Wave breaking in the short-pulse equation

Transcription

1 Dynamics of PDE Vol.6 No Wave breaking in the short-pulse equation Yue Liu Dmitry Pelinovsky and Anton Sakovich Communicated by Y. Charles Li received May Abstract. Sufficient conditions for wave breaking are found for the shortpulse equation describing wave packets of few cycles on the ultra-short pulse scale. The analysis relies on the method of characteristics and conserved quantities of the short-pulse equation and holds both on an infinite line and in a periodic domain. Numerical illustrations of the finite-time wave breaking are given in a periodic domain. Contents. Introduction Wave breaking on an infinite line Wave breaking in a periodic domain Numerical evidence of wave breaking 35 eferences 39 The short-pulse equation. Introduction.) u t = u + 6 u3 ) t > is a useful and simple approimation of nonlinear wave packets in dispersive media in the limit of few cycles on the ultra-short pulse scale [ ]. This equation is a dispersive generalization of the following advection equation.2) u t = 2 u2 u t >. According to the method of characteristics the advection equation.2) ehibits wave breaking in a finite time for any initial data u ) = u ) on an 2 Mathematics Subject Classification Key words and phrases. Wave breaking ultra-short pulse short-pulse equation. 29 c 29 International Press

2 292 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH infinite line if u ) is continuously differentiable and there is a point such that u )u ) >. This follows from the implicit solution uξ t) = u ξ) ξ t) = ξ 2 u2 ξ)t t > ξ for any given u ) C ). We say that the finite-time wave breaking occurs if there eists a finite time T ) such that.3) lim sup t T u t)u t) = while lim For the simple advection equation.2) { T = inf ξ sup t T u ξ)u ξ) : u ξ)u ξ) > u t) <. In view of this result we address the question if the dispersion term u in the short-pulse equation.) can stabilize global dynamics of the advection equation.2) at least for small initial data. Local well-posedness of the short-pulse equation on an infinite line was proven in []. Theorem Schäfer & Wayne 24). Let u H 2 ). There eists a T > such that the short-pulse equation.) admits a unique solution }. ut) C[ T) H 2 )) C [ T) H )) satisfying u) = u. Furthermore the solution ut) depends continuously on u. To etend local solutions into a global solution Pelinovsky & Sakovich [9] used the following conserved quantities of the short-pulse equation:.4) E := u 2 d u E := + u 2 ) 2.5) d = + d + u 2 [ )] 2 u E 2 := + u 2 u 2.6) d = d. + u 2 + u 2 )5/2 If ut) is a local solution in Theorem then E E and E 2 are bounded and constant in time for all t [ T). Global well-posedness of the short-pulse equation on an infinite line was proven in [9] for small initial data in H 2 satisfying.7) 2E + E 2 u 2 L + 2 u 2 L <. 2 The condition.7) can be sharpen using the scaling transformation for the shortpulse equation.). Let α + be an arbitrary parameter. If u t) is a solution of.) then ũ t) is also a solution of.) with.8) = α t = α t ũ t) = αu t). The conserved quantities transform as follows: ) Ẽ = + ũ 2 d = α Ẽ 2 = + u 2 ) d = αe ũ 2 u 2 d = α + ũ2 )5/2 + u d = 2 )5/2 α E 2.

3 WAVE BEAKING 293 Finding the minimum of 2Ẽ + Ẽ2 = 2αE + α E 2 in α gives a sharper sufficient condition on global well-posedness [9]. Theorem 2 Pelinovsky & Sakovich 28). Let u H 2 ) and 2 2E E 2 <. Then the short-pulse equation.) admits a unique global solution ut) C + H 2 )) satisfying u) = u. Theorem 2 does not eclude wave breaking in a finite time for large initial data and this paper gives a proof that the wave breaking may occur in the shortpulse equation.). Negating the sufficient condition for global well-posedness in Theorem 2 a necessary condition for the wave breaking follows: the wave breaking may occur in the short-pulse equation.) with the initial data u H 2 only if 2 2E E 2. We shall find a sufficient condition for the wave breaking in the short-pulse equation.). Unlike the previous work in [9] we will not be using conserved quantity E 2 but will rely on the conservation of E E and the energy [ ] E := u) 2.9) 2 u4 d. Here u is defined from a local solution u by u := u t)d = u t)d = 2 ) u t)d thanks to the zero-mass constraint u t)d = for all t T). Note that the initial data u does not have generally to satisfy the zero-mass constraint u )d =.) Thanks to the Sobolev inequality u L 4 C u H for some C > the quantity E is bounded if u H 2 ) Ḣ ) where Ḣ is defined by its norm u Ḣ := u L 2. Note that if u H 2 ) Ḣ ) then u)d = is satisfied.) Our main result on the wave breaking on an infinite line is formulated as follows. Theorem 3. Let u H 2 ) Ḣ ) and T be the maimal eistence time of Theorem. Let F := 2 E 2 + 8E E + E 4 ) /2 ) /2 F := E + E + ) /2 2 2 E F 2 and assume that there eists such that u )u ) > and ) F either u 2 /3 ) > 4F u ) u ) 2 > F + 2F u ) 3 ) /2 2 F 2 or ) F u 2 /3 ) u ) u 4F ) 2 > F.

