Variform Exact One-Peakon Solutions for Some Singular Nonlinear - PDF document

International Journal of Bifurcation and Chaos, Vol. 24, No. 12 (2014) 1450160 (16 pages) � World Scientific Publishing Company c DOI: 10.1142/S0218127414501600 Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind* Jibin Li Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, P. R. China by CITY UNIVERSITY OF HONG KONG on 01/05/15. For personal use only. Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com lijb@zjnu.cn Received June 13, 2014 In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe − α | x − ct | . In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon). Keywords : Peakon; nonlinear wave equation; exact solution; smoothness of wave. 1. Introduction derivatives are positive and negative infinities, respectively, then the wave profile is called a cuspon. In recent years, nonlinear wave equations with non- In our paper [Li & Chen, 2007] and book [Li & smooth solitary wave solutions, such as peaked soli- Dai, 2007] (or more recent book [Li, 2013]), using tons (peakons) and cusped solitons (cuspons), have the dynamical system approach, it has been theo- attracted much attention in the literature. Peakon retically proved that there exists a curve triangle was first proposed by [Camassa & Holm, 1993; including one singular straight line in a phase por- Camassa et al. , 1994] and thereafter other peakon trait of the traveling wave system corresponding to equations were developed (see [Degasperis & Pro- some nonlinear wave equation such that the trav- cesi, 1999; Degasperis et al. , 2002; Qiao, 2006, 2007; eling wave solutions have peaked profiles and lose Li & Dai, 2007; Novikov, 2009], and cited references their smoothness. In fact, the existence of a singular therein). Peakons are the so-called peaked solitons, straight line leads to a dynamical behavior with two i.e. solitons with discontinuous first-order derivative scale variables in a period annulus of a center. For at the peak point. Usually, the profile of a wave a singular nonlinear traveling wave system of the function is called a peakon if at a continuous point its left and right derivatives are finite and have dif- first kind, the following two results hold (see [Li, ferent signs [Fokas, 1995]. But if its left and right 2013]). ∗ This research was partially supported by the National Natural Science Foundation of China (11471289, 11162020). 1450160-1

J. Li For an example, as a shallow water model, the Theorem A (The Rapid-Jump Property of the generalized Camassa–Holm (CH) equation with real Derivative Near the Singular Straight Line). Sup- pose that in a left ( or right ) neighborhood of a sin- parameters k , α gular straight line there exist a family of periodic u t + ku x − u xxt + αuu x = 2 u x u xx + uu xxx (1) orbits. Then , along a segment of every orbit near the straight line , the derivative of the wave function has a one-peakon solution jumps down rapidly on a very short time interval. u ( x, t ) = u ( x − ct ) = φ ( ξ ) = ce − √ α 3 | ξ | , (2) Theorem B (Existence of Finite Time Interval of when α = 3 c ( c − k ) with c > 0, k < c, where c is the Solution with Respect to Wave Variable in the wave velocity. Equation (1) has the traveling system Positive or Negative Direction). For a singular non- linear traveling wave system of the first class with dξ = − y 2 + 2( k − c ) φ + αφ 2 dφ dy possible change of the wave variable , if an orbit dξ = y, , (3) 2( φ − c ) transversely intersects with a singular straight line at a point or it approaches a singular straight line , which has the following first integral: by CITY UNIVERSITY OF HONG KONG on 01/05/15. For personal use only. Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com but the derivative tends to infinity , then it only takes � ( k − c ) φ 2 + 1 � a finite time interval to make the moving point of H ( φ, y ) = ( φ − c ) y 2 − 3 αφ 3 = h. the orbit arrive on the singular straight line. (4) These two theorems tell us that for a nonlinear Figure 1(a) shows the phase portrait of system (3) wave equation, a peakon solution has a determined 3 when α = c ( c − k ). Corresponding to the curve geometric property. It depends on the existence of triangle enclosing the period annulus of the center a curve triangle surrounding a period annulus of a E 1 ( 2( c − k ) , 0), Fig. 1(b) shows the peakon profile of center of the corresponding traveling wave system, α Eq. (1) given by (2). in the neighborhood of a singular straight line (see When k = 0, α = 3, Eq. (1) is the original [Li, 2013]). In fact, the curve triangle are the limit Camassa–Holm equation, it has one-peakon solu- curves of a family of periodic orbits of the traveling tion u ( x, t ) = ce −| x − ct | . On the basis of this solution wave system. It gives rise to a peakon profile of the form, in [Beals et al. , 1999], the authors investigated nonlinear wave equation. 2 1.5 1 0.5 0 –8 –6 –4 –2 2 4 6 8 (a) (b) The phase portraits of (3) and a peakon when α = 3 c ( c − k ). (a) Phase portrait of system (3) when α = 3 Fig. 1. c ( c − k ) and (b) peakon solution. 1450160-2

