Time-dependent defects in integrable soliton equations
aa r X i v : . [ n li n . S I] A ug Time-dependent defects in integrable soliton equations
Baoqiang Xia and Ruguang ZhouSchool of Mathematics and Statistics, Jiangsu Normal University,Xuzhou, Jiangsu 221116, P. R. China,E-mail address: [email protected]; [email protected]
Abstract
We study (1 + 1)-dimensional integrable soliton equations with time-dependent defectslocated at x = c ( t ), where c ( t ) is a function of class C . We define the defect condition asa B¨acklund transformation evaluated at x = c ( t ) in space rather than over the full line. Weshow that such a defect condition does not spoil the integrability of the system. We alsostudy soliton solutions that can meet the defect for the system. An interesting discovery isthat the defect system admits peaked soliton solutions. Keywords: integrable defect, B¨acklund transformation, soliton equations In recent years, there arose some interest in the study of defects, or impurities, in classical (1+1)-dimensional integrable field theories; see for example [1–18] and references therein. The presenceof defects usually spoil the integrability of a system. An interesting case, on the other hand,is that the defect condition is in form of a B¨acklund transformation (BT) frozen at the defectlocation [5–18]. Such a defect condition was found originally by the Lagrangian approach [5–11]and was proved later to preserve the integrability of a system by showing the existence of infiniteset of conserved quantities and by implementing the classical r -matrix method [12–17]. Thesolutions, including soliton and finite-gap solutions, were also derived for this type of integrabledefect systems [9, 18]. We note that the current investigations of the integrable defect problemsfocused mainly on the case of the defect being at a fixed location; the moving defect problemshad received less attention, despite the fact that it was noticed in [7] that the defect can movewith a constant speed.The aim of the present paper is to study time-dependent defects in (1+1)-dimensional in-tegrable soliton equations, including the nonlinear Schr¨odinger (NLS) equation, Korteweg-deVries (KdV) equation and modified KdV (mKdV) equation belonging to the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problems [19]. More precisely, we will consider (1+1)-dimensionalintegrable soliton equations associated with the AKNS system in the presence of a defect at time-dependent location x = c ( t ), where c ( t ) is a function of class C . We define the defect conditionas a BT fixed at the defect location x = c ( t ) in space rather than over the full line. We showthat the resulting defect systems have infinitely many conservation laws. Furthermore we im-plement the classical r -matrix method to establish the Liouville integrability of the resultingdefect systems. Our results extend the results of [13, 14] from the situation of the defect beingfixed to the situation of the defect moving with time.In the present paper, we also study soliton solutions for the time-dependent defect systems.An illustrative example we take is the KdV equation with an integrable defect that moves witha constant speed. We find that such a defect KdV equation admits peaked soliton (peakon)solutions. We note that the peakons were first found in the Camassa-Holm (CH) equation [25,26].Here it is worth pointing out that peakons for the CH type equations and peakons presentedhere should be interpreted in two different senses: the former ones should be interpreted ina suitable weak sense, while the latter ones should be interpreted in the sense that there is atime-dependent defect; see section 6 of the present paper for details.The paper is organized as follows. In section 2, we briefly review the construction of theconservation laws and BTs for integrable soliton equations belonging to the AKNS spectralproblems. In section 3, we present the time-dependent defect system with the defect conditioncorresponding to a BT and show such a defect system admits a Lagrangian description. Insection 4, we study the integrability of the time-dependent defect system. In section 5, wegeneralize the results of section 4 to the case that there are multiple time-dependent defectsin an integrable system. In section 6, we study soliton solutions for the time-dependent defectKdV equation. Some concluding remarks are drawn in section 7. For self-containedness, we start by a brief review of the construction of the conservation lawsand the BTs for integrable soliton equations associated with the AKNS spectral problems.We consider AKNS spectral problems [19]: φ x ( x, t, λ ) = U ( x, t, λ ) φ ( x, t, λ ) , U = − iλ u ( x, t ) v ( x, t ) iλ ! , (2.1a) φ t ( x, t, λ ) = V ( x, t, λ ) φ ( x, t, λ ) , V = V V V − V ! , (2.1b)where λ is a spectral parameter, φ = ( φ , φ ) T , and V jk , j, k = 1 ,
2, are some functions dependon u ( x, t ), v ( x, t ) and on the spectral parameter λ . The compatibility condition of (2.1), namely U t − V x + [ U, V ] = 0 , (2.2)may generate quite a few important integrable nonlinear evolution equations in the solitontheory. For example, if we consider the reduction v = εu ∗ , ε = ±