4 294 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH Then T < so that the solution ut) C[ T) H 2 ) Ḣ )) of the shortpulse equation.) blows up in the sense of lim sup t T u t)u t) = while lim t T u t) L F. emark. The quantities F and F in Theorem 3 can be defined by F := sup u t) L t [T) Note that the scaling transformation.8) gives F := sup u t) L. t [T) ũ t) L = α u t) L ũ t) L = α 2 u t) L so that the sufficient condition of Theorem 3 with new definitions of F and F is invariant in α. We note however that while the bound on F in Theorem 3 scales correctly as F = αf the bound on F is not correctly scaled in α because Ẽ = α 3 E Ẽ = α 5 E. This is an artefact of using Sobolev embedding in Lemma 3 below. To prove Theorem 3 we shall adopt the method of characteristics and proceed with apriori differential estimates. Our results remain valid in a periodic domain where Theorem 3 is replaced by Theorem 4 below. The technique of characteristics and apriori differential estimates has been applied for wave breaking in other nonlinear wave equations see [ ] for an incomplete list of references. We emphasize that unlike the previous work in [9] we avoid using a transformation between the short-pulse equation.) and the integrable sine Gordon equation in characteristic coordinates. Our proof of the wave breaking for the shortpulse equation.) does not suggest that there eists a similar wave breaking for the sine Gordon equation it is rather the breaking of the coordinate transformation between the two equations. On a similar note we do not use the integrability properties of the short-pulse equation.) such as the La pair the inverse scattering transform method the bi-hamiltonian formulation and the eistence of eact soliton solutions. The article is constructed as follows. The proof of Theorem 3 is given in Section 2. Section 3 reports etension of Theorem 3 to a periodic domain. Section 4 contains numerical evidences of the finite-time wave breaking in a periodic domain. Acknowledgement. The work of Y. Liu is partially supported by the NSF grant DMS The work of D. Pelinovsky is supported by the NSEC grant GPIN The work of A. Sakovich is supported by the McMaster graduate scholarship. 2. Wave breaking on an infinite line Let us rewrite the Cauchy problem for the short-pulse equation on an infinite line in the form { ut = 2.) 2 u2 u + u t > u ) = u ) where u := u t)d. In what follows we use both notations ut) and u t) for the same solution of the Cauchy problem 2.). Local eistence of solutions with the conservation of E and E is described by the following result.

5 WAVE BEAKING 295 Lemma. Let u H s ) Ḣ ) s 2. There eist a maimal time T = Tu ) > and a unique solution u t) to the Cauchy problem 2.) such that ut) C[ T) H s ) Ḣ )) C [ T) H s )) satisfying u) = u. Moreover the solution ut) depends continuously on the initial data u and the values of E E and E in.4).5) and.9) are constant on [ T). Proof. If u H s ) Ḣ ) s 2 then u H s+ ) so that u )d =. By the theorem of Schäfer & Wayne [] there eists a solution ut) C[ T) H s )) C [ T) H s )) of the short-pulse equation.) so that ut) := u t 2 u2 u C T) H s )). Therefore ut) C[ T) H s ) Ḣ )) in view of boundness of u Ḣ. Because f H s ) s implies lim f) = the zero-mass constraint holds in the form 2.2) u t)d = t [ T). Let us define 2 ut) := u) t 6 u3. By the zero-mass constraint 2.2) and uniqueness of the solution ut) for any t [ T) we obtain lim u t) = t [ T). 2 Using balance equations for the densities of E E and E we write [ u) 2 ] [ 2 u4 = 2 u) 2 ] 36 u6 u 2 ) t + u 2 ) t = u)2 + 4 u4 ) u 2 ) + u 2. = t 2 Integrating the balance equation in for any t [ T) we complete the proof that E E and E are bounded and constant on [ T). emark 2. The maimal eistence time T > in Lemma is independent of s 2 in the following sense. If u H s ) H s ) Ḣ ) for s s 2 and s s then and ut) C[ T) H s ) Ḣ )) C [ T) H s )) ut) C[ T ) H s ) Ḣ )) C [ T ) H s )) with the same T = T. See Yin [2] for standard arguments. By using the local well-posedness result in Lemma and energy estimates we obtain a precise blow-up scenario of the solutions to the Cauchy problem 2.).

