) and (2.2), we find that the sign of m and u − ux is invariant under the flow. Now we state the main blow-up result. Theorem 2.3. Let m0 ∈ H s (R), s > 21 . Assume that there√exists some x0 ∈ R so that m0 (x) ≥ 0 for all x ≥ x0 and u0 (x0 ) > 0, u0,x (x0 ) ≥ αu0 (x0 ) with α = 5+16 57 . Then the maximal existence time of the 4 1 corresponding solution satisfies T ∗ ≤ 3α−1 − ∥u ∥22 ). ( u2 (x ) 0

0

0

H 1 (R)

Proof . By the assumption on m0 , we have m(t, x) ≥ 0, when x ≥ q(t, x0 ). If x ≥ q(t, x0 ), we have ∫ ∞ ∫ ∞ (u + ux )(t, x) = ex e−y m(t, y)dy ≥ 0, (p + px ) ∗ (u + ux )3 (t, x) = ex e−y (u + ux )3 (t, y)dy ≥ 0. x

x

Due that the initial data satisfies u0 (x0 ) > 0, u0,x (x0 ) ≥ αu0 (x0 ), we also have (p+px )∗(u+ux )3 (t, q(t, x0 )) > 0. Now we claim that u(t, q(t, x0 )) > 0, ux (t, q(t, x0 )) ≥ αu(t, q(t, x0 )) hold for every t ∈ [0, T ∗ ). If u0,x (x0 ) ≥ u0 (x0 ), by the sign-preserving property of u − ux , we have ux (t, q(t, x0 )) ≥ u(t, q(t, x0 )). Restricting (2.1) on the flow x = q(t, x0 ), we find 1 1 4 2 2 d u(t, q(t, x0 )) ≥ − u3x − u3x + u3x = u3x ≥ u3 (t, q(t, x0 )). dt 6 2 3 3 3 Since u0 (x0 ) > 0, we get u(t, q(t, x0 )) > 0. If u0 (x0 ) > u0,x (x0 ) ≥ αu0 (x0 ), we have u(t, q(t, x0 )) > ux (t, q(t, x0 )). Assume that there exists some t0 ∈ [0, T ∗ ) so that ux (t0 , q(t0 , x0 )) ≥ αu(t0 , q(t0 , x0 )) > 0, that is w(t0 , q(t0 , x0 )) ≥ α, then d ⏐⏐ u3 u(t, q(t, x0 )) > (8w3 − 3w2 − 1)(t0 , q(t0 , x0 )). ⏐ dt t=t0 6

(2.5)

As the function f (w) = 8w3 − 3w2 − 1 is increasing on ( 41 , +∞), by the definition of α, we can check that d dt |t=t0 u(t, q(t, x0 )) > 0. On the other hand, it follows from (2.3) that d ⏐⏐ u2 w(t, q(t, x0 )) > (w − 1)(w + 1)(−8w2 + 5w + 1)(t0 , q(t0 , x0 )) ≥ 0, ⏐ dt t=t0 6

(2.6)

in the last inequality we use the fact that α is the maximal root of the function g(w) = −8w2 + 5w + 1 and g(w) ≤ 0 on [α, +∞). Combining with (2.5) and (2.6), a continuity argument can be applied to ensure that ux (t, q(t, x0 )) ≥ αu(t, q(t, x0 )) > 0 holds for all t ∈ [0, T ∗ ). Eventually we can get from (2.1) that, for all t ∈ [0, T ∗ ), d 8α3 − 3α2 − 1 3 3α − 1 3 u(t, q(t, x0 )) ≥ u (t, q(t, x0 )) = u (t, q(t, x0 )). dt 6 8 Then it yields that 2 1 1 1 3α − 1 ≤ ≤ 2 ≤ 2 − t. ∥u0 ∥2H 1 (R) ∥u(t)∥2L∞ (R) u (t, q(t, x0 )) u0 (x0 ) 4 Here we use the embedding H 1 (R) ↪→ L∞ (R) and the conservation of ∥u∥H 1 (R) . From the above inequality, 4 1 we thus have T ∗ ≤ 3α−1 ( u2 (x − ∥u ∥22 ) < ∞. □ ) 0

0

0

H 1 (R)

R. Zheng and Z. Yin / Applied Mathematics Letters 100 (2020) 106014

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Remark 2.4. We give two examples satisfying the assumption of Theorem 2.3. (1) Let m0 ̸≡ 0 be nonnegative and supp m0 ⊆ [x0 , +∞) for some x0 ∈ R. Clearly u0 > 0 everywhere and ∫ x0 x (u0 − u0,x )(x0 ) = e−x0 −∞ e m0 (x)dx = 0. u (x) = ∞, (2) Let m0 be odd and m0 (x) > 0 when x > 0. Then u0 (0) = 0, u0,x (0) > 0 and limx→0+ u0,x 0 which enables us to choose some x0 > 0 close to 0 to meet the condition. 3. Solitary wave solutions In this section we consider solitary wave solutions of Eq. (1.1). We will see that the single solitary wave solution blows up in H 1 (R). Rewrite (1.1) as follows: 1 1 1 2 1 (u − uxx )t = (uu2x + 2u2 ux − u3x ) + ( uu2x + u2 ux + u3x )x − ( uu2x + u2 ux − u3x )xx . 2 2 6 2 3