1, and take V = − iλ − iε | u | λu + iu x ε (2 λu ∗ − iu ∗ x ) 2 iλ + iε | u | ! , (2.3)we then obtain the celebrated NLS equation iu t + u xx − εu | u | = 0 , ε = ± . (2.4)If we consider the reduction v = − V = − iλ + 2 iλu − u x λ u + 2 iλu x − u − u xx − λ + 8 u iλ − iλu + u x ! , (2.5)we then obtain the famous KdV equation u t + u xxx + 6 uu x = 0 . (2.6)If we consider the reduction v = − u and take V = − iλ + 2 iλu λ u + 2 iλu x − u − u xx − λ u + 2 iλu x + 2 u + u xx iλ − iλu ! , (2.7)we then obtain the mKdV equation u t + u xxx + 6 u u x = 0 . (2.8)We will assume, in this paper, the fields u ( x, t ) for the above equations in the bulk aresufficiently smooth and decay as | x | → ∞ or as | t | → ∞ .Let Γ = φ φ , then it follows from (2.1) that Γ satisfies the following x -part and t -part Riccatiequations Γ x = 2 iλ Γ + v − u Γ , (2.9a)Γ t = V − V Γ − V Γ . (2.9b)Moreover, we find from (2.1) that (ln φ ) x = − iλ + u Γ , (ln φ ) t = V + V Γ , (2.10)which in turn generates the following conservation law( u Γ) t = ( V + V Γ) x . (2.11)The functions u Γ and V + V Γ in (2.11) provide the generating functions for the conservationdensities and for the associated fluxes, respectively. We can derive explicit forms of conservationdensities by expanding Γ in terms of negative powers of λ . Indeed, by substituting the expansionΓ = ∞ X n =1 Γ n (2 iλ ) − n (2.12)into (2.9a) and by equating the coefficients of powers of λ , we arrive atΓ = − v, Γ = − v x , (2.13)and the recursion relation:Γ n +1 = (Γ n ) x + u n − X j =1 Γ j Γ n − j , n ≥ . (2.14)Substituting (2.12), (2.13) and (2.14) into (2.11) we finally obtain an infinite set of conservationlaws.We now turn to the construction of BTs for the AKNS system. We consider another copyof the auxiliary problem for ˜ φ with Lax pair ˜ U , ˜ V defined as in (2.1) with the new potentials ˜ u ,˜ v , replacing u , v . We assume that the two systems are related by the gauge transformation˜ φ ( x, t, λ ) = B ( x, t, λ ) φ ( x, t, λ ) . (2.15)A necessary and sufficient condition for (2.15) is that the matrix B ( x, t, λ ) satisfies B x ( x, t, λ ) = ˜ U ( x, t, λ ) B ( x, t, λ ) − B ( x, t, λ ) U ( x, t, λ ) , (2.16a) B t ( x, t, λ ) = ˜ V ( x, t, λ ) B ( x, t, λ ) − B ( x, t, λ ) V ( x, t, λ ) . (2.16b)Transformation (2.15) is actually a Darboux transformation (DT) [20], since it preserves theforms of the Lax pair. Equation (2.16) induces a relation, called a BT [21], between the potentials u , v and ˜ u , ˜ v : B ( u, v, ˜ u, ˜ v ) = 0 . (2.17)For example, for the NLS equation (2.4), we may take B = I + 12 λ α + i Ω − i (˜ u − u ) iε (˜ u − u ) ∗ α − i Ω ! , Ω = ± p β + ε | ˜ u − u | , (2.18)the corresponding BT becomes ˜ u x − u x = iα (˜ u − u ) + Ω (˜ u + u ) , ˜ u t − u t = − α (˜ u x − u x ) + i Ω (˜ u x + u x ) − iε (˜ u − u ) (cid:0) | ˜ u | + | u | (cid:1) , (2.19)where α and β are two arbitrary real constants. For the KdV equation (2.6), we may take B = I + iλ − p β − u + u ) (˜ u + u )1 − p β − u + u ) ! , (2.20)the corresponding BT becomes (˜ u x + u x ) = (˜ u − u ) p β − u + u ) , (˜ u t + u t ) = − (cid:0) u − u ) + (˜ u − u ) xx (cid:1) p β − u + u ) . (2.21)For the mKdV equation (2.8), we may take B = I + i λ − p β − (˜ u − u ) − (˜ u − u ) − (˜ u − u ) − p β − (˜ u − u ) ! , (2.22)the corresponding BT becomes (˜ u x − u x ) = (˜ u + u ) p β − (˜ u − u ) , (˜ u t − u t ) = − (cid:0) u + u ) + (˜ u + u ) xx (cid:1) p β − (˜ u − u ) . (2.23) Let c ( t ) be a function of class C . We study integrable equations with a time-dependent defectplacing at x = c ( t ) in space. We define the defect condition as a BT evaluated at x = c ( t ). Weshow such a defect system admits a Lagrangian description. We suppose that the auxiliary problem (2.1) exists for x > c ( t ), while the one for ˜ U and ˜ V exists for x < c ( t ). At the time-dependent position x = c ( t ), we assume that the two systemsare connected via the condition (2.15) evaluated at x = c ( t ). Definition 1 A (1 + 1) -dimensional integrable equation with a defect at time-dependent location x = c ( t ) in space is described by the following internal boundary problem: • u ( x, t ) and ˜ u ( x, t ) satisfy the equation in the bulk for x > c ( t ) and for x < c ( t ) , respectively; • at x = c ( t ) , u ( c ( t ) , t ) and ˜ u ( c ( t ) , t ) are connected by a condition corresponding to the BTfor u ( x, t ) and ˜ u ( x, t ) . For example, the NLS equation with the above defined time-dependent defect reads iu t + u xx − εu | u | = 0 , ε = ± , x > c ( t ) , (3.1a) i ˜ u t + ˜ u xx − ε ˜ u | ˜ u | = 0 , ε = ± , x < c ( t ) , (3.1b)(˜ u x − u x ) | x = c ( t ) = ( iα (˜ u − u ) + Ω (˜ u + u )) | x = c ( t ) , (3.1c)(˜ u t − u t ) | x = c ( t ) = (cid:0) − α (˜ u x − u x ) + i Ω (˜ u x + u x ) − iε (˜ u − u ) (cid:0) | ˜ u | + | u | (cid:1)(cid:1)(cid:12)(cid:12) x = c ( t ) , (3.1d)where Ω = ± p β + ε | ˜ u − u | . The KdV equation with the time-dependent defect reads u t + u xxx + 6 uu x = 0 , x > c ( t ) , (3.2a)˜ u t + ˜ u xxx + 6˜ u ˜ u x = 0 , x < c ( t ) , (3.2b)(˜ u x + u x ) | x = c ( t ) = (˜ u − u ) p β − u + u ) (cid:12)(cid:12)(cid:12) x = c ( t ) , (3.2c)(˜ u t + u t ) | x = c ( t ) = − (cid:0) u − u ) + (˜ u − u ) xx (cid:1) p β − u + u ) (cid:12)(cid:12)(cid:12) x = c ( t ) . (3.2d)The mKdV equation with the time-dependent defect reads u t + u xxx + 6 u u x = 0 , x > c ( t ) , (3.3a)˜ u t + ˜ u xxx + 6˜ u ˜ u x = 0 , x < c ( t ) , (3.3b)(˜ u x − u x ) | x = c ( t ) = (˜ u + u ) p β − (˜ u − u ) (cid:12)(cid:12)(cid:12) x = c ( t ) , (3.3c)(˜ u t − u t ) | x = c ( t ) = − (cid:0) u + u ) + (˜ u + u ) xx (cid:1) p β − (˜ u − u ) (cid:12)(cid:12)(cid:12) x = c ( t ) . (3.3d) We now show that the time-dependent defect system also admits a Lagrangian description. Wewill fix our ideas on the above mentioned three examples: the defect NLS equation (3.1), thedefect KdV equation (3.2) and the defect mKdV equation (3.3).