6 296 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH Lemma 2. Let u H 2 ) Ḣ ) and ut) be a solution of the Cauchy problem 2.) in Lemma. The solution blows up in a finite time T ) in the sense of lim t T u t) H 2 = if and only if lim sup t T u t)u t) = +. Proof. We only need to prove the necessary condition since the singularity in u t)u t) as t T implies the singularity in u t) H 2 as t T. Assume a finite maimal eistence time T ) and suppose by the contradiction that there is M > such that 2.3) sup u t)u t) M < t [ T). Applying density arguments we approimate the initial value u H 2 ) by functions u n H 3 ) n so that lim n u n = u. Furthermore write u n t) for the solution of the Cauchy problem 2.) with initial data u n. Using the regularity result proved in Lemma for s = 3 it follows from Sobolev s embedding that if u n t) C[ T) H 3 ) Ḣ )) then u n t) is a twice continuously differentiable function of on for any t [ T). It is then deduced from the short-pulse equation.) that d u n dt ) 2 d = u n u n ) 3 d M u n ) 2 d and d u n dt )2 d = 5 u n u n un )2 d 5M u n )2 d. The Gronwall inequality yields for all t [ T) u n t) L 2 un ) L 2e M 2 t u n t) L 2 un ) L 2e 5 2 Mt. Since u n H 2 converges to u H 2 as n we infer from the continuous dependence of the local solution ut) on initial data u that u t) H 2 remains bounded on [ T) for the solution ut) in Lemma. Therefore the contradiction is obtained and either T is not a maimal eistence time or the bound 2.3) is not valid on [ T). We also show that the blow-up of Lemma 2 is the wave breaking in the sense of condition.3). In other words both u t) L and u t) L are uniformly bounded for all t [ T). Lemma 3. Let u H 2 ) Ḣ ) and T > be the maimal eistence time of the solution u t) in Lemma. Then 2.4) u t) L F u t) L F where 2.5) 2.6) t [ T) F := 2 E 2 + 8E E + E 4 ) /2 ) /2 F := E + E + ) /2 2 2 E F 2.

7 WAVE BEAKING 297 Proof. For all t [ T) and the solution u t) we have u 2 t) = uu d uu d u u u 2 + u 2 d E /2 u ) /2 + u 2 )d As a result we obtain E /2 2E + E u t) 2 L ) /2. u t) 4 L 2E E + E 2 u t) 2 L so that bound 2.5) is found from the quadratic equation on u t) 2 L. On the other hand we have u t) 2 H = u t) 2 L + 2 u t) 2 L 2 = E + E + 2 u t) 4 L 4 E + E + 2 E u t) 2 L. Using the Sobolev inequality u L 2 u H and bound 2.5) we obtain bound 2.6). Let us introduce a continuous family of characteristics for solutions of the Cauchy problem 2.). Let ξ t [ T) where T is the maimal eistence time in Lemma and denote 2.7) = Xξ t) u t) = Uξ t) u t) = Gξ t) so that 2.8) { Ẋt) = 2 U2 X) = ξ { Ut) = G U) = u ξ) where dots denote derivatives with respect to time t on a particular characteristics = Xξ t) for a fied ξ. Applying classical results in the theory of ordinary differential equations we obtain the following useful result about the solutions of the initial-value problem 2.8). Conserved quantities E and E of the Cauchy problem 2.) are used to control values of U and G on the family of characteristics. Lemma 4. Let u H 2 ) Ḣ ) and T > be the maimal eistence time of the solution ut) in Lemma. Then there eists a unique solution Xξ t) C [ T)) to the initial-value problem 2.8). Moreover the map X t) : is an increasing diffeomorphism for any t [ T) with t ) ξ Xξ t) = ep uxξ s) s)u Xξ s) s)ds > t [ T) ξ.