(3.1)

1,3 1,3 Let I be an interval. For initial data u0 ∈ Wloc (R), we say that a function u ∈ C(I; Wloc (R)) is a weak solution of (1.1), if it satisfies (3.1) in the sense of distributions. When u is smooth, (1.1) and (3.1) are equivalent. Consider the multi-solitary wave of the form

u(t, x) =

n ∑

pi (t)e−|x−qi (t)| .

(3.2)

i=1

Following the method in [36], we have the following result. Theorem 3.1. Assume that q1 (t) < q2 (t) < · · · < qn (t). The multi-solitary wave (3.2) is a weak solution of (1.1) if and only if pi , qi satisfy the systems: { q˙i = − 31 p2i − (ai + 3bi )pi + 2ai (ai − 3bi ), (3.3) p˙i = 31 p3i + (ai − bi )p2i + 2ai (ai − bi )pi , where ai =

∑

j
pj eqj −qi and bi =

∑

j>i

pj eqi −qj . The quantity E ≜

∑n

i=1

pi eqi is conserved.

Proof . For 0 ≤ i ≤ n, we introduce the functions ui (t, x) =

i ∑

pj (t)eqj (t)−x +

j=1

n ∑

pj (t)ex−qj (t) ,

j=i+1

which are C ∞ in the space variable. Then (3.2) can be rewritten as u(t, x) =

n ∑

ui (t, x)χi (x),

i=0

where χi denotes the characteristic function of the interval [qi , qi+1 ) with the convention that q0 = −∞ and qn+1 = ∞. As χi have disjoint supports, we have (uu2x )x = =

n n n n−1 ∑ ∑ ∑ ∑ (ui u2i,x χi )x = (ui u2i,x )x χi + (ui u2i,x )(qi )δqi − (ui u2i,x )(qi+1 )δqi+1 i=0

i=0

n ∑

n ∑

i=0

i=1

(ui u2i,x )x χi +

i=1

(ui u2i,x − ui−1 u2i−1,x )(qi )δqi =

i=0 n ∑

n ∑

i=0

i=1

(ui u2i,x )x χi +

[uu2x ]qi δqi ,

R. Zheng and Z. Yin / Applied Mathematics Letters 100 (2020) 106014

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where [v]qi denotes the jump of v across qi , that is [v]qi = v(qi+ ) −v(qi− ). Since u is piecewise smooth, [uu2x ]qi is well defined. Differentiating once more, we get (uu2x )xx

n n n ∑ ∑ ∑ 2 2 = (ui uix )xx χi + [(uux )x ]qi δqi + [uu2x ]qi δq′ i i=0

i=1

i=1

n n n ∑ ∑ ∑ = (ui u2ix )xx χi + ([u3x ]qi + [u2 ux ]qi )δqi + [uu2x ]qi δq′ i . i=0

i=1

i=1

The last step holds because u = ui = ui,xx on every interval (qi , qi+1 ). In a same manner, we can compute the other terms in (3.2) and we end up with 1 1 1 2 1 (uu2x + 2u2 ux − u3x ) + ( uu2x + u2 ux + u3x )x − ( uu2x + u2 ux − u3x )xx 2 2 6 2 3 n ( ) ∑ 1 1 1 1 2 = (ui u2i,x + 2u2i ui,x − u3i,x ) + ( ui u2i,x + u2i ui,x + u3i,x )x − ( ui u2i,x + u2i ui,x − u3i,x )xx χi 2 2 6 2 3 i=0 +

n ∑ 1 1 1 2 ( [uu2x ]qi − [u3x ]qi )δqi − ( [uu2x ]qi + [u2 ux ]qi − [u3x ]qi )δq′ i , 2 3 2 3 i=1

and ut − uxxt =

(3.4)

n n ∑ ∑ (ui,t − ui,xxt )χi − ([uxt ]qi δqi + [ut ]qi δq′ i ). i=0

(3.5)

i=1

As the ui are smooth and ui = ui,xx from the definition, we have 1 1 1 2 1 ui,t − ui,xxt = (ui u2i,x + 2u2i ui,x − u3i,x ) + ( ui u2i,x + u2i ui,x + u3i,x )x − ( ui u2i,x + u2i ui,x − u3i,x )xx . 2 2 6 2 3 Assume the qi are distinct. Comparing (3.4) with (3.5), we find that the multi-solitary wave (3.2) is a weak solution if and only if 1 2 1 1 [uu2x ]qi + [u2 ux ]qi − [u3x ]qi , [uxt ]qi = − [uu2x ]qi + [u3x ]qi . 2 3 2 3 ∑ and bi = j>i pj eqi −qj . By the expression of u in (3.2), we have

[ut ]qi = Let ai =

∑

j
pj eqj −qi

[ut ]qi = 2pi q˙i ,

[uxt ]qi = −2p˙i ,

(3.6)

[u2 ux ]qi = −2pi (ai + pi + bi )2 ,

[uu2x ]qi = 4pi (ai − bi )(ai + pi + bi ),

[u3x ]qi = −2pi (p2i + 3(bi − ai )2 ).