Variform Exact One-Peakon Solutions for Some Singular NTW Equations the N -soliton solution of CH-equation of the form (i) The generalized Camassa–Holm equation N u t + 2 ku x − u xxt + 1 2[ αu 2 + βu 3 ] x � p j ( t ) e −| x − q j ( t ) | , u ( x, t ) = (5) j =1 = 2 u x u xx + uu xxx . (9) where the positions q j and amplitudes p j satisfy the When β = 0 , Eq. (9) is just Eq. (1). following system: (ii) The nonlinear dispersion equation K ( m, n ), N i.e. � p k e −| q j − q k | , q j = ˙ u t + a ( u m ) x + ( u n ) xxx = 0 , m, n ≥ 1 , (10) k =1 N (6) where m, n are integers, a is a real parameter � p k sgn( q j − q k ) e −| q j − q k | , p j = p j ˙ (see [Rosenau, 1997; Li & Liu, 2002]). k =1 (iii) The two-component Hunter–Saxton (HS) sys- tem with real parameters A, σ (see [Moon, for j = 1 , . . . , N. 2013]): by CITY UNIVERSITY OF HONG KONG on 01/05/15. For personal use only. Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com In [Hone & Wang, 2008], the authors considered u txx + 2 σu x u xx + σuu xxx − ρρ x + Au x = 0 , the N -soliton solution of form (5) of the Novikov equation [Novikov, 2009]: ρ t + ( ρu ) x = 0 , u t − u xxt + 4 u 2 u x = uu x u xx + u 2 u xxx , (7) (11) where where σ ∈ R and A ≥ 0. System (11) is the short wave (or high-frequency) limit of the gen- N � p k p l e −| q j − q k |−| q j − q l | , eralized two-component form of the Camassa– q j = ˙ Holm shallow water equations. k =1 (iv) The two-component Camassa–Holm system N (8) with real parameters k, α, e 0 = ± 1 (see � p k p l sgn( q j − q k ) e −| q j − q k |−| q j − q l | , p j = p j ˙ [Olver & Rosenau, 1996; Chen et al. , 2006; k =1 Chen et al. , 2011; Li & Qiao, 2013]): for j = 1 , . . . , N. m t + σum x − Au xx + 2 σmu x Unfortunately, we have showed in [Li, 2014] that even though φ = pe ( x − ct ) and φ = pe − ( x − ct ) are + 3(1 − σ ) uu x + e 0 ρρ x = 0 , (12) two traveling wave solutions of Eq. (7), they cannot ρ t + ( ρu ) x = 0 , be combined to become the solution φ = pe −| x − ct | , i.e. an one-peakon solution of Eq. (7). where m = u − α 2 u xx − k 2 . In this paper, we shall show the following two conclusions: The corresponding traveling wave systems of Eqs. (9)–(12) have one or two singular straight (1) Under different parameter conditions, one non- lines, respectively (see next sections below). Under linear wave equation can have different exact some particular parameter conditions, there exist one-peakon solutions. at least one family of periodic orbits surround- (2) Different nonlinear wave equations can have dif- ing a center such that the boundary curves of the ferent explicit exact one-peakon solutions. period annulus are a curve triangle including a Namely, there are various exact explicit one- singular straight line (see the phase portraits in peakon solutions, which are different from the one- the next sections). Applying the classical analy- peakon solution given by (2). Therefore, to investi- sis method, we can obtain the parametric repre- gate N -peakon solutions for a given nonlinear wave sentations for these boundary curves. When we equation, we may need to consider other forms of take these curve triangles into account as the limit exact solutions, which is different from (5). curves of period annulus, these exact parametric We consider the following four nonlinear wave representations provide very good understanding of equations as examples. the occurrence of peaked traveling wave solutions. 1450160-3