8 298 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH Proof. Eistence and uniqueness of Xξ t) C [ T)) follows from the integral equation Xξ t) = ξ 2 t U 2 ξ s)ds t [ T) ξ since Uξ t) C [ T)) for the solution ut) in Lemma. Using the chain rule we obtain t ) ξ Ẋξ t) = Wξ t) ξ Xξ t) ξ Xξ t) = ep Wξ s)ds where Wξ t) = uxξ t) t)u Xξ t) t) C [ T)). Therefore ξ Xξ t) > for all t [ T) and ξ. Let V ξ t) = u Xξ t) t) Wξ t) = uxξ t) t)u Xξ t) t) Uξ t)v ξ t) and compute their rate of changes along the family of characteristics { V = V W + U 2.9) Ẇ = W 2 + V G + U 2. Let F F > be fied in terms of conserved quantities E E and E as in Lemma 3 and assume that there eists ξ such that Wξ ) > and ) F 2 /3 either V ξ ) > 4F V ξ ) Wξ ) > F + 2F V ξ ) 3 2 F 2 ) /2 or ) F 2 /3 V ξ ) V ξ ) Wξ ) > F. 4F Because of the invariance of system 2.9) with respect to G G U U V V W W it is sufficient to consider the case with V ξ ) >. We will prove that under the conditions above V ξ t) and Wξ t) remain positive and monotonically increasing functions for all t > for which they are bounded so that V ξ t) and Wξ t) satisfy the apriori differential estimates { V V W F 2.) Ẇ W 2 V F. In what follows we use V t) and Wt) instead of V ξ t) and Wξ t) for a particular ξ. The following lemma establishes sufficient conditions on the initial point V ) W)) that ensure that a lower solution satisfying { V = V W F 2.) Ẇ = W 2 V F goes to infinity in a finite time.

9 WAVE BEAKING 299 Lemma 5. Assume that the initial data for system 2.) satisfy 2.2) ) F 2 /3 either V ) > 4F V )W) > F + 2F V 3 ) ) /2 2 F 2 2.3) or ) F 2 /3 < V ) V )W) > F. 4F Then the trajectory of system 2.) blows up in a finite time t ) such that V t) and Wt) are positive and monotonically increasing for all t [ t ) and there is C > such that 2.4) lim t t V t) = Moreover t is bounded by lim t t Wt) = and lim t t t t)v t) = C. V ) 2.5) t { ) /2 }. min V ) V 2 ) 2F V 3 ) + 2 F 2 Proof. Let us first consider the homogeneous version of system 2.) for F = that is { V = V W 2.6) Ẇ = W 2 V F. Of course F is never zero otherwise E =. This case is used merely for illustration since eplicit solutions can be obtained for F = whereas qualitative analysis has to be developed for F. System 2.6) is integrable since W = V V d dt V V 2 ) = F V = V 2 C F t) where C = W)/V ). Integrating the last equation for V t) we obtain the eplicit solution of the truncated system V t) = V ) CV )t + 2 F V )t 2 Wt) = C F t)v t). The solution reaches infinity in a finite time t ) if V ) > and C 2 V ) > 2F. Note that these conditions coincide with condition 2.2) for F =. Also note that t is the first positive root of CV )t + 2 F V )t 2 = so that Ut ) = C F t > and t = W) W 2 ) 2F V ) = F V ) 2V ) V ) V ) + V 2 ) 2F V 3 ) V 2 ) 2F V ). 3 Consider now the full system 2.). Let V = W = y