The NLS equation (2.4) in the bulk is described by the Lagrangian L = Z ∞−∞ dx (cid:18) i u ∗ u t − uu ∗ t ) − | u x | − ε | u | (cid:19) . (3.4)To describe the defect NLS equation (3.1), we modify the Lagrangian as follows L = Z c ( t ) −∞ dx L (˜ u ) + D + Z ∞ c ( t ) dx L ( u ) , (3.5)where L ( u ) = i u ∗ u t − uu ∗ t ) − | u x | − ε | u | (3.6)is the Lagrangian density of the bulk system for x > c ( t ), L (˜ u ) = i u ∗ ˜ u t − ˜ u ˜ u ∗ t ) − | ˜ u x | − ε | ˜ u | (3.7)is the Lagrangian density of the bulk system for x < c ( t ), D = − i εω ˙˜ u − ˙ u ˜ u − u − ˙˜ u ∗ − ˙ u ∗ ˜ u ∗ − u ∗ ! − εω + ω (cid:0) | ˜ u | + | u | + εα − εαc ′ ( t ) (cid:1) + ( i c ′ ( t ) − iα ) (˜ u ∗ u − ˜ uu ∗ ) (3.8)is the defect contribution at x = c ( t ). In (3.8), we used the following abbreviated expression u = u ( c ( t ) , t ) , ˜ u = ˜ u ( c ( t ) , t ) , ω = ± p β + ε | ˜ u − u | , (3.9a) u = u x ( x, t ) | x = c ( t ) , ˜ u = ˜ u x ( x, t ) | x = c ( t ) , (3.9b) u = u t ( x, t ) | x = c ( t ) , ˜ u = ˜ u t ( x, t ) | x = c ( t ) , (3.9c)˙ u = du ( c ( t ) , t ) dt = c ′ ( t ) u + u , ˙˜ u = d ˜ u ( c ( t ) , t ) dt = c ′ ( t )˜ u + ˜ u . (3.9d) Claim 1
The defect NLS equation (3.1) can be described by the Lagrangian (3.5).
Indeed, we consider the complete action A = Z ∞−∞ dt (Z c ( t ) −∞ dx L (˜ u ) + D + Z ∞ c ( t ) dx L ( u ) ) . (3.10)The variation of A with respect to u ∗ gives δ A = Z ∞−∞ dt (Z ∞ c ( t ) dx (cid:18) ∂ L ( u ) ∂u ∗ δu ∗ + ∂ L ( u ) ∂u ∗ x δu ∗ x + ∂ L ( u ) ∂u ∗ t δu ∗ t (cid:19) + ∂D∂ u ∗ δ u ∗ + ∂D∂ ˙ u ∗ δ ˙ u ∗ ) . (3.11)Integrating the second term in (3.11) by parts with respect to x , we find Z ∞−∞ dt Z ∞ c ( t ) dx (cid:18) ∂ L ( u ) ∂u ∗ x δu ∗ x (cid:19) = − Z ∞−∞ dt ( (cid:18) ∂ L ( u ) ∂u ∗ x δu ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = c ( t ) + Z ∞ c ( t ) dx (cid:18)(cid:18) ∂ L ( u ) ∂u ∗ x (cid:19) x δu ∗ (cid:19)) = Z ∞−∞ dt ( u δ u ∗ − Z ∞ c ( t ) dx (cid:18)(cid:18) ∂ L ( u ) ∂u ∗ x (cid:19) x δu ∗ (cid:19)) . (3.12)Using the identity ddt Z ∞ c ( t ) dx (cid:18) ∂ L ( u ) ∂u ∗ t δu ∗ (cid:19)! = − c ′ ( t ) ∂ L ( u ) ∂u ∗ t δu ∗ (cid:12)(cid:12)(cid:12)(cid:12) x = c ( t ) + Z ∞ c ( t ) dx (cid:18)(cid:18) ∂ L ( u ) ∂u ∗ t (cid:19) t δu ∗ + ∂ L ( u ) ∂u ∗ t δu ∗ t (cid:19) = i c ′ ( t ) u δ u ∗ + Z ∞ c ( t ) dx (cid:18)(cid:18) ∂ L ( u ) ∂u ∗ t (cid:19) t δu ∗ + ∂ L ( u ) ∂u ∗ t δu ∗ t (cid:19) , (3.13)the third term in (3.11) can be written as Z ∞−∞ dt Z ∞ c ( t ) dx (cid:18) ∂ L ( u ) ∂u ∗ t δu ∗ t (cid:19) = − Z ∞−∞ dt ( i c ′ ( t ) u δ u ∗ + Z ∞ c ( t ) dx (cid:18)(cid:18) ∂ L ( u ) ∂u ∗ t (cid:19) t δu ∗ (cid:19)) . (3.14)Integrating the last term in (3.11) by parts with respect to t , we have Z ∞−∞ dt (cid:18) ∂D∂ ˙ u ∗ δ ˙ u ∗ (cid:19) = − Z ∞−∞ dt (cid:18) δ u ∗ ddt (cid:18) ∂D∂ ˙ u ∗ (cid:19)(cid:19) . (3.15)Inserting (3.12), (3.14) and (3.15) into (3.11) and requiring the variation to be stationary, weobtain 0 = Z ∞−∞ dt Z ∞ c ( t ) dx (cid:20) δu ∗ (cid:18) ∂ L ( u ) ∂u ∗ − ∂∂x (cid:18) ∂ L ( u ) ∂u ∗ x (cid:19) − ∂∂t (cid:18) ∂ L ( u ) ∂u ∗ t (cid:19)(cid:19)(cid:21) + Z ∞−∞ dt (cid:20) δu ∗ (cid:18) u − i c ′ ( t ) u + ∂D∂ u ∗ − ddt (cid:18) ∂D∂ ˙ u ∗ (cid:19)(cid:19)(cid:21) . (3.16)Similarly, requiring the variation of (3.10) with