Substituting the above into (3.6), it eventually reduces to (3.3). Finally, to see that the quantity E is ∑n qi conserved, it is enough to prove dE i=1 (p˙i + pi q˙i )e = 0. From (3.3), we have dt = n

n

∑ 1∑ (p˙i + pi q˙i )eqi = (−bi p2i + ai (ai − 2bi )pi )eqi 4 i=1 i=1 ∑ ∑ ∑ = −p2i pj e2qi −qj + pi pj pk eqj +qk −qi − 2 pi pj pk eqi +qj −qk i
=

∑

=

∑

(

+

k
−p2i pj e2qi −qj

i
+

∑

∑

□

∑ )

pi pj pk eqi +qj −qk = 0,

pi pj pk e

qj +qk −qi

+2

−2

∑ j
k=j
pi p2j e2qj −qi

j
j
which completes the proof.

∑

+ 2

i
−2

j
k
−p2i pj e2qi −qj

∑ k
pi pj pk e

qj +qk −qi

pi pj pk eqi +qj −qk

R. Zheng and Z. Yin / Applied Mathematics Letters 100 (2020) 106014

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Corollary 3.2. calculation. (1) n = 1,

Some solitary wave solutions with specific expression can be obtained after the direct

u(t, x) = ± √

1 a−

e−|x−b−ln

√

a− 2t 3 |

.

(3.7)

2t 3

−

b−x When t → 3a and ∥u(t)∥H s (R) → ∞ for any s < 32 , (Especially, u blows up in H 1 (R)). 2 , u(t, x) → e (2) n = 2, when E = 0,

( u(t, x) = ± √

8 t)− 1 ln(1+ −|x−b+ln(a− 3 2

1 3

e

(a − 83 t) + c(a − 83 t) 4

c 1 )| 4 (a− 8 3 t)

−√

1

) 8 e−|x−b+ln(a− 3 t)| .

a − 83 t

Here a, b ∈ R, c < 0 are arbitrary constants. Corollary 3.3. For any 0 < T < ∞, there exists initial data m0 ∈ M+ (R) such that the corresponding weak solution m of (1.1) is not unique in L∞ (0, T ; M+ (R)). Proof . For any fixed T < ∞, the solitary wave solution given by (3.7), denoted by u1 , satisfies m1 = u1 − u1xx ∈ L∞ (0, T ; M+ (R)) if choosing a > 0 such that T < 3a 2 , and ∥u1 (t)∥H 1 (R) is strictly increasing as t increases. However, for the initial data m1 (0), as known from the introduction, we have a global weak + + solution m2 ∈ L∞ loc (R ; M (R)) with ∥u2 ∥L∞ (R+ ;H 1 (R)) ≤ ∥u1 (0)∥H 1 (R) . Thus m1 ̸= m2 on (0, T ). □ Acknowledgments This work was partially supported by NNSFC (No. 11671407), FDCT (No. 0091/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004). The authors thank the referees for their valuable comments and suggestions. References [1] V. Novikov, Generalizations of the Camassa–Holm equation, J. Phys. A 42 (2009) 342002, 14pp. [2] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664. [3] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186. [4] A. Boutet de Monvel, A. Kostenko, D. Shepelsky, G. Teschl, Long-time asymptotics for the Camassa–Holm equation, SIAM J. Math. Anal. 41 (2009) 1559–1588. [5] A. Constantin, On the scattering problem for the Camassa–Holm equation, Proc. R. Soc. Lond. Ser. A 457 (2001) 953–970. [6] A. Constantin, V.S. Gerdjikov, R.I. Ivanov, Inverse scattering transform for the Camassa–Holm equation, Inverse Problems 22 (2006) 2197–2207. [7] A. Fokas, B. Fuchssteiner, Symplectic structures, their B¨ acklund transformation and hereditary symmetries, Physica D 4 (1981) 47–66. [8] A. Constantin, W.A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610. [9] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535. [10] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math. 77 (2012) 293–307. [11] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431. [12] A. Constantin, J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math. 173 (2011) 559–568. [13] T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst. 34 (2014) 3095–3107. [14] J.F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996) 1–48. [15] A. Constantin, J. Escher, Global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998) 475–504. [16] R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988.

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