10 3 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH and rewrite the system in the form { ẋ = F 2.7) 2 y ẏ = F y F. Epressing y from the system we can rewrite it in the scalar form 2.8) y = F 2 ẋ ẍ = f) + 3F ẋ where f) = F F 2 3. The only critical point of equation 2.8) is ẋ) = ) where = F /F 2 ) /3 is the root of f). We need to show that there is a domain D 2 in the phase plane ẋ) so that all initial data in D generate trajectories in D that cross the vertical line = in a finite time. To do so we construct a Lyapunov function for equation 2.8) in the form E ẋ) = 2ẋ2 F + 4 F 2 4. The function E ẋ) has a global minimum at ). For any solution t) we have d dt E ẋ) = 3F ẋ 2 > for >. The zero level of the Lyapunov function E ẋ) = passes through the points ) and ) where = 4 /3 > see Figure ). It is clear that E ẋ) > in the domain D = { ẋ) : > ẋ < σ)} where σ) = { 2F 2 F 2 4) /2 ) [ ). We note that the condition ẋ) D is equivalent to the condition { < < y > F 2 + 2F 2 F 24) /2 > y > F 2 which is nothing but the set of conditions 2.2) and 2.3) at t =. By continuity if ẋ) D at t = then ẋ) remains in D for some time t >. No critical points of system 2.8) are located in D and t) is decreasing function for any t > as long as the trajectory stays in D. ecall that E ẋ) > and d dte ẋ) > for any ẋ) D. A trajectory in D can not cross ẋ = σ) because E σ)) = for ) and ẍ = f) + 3F ẋ < for > and ẋ <. Therefore the trajectory either reaches = in a finite time t ) or escapes to ẋ = for >. To eliminate the last possibility we note that so that d dt ẋ 3F 2 2 ) = f) ẋt) ẋ) 3F 2 2 ) + tf)) > for any finite time interval. Moreover ẋ is bounded from zero in D by the level curve E ẋ) = E) ẋ)) which is a conve curve in D. Therefore ẋt) ma{ẋ) ρ} t > as long as >

11 WAVE BEAKING 3 ẋ E ẋ) = σ) D Figure. Domain D in the phase plane ẋ) of equation 2.8). where ρ < is uniquely found from E ρ) = E) ẋ)) that is from the point of intersection of the level curve of E ẋ) = E) ẋ)) with the negative ẋ-ais. Therefore t) ) + t ma{ẋ) ρ} so that t) reaches = in a finite time t ) for any trajectory in D. Moreover finding ρ eplicitly gives the bound on the blow-up time < < t > ) ẋ 2 ) 2F )+ 2 F2 4 )) /2 t ) ẋ) which becomes bound 2.5) after the return back to variable V t). Since V = W = y and y = F 2 ẋ > we have lim t t V t) = and lim t t Wt) = for any trajectories in D. Since ẋ < for the trajectory in D we also have t) t t) as t t so that there eists C > such that lim t t t t)v t) = C. It remains to show that V t) and Wt) are monotonically increasing functions on [ t ). To do so we write where V = V W F = ẋ 2 Ẇ = W 2 F V = g ẋ) 2 g ẋ) = ẋ 2 2F 2 ẋ f). For any trajectory in D ẋt) < so that V t) >. Furthermore since g ẋ) is zero at a curve outside the domain D because g σ)) = F + 2 F F 2 σ) > ] then Ẇt) > for any trajectory in D. ecalling that W = UV in system 2.9) and using the bound 2.4) we obtain the upper bound for any solution at the family of characteristics 2.9) V = UV 2 + U F V 2 + F. We can now show that any upper solution with V ) > goes to infinity in a finite time.

12 32 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH Lemma 6. Consider 2.2) V t) = F V 2 + F with V ) >. There eists t ) such that V t) is positive monotonically increasing for all t [ t ) and there eists C > such that Moreover t /W). limt t)v t) = C. t t Proof. Since V > for any V V t) is monotonically increasing function. To show that V t) reaches in a finite time one can integrate the separable equation 2.2) eplicitly and obtain so that t = since sup + cot ). V t) = tanarctanv ) + F t) π/2 arctanv ) F F V ) W) Applying results of Lemmas 5 and 6 we conclude the proof of Theorem 3. Proof of Theorem 3. Let V W) satisfy system 2.9) corresponding to the characteristics with ξ. Let V W) be the lower solution of system 2.) in Lemma 5 with V ) = V ) and W) = W). Let V be the upper solution of equation 2.2) in Lemma 6 with V ) = V ). Let t be the blow-up time of the lower solution and t be the blow-up time of the upper solution. The upper bound for the solution of system 2.9) follows from the comparison principle for the differential equations since V W + U = V 2 + ) U V 2 + )F which implies that V t) V t) for all t [ t ) for which V t) remains bounded. To obtain the lower bound we note that { V V W F V W F V V W W = V Ẇ + F V W 2 W 2 = Ẇ + F V. Let V = [V W] T V = [V W] T and L = [ F be a nilpotent matri of order one so that e tl = I + tl. Thus we write V V W W d e tl V ) d e tl V ). dt dt Integrating this equation in t > we infer that ] e tl Vt) e tl Vt) and since e tl is invertible for any t we conclude that V t) V t) Wt) Wt) for all t [ T) [ t ) for which V W) remain finite. Therefore V W) become infinite as t T and T [t t ].