respect to ˜ u ∗ to be stationary gives0 = Z ∞−∞ dt Z c ( t ) −∞ dx (cid:20) δ ˜ u ∗ (cid:18) ∂ L (˜ u ) ∂ ˜ u ∗ − ∂∂x (cid:18) ∂ L (˜ u ) ∂ ˜ u ∗ x (cid:19) − ∂∂t (cid:18) ∂ L (˜ u ) ∂ ˜ u ∗ t (cid:19)(cid:19)(cid:21) + Z ∞−∞ dt (cid:20) δ ˜ u ∗ (cid:18) − ˜ u + i c ′ ( t )˜ u + ∂D∂ ˜ u ∗ − ddt (cid:18) ∂D∂ ˙˜ u ∗ (cid:19)(cid:19)(cid:21) . (3.17)Formulae (3.16) and (3.17) yield ∂ L ( u ) ∂u ∗ − ∂∂x (cid:18) ∂ L ( u ) ∂u ∗ x (cid:19) − ∂∂t (cid:18) ∂ L ( u ) ∂u ∗ t (cid:19) = 0 , x > c ( t ) , (3.18a) ∂ L (˜ u ) ∂ ˜ u ∗ − ∂∂x (cid:18) ∂ L (˜ u ) ∂ ˜ u ∗ x (cid:19) − ∂∂t (cid:18) ∂ L (˜ u ) ∂ ˜ u ∗ t (cid:19) = 0 , x < c ( t ) , (3.18b) u − i c ′ ( t ) u + ∂D∂ u ∗ − ddt (cid:18) ∂D∂ ˙ u ∗ (cid:19) = 0 , (3.18c)˜ u − i c ′ ( t )˜ u − ∂D∂ ˜ u ∗ + ddt (cid:18) ∂D∂ ˙˜ u ∗ (cid:19) = 0 . (3.18d)Equations (3.18a) and (3.18b) give nothing but (3.1a) and (3.1b), while (3.18c) and (3.18d),after some algebra, give exactly the defect conditions (3.1c) and (3.1d) at the defect location. For the KdV equation, the setting u = q x is suitable for a Lagrangian description. In thissetting, the defect KdV equation (3.2) can be rewritten as q t + q xxx + 3 q x = 0 , x > c ( t ) , (3.19a)˜ q t + ˜ q xxx + 3˜ q x = 0 , x < c ( t ) , (3.19b)(˜ q x + q x ) | x = c ( t ) = − α −
12 (˜ q − q ) (cid:12)(cid:12)(cid:12) x = c ( t ) , (3.19c)(˜ q t + q t ) | x = c ( t ) = (cid:2) (˜ q xx − q xx ) (˜ q − q ) − (cid:0) (˜ q x ) + ( q x ) + ˜ q x q x (cid:1)(cid:3)(cid:12)(cid:12) x = c ( t ) . (3.19d)where α = − β . Regarding this defect system, we introduce the Lagrangian L = Z c ( t ) −∞ dx L (˜ q ) + D + Z ∞ c ( t ) dx L ( q ) , (3.20)where L ( q ) = 12 q x q t + ( q x ) −
12 ( q xx ) (3.21)is the Lagrangian density of the bulk system for x > c ( t ), L (˜ q ) = 12 ˜ q x ˜ q t + (˜ q x ) −
12 (˜ q xx ) (3.22)is the Lagrangian density of the bulk system for x < c ( t ), D = 14 (cid:16) q ˙˜ q − ˜ q ˙ q (cid:17) −
940 (˜ q − q ) − (˜ q − q ) (cid:18) α + 34 (˜ q + q ) (cid:19) + 14 (˜ q − q ) (˜ q − q ) − (˜ q − q ) (cid:0) (˜ q ) + ( q ) + ˜ q q + 3 α (˜ q + q ) + 6 α (cid:1) + 12 (˜ q − q ) (˜ q + q + 2 α ) − c ′ ( t ) (˜ q − q ) (cid:18) α + 112 (˜ q − q ) (cid:19) (3.23)0is the defect contribution at x = c ( t ). In (3.23), we have used the following abbreviated expres-sions: q = q ( c ( t ) , t ) , ˜ q = ˜ q ( c ( t ) , t ) , (3.24a) q = q x ( x, t ) | x = c ( t ) , ˜ q = ˜ q x ( x, t ) | x = c ( t ) , (3.24b) q = q xx ( x, t ) | x = c ( t ) , ˜ q = ˜ q xx ( x, t ) | x = c ( t ) , (3.24c) q = q t ( x, t ) | x = c ( t ) , ˜ q = ˜ q t ( x, t ) | x = c ( t ) , (3.24d)˙ q = dq ( c ( t ) , t ) dt = c ′ ( t ) q + q , ˙˜ q = d ˜ q ( c ( t ) , t ) dt = c ′ ( t )˜ q + ˜ q . (3.24e)In analogy to the case of NLS equation, by requiring the variation of the complete action R ∞−∞ dtL to be stationary with respect to q or ˜ q , we find the following defect conditions:12 ˙ q = − ∂D∂ q + ddt (cid:18) ∂D∂ ˙ q (cid:19) , q + ∂D∂ q = 0 , ∂D∂ q = 0 , (3.25a)12 ˙˜ q = ∂D∂ ˜ q − ddt (cid:18) ∂D∂ ˙˜ q (cid:19) , − ˜ q + ∂D∂ ˜ q = 0 , ∂D∂ ˜ q = 0 . (3.25b)Using (3.23) the above defect conditions are exactly equivalent to (3.19c) and (3.19d). To sumup, we find Claim 2
The defect KdV equation (3.19) can be described by the Lagrangian (3.20).