13 WAVE BEAKING 33 emark 3. The bounds on t and t in Lemmas 5 and 6 are inconclusive to compare T with the eact time of blow-up T := W) along a particular characteristic of the dispersionless advection equation.2). 3. Wave breaking in a periodic domain Consider now the Cauchy problem for the short-pulse equation.) in a periodic domain { ut = 3.) 2 u2 u + u S t > u ) = u ) S where S is a unit circle equipped with periodic boundary conditions and mean-zero anti-derivative in the form u := u t)d u t)d d. S is the Local well-posedness and useful conserved quantities for the Cauchy problem 3.) in a periodic domain are obtained in the following lemma. Lemma 7. Assume that u H s S) s 2 and S u )d =. Then there eist a maimal time T > such that the Cauchy problem 3.) admits a unique solution ut) C[ T) H s S)) C [ T) H s S)) satisfying u ) = u ) and S u t)d = for all t [ T). Moreover the solution ut) depends continuously on the initial data u and the quantities E = u 2 d E = + u 2 d are constant for all t [ T). S Proof. Eistence of the solution u t) and continuous dependence on u is proved on S similarly to what is done in Lemma on. To prove the zero-mass constraint we note u t t) C T) H s S)) S u 2 u t) C[ T) H s S)) so that for all t T) we have u t)d = u t d + u 2 u ) d =. S 2 S S Initial values of E and E are bounded if u H s S) s 2. Conservation of E and E on [ T) follows from the balance equations u 2 ) = t u) 2 + ) 4 u4 ) ) + u 2 = t 2 u2 + u 2 thanks to the continuity and the periodic boundary conditions for ut) C T) H s+ S)) ut) C T) H s S)) and u t) C T) H s S)) in on S if s 2.

14 34 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH emark 4. The assumption S u )d = on the initial data u in the periodic domain S is necessary as it follows from the following apriori estimate u t)d u )d ut) u L 2 S) t T). S S Note that S u t)d = for all t T) and ut) C[ T) Hs S)) s 2. Hence the above estimate implies that S u )d =. Note that no zero-mass constraint is necessary on an infinite line in Theorem. The blow-up scenario for the solutions to the Cauchy problem 3.) coincides with the one in Lemma 2 after the change of by S. The main result of this section is the proof of the finite-time wave breaking in a periodic domain according to the following theorem. Theorem 4. Let u H 2 S) and S u )d =. Assume that there eists such that u )u ) > and ) /3 either u E 2 ) > 4E /2 u ) u ) 2 > E + 2E /2 u ) 3 /2 2 ) E2 or u ) ) /3 E 2 u 4E /2 ) u ) 2 > E. Then there eists a finite time T ) such that the solution ut) C[ T) H 2 S)) of the Cauchy problem 3.) blows up with the property lim sup t T S u t)u t) = + while lim t T u t) L E. Proof. Let T > be the maimal time of eistence of the solution ut) C[ T) H 2 S)) to the Cauchy problem 3.) constructed in Lemma 7. Since S u t)d = for each t [ T) there is a ξ t [ ] such that uξ t t) =. Then for S and t [ T) we have u t) = u t)d u t) d E. ξ t Since ut) C[ T) H3 S)) is the mean-zero periodic function of for each t [ T) there eists another ξ t [ ] such that u ξ t t) =. Then for S and t [ T) we have u t) = u t)d u t) d E. ξ t Therefore bounds 2.4) are rewritten with F := E F := E S S The rest of the proof follows the proof of Theorem 3.

15 WAVE BEAKING Numerical evidence of wave breaking The goal of this section is to complement the analytic results by several eamples and numerical computations. More specifically we first show that the sufficient condition for wave breaking in Theorem 3 is not satisfied for the eact modulated pulse solution to the short-pulse equation which is known to remain globally bounded in space and time. Then we consider the interplay between global wellposedness and wave breaking of Theorems 2 and 3 for a class of decaying data on an infinite line. Finally we perform numerical simulations in a periodic domain for a simple harmonic initial data and thus give illustrations to the sufficient condition for wave breaking in Theorem 4. Theorem 3 gives a sufficient condition for formation of shocks in the short-pulse equation.) on the infinite line. Let us show that this condition is not satisfied for eact modulated pulse solutions obtained in [8 ]. The simplest one-pulse solution is given in the parametric form where 4.) u t) = Uy t) = Xy t) m sinψ sinhφ + n cosψ coshφ Uy t) = 4mn m 2 sin 2 ψ + n 2 cosh 2 φ m sin2ψ n sinh 2φ Xy t) = y + 2mn m 2 sin 2 ψ + n 2 cosh 2 φ m ) is an arbitrary parameter n = m 2 and φ = my + t) ψ = ny t). The pulse solution enjoys the periodicity property { Uy t) = U y π m t + π m) Xy t) = X y π m t + π m) + π m y t) 2. y t) 2 and an eponential decay in any direction transverse to the anti-diagonal on the y t)-plane. Since X y = 8m2 n 2 sin 2 ψ cosh 2 φ m 2 sin 2 ψ + n 2 cosh 2 φ) = cos 4 arctan m sin ψ ) 2 n coshφ the function = Xy t) is invertible in y for all t if m sin ψ n coshφ < tan π m 8 n tan π 8 that is for all m m cr ) where m cr = sin π For these values of m the pulse solution u t) is analytic in variables t) has the space-time periodicity u t) = u π m t + π ) t) m and the eponential decay in the transverse direction to the anti-diagonal in the t)-plane. The graph of a nonsingular pulse solution for m =.32 is shown on Figure 2 left).

16 36 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH emark 5. Coordinate y in the eact solution 4.) is different from coordinate ξ in the method of characteristics because Xy ) y. Nevertheless Xy t) and Uy t) satisfy the same set of equations X t = U 2 U2 y t) t = u =Xyt) so that ξ and y are uniquely related by the representation ξ = Xy ). If y is found as a function of ξ the initial data of the Cauchy problem 2.) is found from u ξ) = Uy ). Since u t) is analytic in for any fied t and decays to zero eponentially fast at infinity it is clear that u t) H 2 ). Furthermore since u = u t 2 u2 u it also follows that u t) Ḣ ). We compute numerically bounds F and F using the eact solution 4.) and new definitions F := sup u t) L t F := sup u t) L. t It follows from emark that bounds F and F defined above preserve the sufficient condition of Theorem 3 with respect to the scaling transformation.8). Let us define 4.2) where [ f := sup u ) u ) 2 ] F I [ f 2 := sup I 2 I = I 2 = u ) u ) 2 F { F : u 2 ) 4F { : u ) > F 2 4F 2F u ) 3 ) ] /2 2 F 2 ) /3 u )u ) > } ) /3 u )u ) > }. According to Theorem 3 wave breaking occurs if either f or f 2 is positive. For the eact modulated pulse solution 4.) at t = the numerical calculations show that the set I 2 is empty and the quantity f is strictly negative for any m m cr ) see Figure 2 right). Therefore the sufficient condition for the wave breaking in Theorem 3 is not satisfied which corresponds to our understanding that the eact modulated pulse solutions 4.) remain bounded for all t) 2. We note however that the sufficient condition for the global well-posedness in Theorem 2 is satisfied only for pulses with m 32) since 2 2E E 2 = 32m. This computation shows that the sufficient condition of Theorem 2 is not sharp. Net we compare the sufficient conditions for the global well-posedness and wave breaking in Theorems 2 and 3 for a class of initial data 4.3) u ) = a 2b 2 )e b2 a > b > where parameters a and b determine the amplitude and steepness of u. Note that the zero-mass constraint 2.2) is satisfied by u and it is clear that u H 2 ) Ḣ ). The conserved quantities E and E can be computed analytically E = a2 π a 2 b ) 248 E = 3a2 2π b 3 8 b