For the mKdV equation in the setting u = q x , an alternative Darboux matrix B can be takenas B = I + iβ λ cos(˜ q − q ) − sin(˜ q − q ) − sin(˜ q − q ) − cos(˜ q − q ) ! , (3.26)and the corresponding BT becomes [9] (˜ q x + q x ) = β sin(˜ q − q ) , (˜ q t + q t ) = − β (cid:2) (˜ q xx − q xx ) cos(˜ q − q ) + (˜ q x + q x ) sin(˜ q − q ) (cid:3) . (3.27)Then the time-dependent defect mKdV equation in the potential q reads q t + q xxx + 2 ( q x ) = 0 , x > c ( t ) , (3.28a) q t + q xxx + 2 ( q x ) = 0 , x < c ( t ) , (3.28b)(˜ q x + q x ) | x = c ( t ) = β sin(˜ q − q ) | x = c ( t ) , (3.28c)(˜ q t + q t ) | x = c ( t ) = − β (cid:2) (˜ q xx − q xx ) cos(˜ q − q ) + (˜ q x + q x ) sin(˜ q − q ) (cid:3)(cid:12)(cid:12) x = c ( t ) . (3.28d)1We introduce the Lagrangian L = Z c ( t ) −∞ dx L (˜ q ) + D + Z ∞ c ( t ) dx L ( q ) , (3.29)where L ( q ) = 12 q x q t + 12 ( q x ) −
12 ( q xx ) (3.30)is the Lagrangian density of the bulk system for x > c ( t ), L (˜ q ) = 12 ˜ q x ˜ q t + 12 (˜ q x ) −
12 (˜ q xx ) (3.31)is the Lagrangian density of the bulk system for x < c ( t ), D = 14 (cid:16) q ˙˜ q − ˜ q ˙ q (cid:17) + 12 (˜ q − q ) (˜ q + q − β sin(˜ q − q ))+ β q − q ) h (˜ q ) + ( q ) − q q + β (˜ q + q ) sin(˜ q − q ) + β i − βc ′ ( t ) cos(˜ q − q )(3.32)is the defect contribution at x = c ( t ). In (3.32) we have used the same abbreviated expressionsas used in the case of defect KdV equation (see (3.24)). In analogy to the case of defect KdVequation, we find Claim 3
The defect mKdV equation (3.28) can be described by the Lagrangian (3.29).
Remark 1.
Taking c ′ ( t ) = 0 in (3.5), (3.20) and (3.29) respectively, from our Lagrangianformulations for the time-dependent defect systems we can recover the corresponding Lagrangianformulations for the defect systems in the situation of the defect being fixed [9]. In this section, we will establish the integrability of the defect system both by constructingan infinite set of conserved densities and by implementing the classical r -matrix method. Thisanalysis is based on an extension of the results of [13, 14] from the situation of the defect beingfixed to the situation of the defect moving with time. By generalizing the analogous result of [13], we find the following conservation densities for thetime-dependent defect system.2
Proposition 1
The generating function for the integrals of motion reads I ( λ ) = I leftbulk ( λ ) + I rightbulk ( λ ) + I defect ( λ ) , (4.1) where I leftbulk ( λ ) = Z c ( t ) −∞ ˜ u ˜Γ dx, (4.2) I rightbulk ( λ ) = Z ∞ c ( t ) u Γ dx, (4.3) I defect ( λ ) = − ln( B + B Γ) | x = c ( t ) , (4.4) and B jk , j, k = 1 , , is the jk -entry of the defect matrix B . Proof
From (2.11), we have ( u Γ) t = ( V + V Γ) x , x > c ( t ) , (4.5a) (cid:16) ˜ u ˜Γ (cid:17) t = (cid:16) ˜ V + ˜ V ˜Γ (cid:17) x , x < c ( t ) , (4.5b)where ˜Γ = ˜ φ ˜ φ . Using (4.5) and the rapid decay of the fields u ( x, t ), v ( x, t ), ˜ u ( x, t ), ˜ v ( x, t ), wefind Z c ( t ) −∞ ˜ u ˜Γ dx + Z ∞ c ( t ) u Γ dx ! t = (cid:16) ˜ V + ˜ V ˜Γ − V − V Γ (cid:17)(cid:12)(cid:12)(cid:12) x = c ( t ) + c ′ ( t ) (cid:16) ˜ u ˜Γ − u Γ (cid:17)(cid:12)(cid:12)(cid:12) x = c ( t ) . (4.6)From (2.15), we have ˜Γ (cid:12)(cid:12)(cid:12) x = c ( t ) = B + B Γ B + B Γ (cid:12)(cid:12)(cid:12)(cid:12) x = c ( t ) . (4.7)Using (2.9b), (2.16b) and (4.7), we obtain (cid:16) ˜ V + ˜ V ˜Γ − V − V Γ (cid:17)(cid:12)(cid:12)(cid:12) x = c ( t ) = ( B + B Γ) t B + B Γ (cid:12)(cid:12)(cid:12)(cid:12) x = c ( t ) . (4.8)Using (2.9a), (2.16a) and (4.7), we obtain (cid:16) ˜ u ˜Γ − u Γ (cid:17)(cid:12)(cid:12)(cid:12) x = c ( t ) = ( B + B Γ) x B + B Γ (cid:12)(cid:12)(cid:12)(cid:12) x = c ( t ) . (4.9)Substituting (4.8) and (4.9) into (4.6), we obtain Z c ( t ) −∞ ˜ u ˜Γ dx + Z ∞ c ( t ) u Γ dx ! t = (cid:16) ln( B + B Γ) | x = c ( t ) (cid:17) t . (4.10)3Thus ( I ( λ )) t = 0 . (4.11)This completes the proof. (cid:3) Remark 2.
Proposition 1 implies that the integrals of motion for the time-dependent defectsystem take a very similar form as those for the system with a defect being fixed [13]. However,the proof for proposition 1 is technically more involved than analogous proof for the case thatthe defect being fixed (see section 1.2 in [13]); we need to pay more attention to the t -derivativesof the associated quantities. r -matrix approach Canonical properties of BTs to integrable nonlinear evolution equations have been establishedin [27, 28]. Recently, by introducing a new Poisson bracket (called equal-space bracket), it wasshown in [14] that a defect condition described by a frozen BT can be interpreted naturally asa canonical transformation of the system. As a consequence, the classical r -matrix approach[22–24] can be implemented to establish Liouville integrability for the defect system with a defectat a fixed location. Here we show that analogous discussions can be adapted to the case of thetime-dependent defect systems.To fix ideas, we concentrate on the NLS equation. Let us first recall some important resultsregarding the multi-symplectic formalism of the NLS equation [14]. The key observation in [14]is to introduce the following new equal-space Poisson bracket { u ( x, t ) , u ∗ x ( x, τ ) } = − δ ( t − τ ) , { u x ( x, t ) , u ∗ ( x, τ ) } = δ ( t − τ ) , { u ( x, t ) , u ( x, τ ) } = { u ( x, t ) , u ∗ ( x, τ ) } = { u x ( x, t ) , u ( x, τ ) } = { u x ( x, t ) , u x ( x, τ ) } = 0 . (4.12)With this Poisson bracket, the NLS equation (2.4) can be written in the following Hamiltonianform u xx = { u x , H T } , (4.13)where the new Hamiltonian H T are given by H T = Z ∞−∞ dτ (cid:18) −| u x | − i u ∗ u τ − u ∗ τ u ) + ε | u | (cid:19) . (4.14)We construct a transition matrix from the time-part of the Lax pair: Transition matrix M T ( x, t, λ )is defined as the fundamental solution of the auxiliary linear problem (2.1b) with M T ( x, −∞ , λ ) = I (here I denotes identity matrix), M T ( x, t, λ ) = x exp Z t −∞ V ( x, τ, λ ) dτ. (4.15)4Using the Poisson bracket (4.12), one can check directly that M T ( x, t, λ ) satisfies the following r -matrix relation [14]: { M T ( x, t, λ ) , M T ( x, t, µ ) } = [ r ( λ − µ ) , M T ( x, t, λ ) ⊗ M T ( x, t, µ )] , (4.16)where M T ( x, t, λ ) = M T ( x, t, λ ) ⊗ I , M T ( x, t, µ ) = I ⊗ M T ( x, t, µ ), and r ( λ − µ ) = ε λ − µ ) . (4.17)As an application, we can deduce that integrals of motion constructed from the trace of themonodromy matrix M T ( x, ∞ , λ ) are in involution with respect to the Poisson bracket (4.12).Thus, Liouville integrability of the NLS equation with respect to the Poisson bracket (4.12)is proved. For a system without a defect, the above argument for Liouville integrability isequivalent to the standard argument with respect to the usual (equal-time) Poisson bracket(see [24] for details). The advantage of the above argument is that it can be applied to establishLiouville integrability of a system with a defect; see [14] for the case of the NLS equation with adefect being fixed at x = x and see the following discussions for the system with time-dependentdefect.We now adapt the arguments of [27, 28] about canonical transformations to the above newPoisson bracket: Transformation, which maps u to ˜ u , is canonical if the following Pfaffian formis relative integrable invariant Z ∞−∞ dt ( u ∗ x du + u x du ∗ ) + H T dx. (4.18)That is Z ∞−∞ dt (˜ u ∗ x d ˜ u + ˜ u x d ˜ u ∗ ) + ˜ H T dx = Z ∞−∞ dt ( u ∗ x du + u x du ∗ ) + H T dx − dW. (4.19)Here W ( u, u ∗ , ˜ u, ˜ u ∗ ; x ) = F ( u, u ∗ , ˜ u, ˜ u ∗ ) − Ex (4.20)(with E being a real constant) is called a generator of the transformation. From (4.19), weobtain the transformation equations: u x = δFδu ∗ , u ∗ x = δFδu , ˜ u x = − δFδ ˜ u ∗ , ˜ u ∗ x = − δFδ ˜ u . (4.21)5For the NLS equation, we find that F can be taken as F = Z ∞−∞ dt (cid:18) i ε Ω (cid:18) ˜ u t − u t ˜ u − u − ˜ u ∗ t − u ∗ t ˜ u ∗ − u ∗ (cid:19) + 13 ε Ω − Ω (cid:0) | u | + | ˜ u | + εα (cid:1) + iα ( u ˜ u ∗ − u ∗ ˜ u ) (cid:19) . (4.22)Then, the transformation equation (4.21) becomes nothing but the BT (2.19) of the NLS equa-tion. Hence, the defect condition for the NLS equation can be interpreted as a canonical trans-formation with respect to the Poisson bracket (4.12).We now turn to the implementation of the classical r -matrix approach to the NLS equationin the presence of the time-dependent defect. For the time-dependent defect NLS system, wedefine the transition matrix as follows M ( x, t, λ ) = ( f M T ( x, t, λ ) , −∞ < x < c ( t ) ,M T ( x, t, λ ) , c ( t ) ≤ x < ∞ , (4.23)where f M T ( x, t, λ ) is the analogous matrix of M T ( x, t, λ ) but defined by the new canonical variable˜ u . Due to the canonical property of the transformation, we immediately conclude that M ( x, t, λ )satisfies the same r -matrix relation as that of M T ( x, t, λ ), that is {M ( x, t, λ ) , M ( x, t, µ ) } = [ r ( λ − µ ) , M ( x, t, λ ) ⊗ M ( x, t, µ )] . (4.24)As a result, the trace of the monodromy matrix M ( x, ∞ , λ ) provides a generating function forthe conserved quantities that are in involution with respect to the Poisson bracket (4.12). Thus,we establish Liouville integrability of the time-dependent defect NLS system (3.1). In this section, we generalize the above arguments for integrability to the case that there aremultiple time-dependent defects in a system.Let us first fix some notations. We assume that c ( t ), c ( t ), · · · , c n ( t ) are n functions ofclass C such that c ( t ) < c ( t ) < · · · < c n ( t ). We consider n + 1 auxiliary problems for φ ( j ) , j = 0 , · · · , n , with Lax pair U ( j ) , V ( j ) defined as in (2.1) with the fields u ( j ) , v ( j ) , replacing u , v .We assume that the auxiliary problem for U (0) , V (0) exists for x < c ( t ), the one for U ( j ) , V ( j ) exists for c j ( t ) < x < c j +1 ( t ), j = 1 , · · · , n −
1, and the one for U ( n ) , V ( n ) exists for x > c n ( t ).At x = c j ( t ), j = 1 , · · · , n , we assume that the two systems are connected via the condition φ ( j − ( c j ( t ) , t, λ ) = B ( j ) ( c j ( t ) , t, λ ) φ ( j ) ( c j ( t ) , t, λ ) , (5.1)where B ( j ) ( x, t, λ ), j = 1 , · · · , n , satisfy B ( j ) x ( x, t, λ ) = U ( j − ( x, t, λ ) B ( j ) ( x, t, λ ) − B ( j ) ( x, t, λ ) U ( j ) ( x, t, λ ) , (5.2a) B ( j ) t ( x, t, λ ) = V ( j − ( x, t, λ ) B ( j ) ( x, t, λ ) − B ( j ) ( x, t, λ ) V ( j ) ( x, t, λ ) . (5.2b)6Given these notations, we find the following proposition, whose proof is similar to that ofproposition 1. Proposition 2
In the presence of multiple time-dependent defects, the generating function forthe integrals of motion reads I ( λ ) = Z c ( t ) −∞ u (0) Γ (0) dx + n − X j =1 Z c j +1 ( t ) c j ( t ) u ( j ) Γ ( j ) dx + Z ∞ c n ( t ) u ( n ) Γ ( n ) dx + I defect ( λ ) , (5.3) where I defect ( λ ) = − n X j =1 ln( B ( j )11 + B ( j )12 Γ ( j ) ) (cid:12)(cid:12)(cid:12) x = c j ( t ) . (5.4) Here B ( j )11 and B ( j )12 denote respectively the -entry and -entry of the defect matrix B ( j ) . The classical r -matrix approach can also be implemented to the system with multiple defects.Indeed, we define the transition matrix for the defect NLS equation as follows M ( x, t, λ ) = M (0) T ( x, t, λ ) , −∞ < x < c ( t ) ,M ( j ) T ( x, t, λ ) , c j ( t ) ≤ x < c j +1 ( t ) , j = 1 , · · · , n − ,M ( n ) T ( x, t, λ ) , c n ( t ) ≤ x < ∞ , (5.5)where M ( j ) T ( x, t, λ ) = x exp Z t −∞ V ( j ) ( x, τ, λ ) dτ, j = 0 , , · · · , n. (5.6)Then M ( x, t, λ ) satisfies the same r -matrix relation as that of (4.24). This fact immediatelyyields the Poisson commutativity of the motion integrals that generated from the trace of themonodromy matrix M ( x, ∞ , λ ). It is now our aim to seek soliton solutions that can meet the defect of a system. To fix ourideas, we consider the KdV equation as an illustrative example. We will focus on an interestingcase: the defect moves at a constant speed. We will show that the KdV equation with such atime-dependent defect admits peakon solutions.Recall that a single-soliton for the KdV equation is given by u ( x, t ) = 8 k (cid:0) α exp( ξ ) + α − exp( − ξ ) (cid:1) , ξ = k ( x − k t ) , k > , (6.1)7where α is a positive constant. We assume that the defect takes place at x = 4 k t (i.e. thespeed of the defect is in coincidence with the wave speed). In the presence of such a defect, wetake the soliton on the other side of the defect in a similar form˜ u ( x, t ) = 8 k (cid:0) α exp( ξ ) + α − exp( − ξ ) (cid:1) , ξ = k ( x − k t ) , k > , (6.2)where α is a parameter to be determined by the defect condition. By applying the defectcondition (3.2c) and (3.2d), we find that α is determined by2 k α − α − (cid:0) α + α − (cid:1) + α − α − (cid:0) α + α − (cid:1) ! = (cid:0) α + α − (cid:1) − (cid:0) α + α − (cid:1) ! × vuut β − k (cid:0) α + α − (cid:1) + 1 (cid:0) α + α − (cid:1) ! . (6.3)We further restrict our attention to find a solution such that there is no discontinuity at thedefect. This requirement implies that (cid:0) α + α − (cid:1) = (cid:0) α + α − (cid:1) . (6.4)A nontrivial solution for α satisfying both (6.3) and (6.4) is that α = α − . Let α = exp γ .The solutions (6.1) and (6.2) on each side of the defect can be written in a uniform form: u ( x, t ) = 2 k sech ( | ξ | + γ ). To sum up, we find Proposition 3
Let the defect move at a constant speed x = 4 k t , and let the defect conditionbe defined by (3.2c) and (3.2d) with x = c ( t ) replaced by x = 4 k t . The KdV equation with sucha defect admits the following single-peakon solution u ( x, t ) = 2 k sech ( | ξ | + γ ) , ξ = k ( x − k t ) . (6.5)If γ >
0, (6.5) presents a peakon wave with discontinuous first derivative at the peak (at ξ = 0); see figure 1 for a profile of this wave. If γ <
0, (6.5) presents a wave with two peaks (at ξ = ± γ ) pointing upwards and one peak (at ξ = 0) pointing downwards (called an anti-peakonsimply), where the first derivative is discontinuous at ξ = 0 (the position of the anti-peakon);see figure 2 for a profile of this wave. We note that the existence of peakon solutions was knownas a typical feature of the CH type equations [25, 26]. Here our results show that the usualsoliton equations in the presence of time-dependent defects (such as the defect KdV equationdiscussed above) can also admit peakon solutions. We should emphasize that peakon solutionsfor the CH type equations and peakon solutions presented here should be interpreted in twodifferent senses: the former ones should be interpreted in a suitable weak sense, while the latterones should be interpreted in the sense that there is a time-dependent defect.8Figure 1: The single peakonsolution determined by (6.5)with parameters k = γ = 1. Figure 2:
The solution deter-mined by (6.5) with parameters k = − γ = 1. Figure 3:
The M-shape peakonsolution determined by (6.6)with N = 1 and parameters k = γ = 1. We now extend the above results to the case of the KdV equation with multiple defectslocated at different positions. For clarity, we will assume, in the following, γ > ξ = jγ , j = 0 , ± , · · · , ± N . As above, we restrict our attention to asolution which is continuous at the defects. In analogy with the analysis used above, we find Proposition 4
Assume that γ > and the defects locate at ξ = jγ , j = 0 , ± , · · · , ± N . Letthe defect conditions be defined by (3.2c) and (3.2d) with x = c ( t ) replaced by x = 4 k t + jk − γ , j = 0 , ± , · · · , ± N . The KdV equation with such multiple defects admits the following multi-peakon solution u ( x, t ) = k sech ( | ξ + N γ | + γ ) , −∞ < ξ ≤ − ( N − γ, k sech ( | ξ + ( N − j ) γ | + γ ) , (2 j − − N ) γ < ξ ≤ (2 j + 1 − N ) γ, j = 1 , · · · , N, k sech ( | ξ − N γ | + γ ) , ( N + 1) γ < ξ < ∞ , (6.6) where ξ = k ( x − k t ) . The above solution (6.6) represents a wave which has ( N + 1) peakons at ξ = (2 m − N ) γ , m = 0 , , · · · N , and N anti-peakons at ξ = (2 m − − N ) γ , m = 1 , · · · N . For example, for N = 1, it has two peakons at ξ = ± γ , and one anti-peakon at ξ = 0, and it looks like a “M”shape wave; see figure 3 for a profile of this M-shape wave solution. Remark 3.
The above solutions are derived by a direct ansatz for the fields to eitherside of the defect tuned to satisfy the defect condition. The fact that the defect condition iscorresponding to a BT implies that we can systematically construct the solution of the defectsystem in the following way. Given a solution u ( x, t ) of the bulk system for x ∈ ( c ( t ) , ∞ ), we9first implement a BT for all x , t to find ˜ u ( x, t ). Then we define ˜ u ( x, t ) as the solution of thebulk system for x ∈ ( −∞ , c ( t )). The solution constructed in such a manner solves the equationin the bulk as well as satisfies the defect condition at x = c ( t ), thus it provides a solution of thedefect system. For the case of the defect being fixed, this strategy has been employed recentlyin [18] to construct finite-gap solutions for the defect KdV and sine-Gordon equations. For thecase of the defect moving with time as presented in this paper, similar considerations will beinvestigated in the future. We have studied (1 + 1)-dimensional integrable soliton equations associated with the AKNS sys-tem in the presence of time-dependent defects. We defined the defect condition as a B¨acklundtransformation evaluated at the time-dependent defect location. We demonstrated that theresulting defect systems admit Lagrangian descriptions and established the integrability of theresulting defect systems both by constructing an infinite set of conserved densities and by imple-menting the classical r -matrix method. We also studied soliton solutions for the defect systems.Although our results are presented for integrable soliton equations in continuous case, it is clearthat analogous results can be applied to integrable soliton equations in discrete case, such asthe integrable discrete NLS equation and the Toda lattice equation. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos.11771186 and 11671177).