17 WAVE BEAKING 37 ut) f t m Figure 2. The eact modulated pulse solution 4.) of the shortpulse equation.) for m =.32 left). The quantity f is negative for any m m cr ) right). Figure 3. Boundaries of the global well-posedness and the wave breaking in the Cauchy problem 2.) with initial data 4.3): the global well-posedness occurs below the lower curve and the wave breaking occurs above the upper curve. whereas the conserved quantities E and E 2 are not epressed in a closed form. Using numerical approimations of the integrals we determine the boundary of the well-posedness region in the a b)-plane by finding the parameters a and b from the condition 2 2E E 2 =. We also compute the boundary of the wave breaking region in the a b)-plane by computing f and f 2 in 4.2). Unlike the case of modulated pulses we find that the set I is empty and f 2 may change the sign along the curve on the a b)-plane. The two boundaries are shown on Figure 3 where we can see that the two regions of global well-posedness and wave breaking are disjoint. Finally we perform numerical simulations of the periodic Cauchy problem 3.) with the -periodic initial data 4.4) u ) = a cos2π a >.

18 38 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH.8 ut) supu u ).2 t t ut) supu u ) t T t Figure 4. Solution surface u t) left) and the supremum norm Wt) right) for a =.2 top) and a =.5 bottom). The dashed curve on the bottom right picture shows the linear regression with C =.72 T = T 2.3 C a a Figure 5. Time of wave breaking T versus a left). Constant C of the linear regression versus a right). The two conserved quantities E and E in Lemma 7 are computed analytically as E = 2 a2 E = 2 π E2πai) where E stands for a complete elliptic integral. Using the above conserved quantities we find out that the sufficient condition for the wave breaking in Theorem 4 is satisfied for a >.53.

19 WAVE BEAKING 39 To illustrate the behaviour of a solution u t) to the Cauchy problem 3.) we perform numerical simulations using a pseudospectral method. When the parameter a is sufficiently small the value of Wt) := sup u t)u t) S remains bounded as shown on the top panel of Figure 4 for a =.2. On the other hand when a becomes larger the wave breaking occurs as on the bottom panel of Figure 4 for a =.5. On the bottom right panel of Figure 4 we show using the linear regression that the curve W t) is fitted well with the straight line A + Bt for some coefficients A B). Thus we make a conclusion that Wt) C T t for < T t where C = B and T = AB. Using the linear regression we also obtain pairs T C) for different values of a. The results are shown on Figure 5. Note that the constant C approach as a gets larger. This observation is consistent with the eact blow-up law Wt) = T t obtained for the advection equation.2) using the method of characteristics. eferences [] Y. Chung C.K..T. Jones T. Schäfer and C.E. Wayne Ultra-short pulses in linear and nonlinear media Nonlinearity ) [2] A. Constantin J. Escher Well-posedness global eistence and blowup phenomenon for a periodic quasi-linear hyperbolic equation Comm. Pure Appl. Math ) [3] A. Constantin and J. Escher Wave breaking for nonlinear nonlocal shallow water equations Acta Mathematica ). [4] D. Chae A. Cordoba D. Cordoba and M.A. Fontelos Finite time singularities in a D model of the quasi-geostrophic equation Adv. Math ). [5] J. Hunter Numerical solutions of some nonlinear dispersive wave equations Lectures in Appl. Math ) [6] Y. Liu and Z.Y. Yin Global eistence and blow-up phenomena for the Degasperis Procesi equation Comm. Math. Phys ). [7] Y. Liu and Z.Y. Yin On the blow-up phenomena for the Degasperis Procesi equation Inter. Math. es. Not pages 27). [8] Y. Matsuno Periodic solutions of the short pulse model equation J. Math. Phys pp. 28) [9] D. Pelinovsky A. Sakovich Global well-posedness of the short-pulse and sine Gordon equations in energy space arxiv: ) 4 pages. [] A. Sakovich and S. Sakovich Solitary wave solutions of the short pulse equation J. Phys. A: Math. Gen. 39 L36 L367 26). [] T. Schäfer and C. E. Wayne Propagation of ultra-short optical pulses in cubic nonlinear media Physica D 96 24) 9 5. [2] Z. Yin On the Cauchy problem for an integrable equation with peakon solutions Illinois J. Math ).

20 3 YUE LIU DMITY PELINOVSKY AND ANTON SAKOVICH Department of Mathematics University of Teas at Arlington Arlington TX 769 USA address: Department of Mathematics McMaster University Hamilton ON L8S 4K Canada address: Department of Mathematics McMaster University Hamilton ON L8S 4K Canada address: