Localized structures on librational and rotational travelling waves in the sine-Gordon equation
LLocalized structures on librational and rotationaltravelling waves in the sine–Gordon equation
Dmitry E. Pelinovsky and Robert E. White
Department of Mathematics, McMaster University,Hamilton, Ontario, Canada, L8S 4K1
Abstract
We derive exact solutions to the sine–Gordon equation describing localized structureson the background of librational and rotational travelling waves. In the case of librationalwaves, the exact solution represents a localized spike in space-time coordinates (a roguewave) which decays to the periodic background algebraically fast. In the case of rotationalwaves, the exact solution represents a kink propagating on the periodic background anddecaying algebraically in the transverse direction to its propagation. These solutionsmodel the universal patterns in the dynamics of fluxon condensates in the semi-classicallimit. The different dynamics is related to different outcomes of modulational stability ofthe librational and rotational waves.
This paper is inspired by the series of works [3, 4, 18] on dynamics of the sine-Gordonequation in the semi-classical limit. This physical regime is relevant for propagation of themagnetic flux along superconducting Josephson junctions [20]. Other physical applicationsof the sine–Gordon equation include crystal dislocations, DNA double helix, fermions in thequantum field theory, and structures in galaxies (see reviews in [2, 21]).The sine–Gordon equation in the semi-classical limit can be written in the form: (cid:15) u T T − (cid:15) u XX + sin( u ) = 0 , (1.1)where the subscripts denote partial derivatives of u = u ( X, T ) and the parameter (cid:15) is small. Byusing the initial data with zero displacement and large velocity, u ( X,
0) = 0 and (cid:15)u T ( X,
0) = G ( X ), the authors of [3, 4, 18] studied the sequence { (cid:15) N } N ∈ N with (cid:15) N → N → ∞ , where (cid:15) N is defined from the N -soliton (reflectionless) potential associated with the (cid:15) -independent velocityprofile G ( X ). The sequence of solutions was termed as the fluxon condensate . The regime ofrotational waves with (cid:107) G (cid:107) L ∞ > (cid:107) G (cid:107) L ∞ < θ = θ ( t ) satisfying θ (cid:48)(cid:48) ( t ) + sin( θ ( t )) = 0 . (1.2)1 a r X i v : . [ n li n . S I] J un t was suggested in [4] that the dynamics of fluxon condensates was different between therotational and librational regimes. In both cases, the initial evolution in the semi-classical limitcan be modeled by the travelling wave with slowly varying parameters. Dynamics of librationalwaves is affected by the gradient catastrophe and the emergence of a universal pattern of roguewaves (localized spikes in space-time on a distributed background) [18]. Dynamics of rotationalwaves is accompanied by the emergence of a universal pattern of propagating kinks and antikinksat the interface between the rotational and librational motion of the fluxon condensate [3].Figure 1 (reproduced from [4]) shows the dynamics of cos( u ) in the sine–Gordon equation(1.1) with (cid:15) = (cid:15) N for N = 4 , ,
16. The top panel shows the regime of librational waves inducedby the initial data u ( X,
0) = 0 and (cid:15)u T ( X,
0) = G ( X ) with (cid:107) G (cid:107) L ∞ <
2. The bottom panelshows the regime of rotational waves, for which (cid:107) G (cid:107) L ∞ > to develop a short and simple algebraic method , which allows us to construct theexact solutions for the principal waveforms that make dynamics of librational and rotationalwaves so different. In the case of librational waves, we derive a closed-form solution for arogue wave decaying algebraically to the periodic background in all directions. In the case ofrotational waves, we derive a closed-form solution for propagating kinks and antikinks thatdecay algebraically to the periodic background in the transverse direction to their propagation.These solutions with localized structures on librational and rotational waves are associated withthe particular eigenvalues in the Lax spectrum for which the eigenfunctions are bounded andperiodic in space-time coordinates. Since we are not dealing with the initial-value problem inthe semi-classical limit, we can scale the space-time coordinates and consider the normalizedsine–Gordon equation: u tt − u xx + sin( u ) = 0 , (1.3)where u = u ( x, t ).Although the algebraic method used for librational and rotational waves is similar, theoutcomes are different dynamically. This difference is explained by the different types of spec-tral stability of the travelling periodic waves [15, 16, 19] (see also [12, 13] and [11] for recentcontributions). In the superluminal regime (which is the only regime we are interested in),the librational periodic waves are spectrally unstable and the Floquet-Bloch spectrum forms afigure eight intersecting at the origin. Such instability is usually referred to as modulationalinstability [15, 16]. On the other hand, the rotational periodic waves are modulationally stablein the sense that the only Floquet–Bloch spectrum near the origin is represented by the verticalbands along the purely imaginary axis. The rotational waves are still spectrally unstable in thesuperluminal regime but the unstable band is given by bubbles away from the origin (see Fig.2in [15], Figs.1-2 in [19], Fig. 6 in [12], or Fig. 1 in [11]).We develop the algebraic method which was previously applied to the modified KdV equa-tion in [7, 8] and the focusing cubic NLS equation in [6, 9, 10]. The travelling periodic wavesand the periodic eigenfunctions in space-time coordinates are characterized by using nonlin-earization of the Lax equations [5]. This method allows us to find particular eigenvalues inthe Lax spectrum, for which the first solutions to the Lax equations are bounded and periodicwhereas the second, linearly independent solutions are unbounded and non-periodic. When the2igure 1: Surface plots of cos( u ) in space-time coordinates for the dynamics of fluxon conden-sates in the semi-classical limit. Top: librational waves. Bottom: rotational waves. Reproducedfrom [4] with permission of the authors.second solutions of the Lax equations are used in the Darboux transformation, new solutionsof integrable equations are generated from the travelling periodic wave solutions. The newsolutions represent algebraically localized structures on the background of travelling periodicwaves. Similar solutions but in a different functional-analytic form were obtained in [14] for3he NLS equation and in [17] and [18] for the sine–Gordon equation.The algebraic method can be applied similarly to what was done in [7, 8] because the sine–Gordon equation is related to the same Lax spectral problem as the modified KdV, the cubicNLS, and other integrable equations considered in the seminal work [1]. In order to enable thisapplication, we have to rewrite the sine–Gordon equation in the characteristic coordinates anduse the chain rule for the inverse transformation of variables. Since many computational detailsare similar, we will omit many computations and refer to [7, 8] or to [22] where computationaldetails can be found.Rogue waves on the background of librational waves are displayed on Figure 2, where surfaceplots of sin( u ) are plotted versus ( x, t ). The wave patterns are very similar to the solutionsfrom Appendix D of [18]. This confirms that rogue waves on a background of librational wavesmodel defects in the fluxon condensate obtained in [18] from the Riemann–Hilbert problem.Figure 2: Surface plots of sin( u ) versus ( x, t ) for rogue waves on the background of librationalwaves for different values of k . Top left: k = sin( π ). Top right: k = sin( π ). Bottom left: k = sin( π ). Bottom right: k = sin( π ).Kinks and antikinks propagating on the background of rotational waves are shown on Fig-ure 3, where the surface plots of sin( u ) are plotted versus ( x, t ). The wave patterns appear4ery similar to the propagation of kinks and antikinks studied for the dynamics of the fluxoncondensate in the semi-classical limit [3, 4].Figure 3: Surface plots of sin( u ) versus ( x, t ) for kinks (left) and antikinks (right) propagatingon the background of rotational waves for different values of k . Top: k = sin( π ). Bottom: k = sin( π ).This article is organized as follows. Travelling periodic waves of the sine–Gordon equationare expressed by elliptic functions in Section 2. Lax equations are introduced for the sine–Gordon equation in characteristic variables in Section 3. The algebraic method is developed inSection 4, where the bounded periodic eigenfunctions in space-time coordinates are explicitlycomputed for particular eigenvalues in the Lax spectrum. The new solutions on the back-ground of the rotational (libratitional) waves are constructed in Section 5 (Section 6). Section7 concludes the paper with the summary. 5 Travelling periodic waves
Travelling wave solutions of the sine-Gordon equation (1.3) are written in the form u ( x, t ) = f ( x − ct ), where c is the wave speed and f ( x ) : R → R is the wave profile satisfying the followingdifferential equation: ( c − f (cid:48)(cid:48) + sin( f ) = 0 , (2.1)where the prime corresponds to differentiation in x (after translation to the right by ct ). Super-luminal motion corresponds to c >
1, in which case the following transformation f ( x ) = ˆ f (ˆ x )with ˆ x = x/ √ c − f (cid:48)(cid:48) + sin( ˆ f ) = 0 , (2.2)where the prime now corresponds to differentiation in ˆ x . In what follows, we drop hats forsimplicity of notations.The reason why the travelling wave solutions to the sine–Gordon equation (1.3) can beexpressed without wave speed c is the following Lorentz transformation for c > c < x = x − ct √ c − , ˆ t = t − cx √ c − , ˆ u = π + u, (2.3)where ˆ u = ˆ u (ˆ x, ˆ t ) satisfies the same sine–Gordon equation (1.3). The time-independent functionˆ u (ˆ x, ˆ t ) = π + ˆ f (ˆ x ) satisfies the differential equation (2.2). -6 -4 -2 0 2 4 6 f -3-2-10123 f ' T o t a l E ne r g y ( E ) Figure 4: Orbits of the second-order equation (2.2) on the phase plane ( f, f (cid:48) ).6he second-order equation (2.2), where hats are now dropped, is integrable with the first-order invariant: E ( f, f (cid:48) ) := 12 ( f (cid:48) ) + 1 − cos( f ) . (2.4)It is straightforward to verify that E ( f, f (cid:48) ) is constant in x along the solutions of the second-order equation (2.2). The level sets of E ( f, f (cid:48) ) represent all solutions to the differential equation(2.2) as orbits on the phase plane ( f, f (cid:48) ). Figure 4 plots the level sets of E ( f, f (cid:48) ). There arethree different cases for f ∈ [ − π, π ]. When E ∈ (0 ,
2) the level curve is a periodic orbit centeredaround (0 ,
0) which corresponds to librational motion. When E = 2 there are two heteroclinicorbits connecting ( − π,
0) to ( π,
0) which are referred to as kinks. Orbits for
E > z = F ( τ, k ) = (cid:90) τ dt (cid:112) − k sin t , where k ∈ (0 ,
1) is the elliptic modulus. The complete elliptic integral is defined as K ( k ) = F ( π , k ). The first two Jacobi elliptic functions are defined by sn( z, k ) = sin τ and cn( z, k ) =cos τ such that sn ( z, k ) + cn ( z, k ) = 1 . (2.5)These functions are smooth, sign-indefinite, and periodic with the period 4 K ( k ). The thirdJacobi elliptic function is defined from the quadratic formuladn ( z, k ) + k sn ( z, k ) = 1 . (2.6)The function dn( z, k ) is given by the positive square root of (2.6), so that it is smooth, positive,and periodic with the period 2 K ( k ). The Jacobi elliptic functions are related by the derivatives: ddz sn( z, k ) = cn( z, k ) dn( z, k ) , ddz cn( z, k ) = − sn( z, k ) dn( z, k ) , ddz dn( z, k ) = − k sn( z, k ) cn( z, k ) . (2.7)For E ∈ (0 , x by cos( f ) = 1 − k sn ( x, k ) , sin( f ) = 2 k sn( x, k )dn( x, k ) f (cid:48) = 2 k cn( x, k ) , (2.8)where E = 2 k ∈ (0 , f ) and sin( f ) are consistent due to (2.7). The period of the librationalwaves (2.8) is L = 4 K ( k ). 7or E ∈ (2 , ∞ ), the rotational waves of the first-order invariant (2.4) are given up to anarbitrary translation in x by cos( f ) = 1 − ( k − x, k ) , sin( f ) = ± k − x, k )cn( k − x, k ) f (cid:48) = ± k − dn( k − x, k ) , (2.9)where E = 2 k − ∈ (2 , ∞ ) and the upper/lower sign corresponds to the orbit in the upper/lowerhalf plane on Fig. 4. Again, the first-order invariant (2.4) is satisfied due to (2.6), the trigono-metric identity is satisfied due to (2.5), and the derivative of cos( f ) and sin( f ) are consistentdue to (2.7). The period of the rotational waves (2.9) is L = 2 kK ( k ). Lax equations for the sine–Gordon equation (1.3) are rather combursome [4, 12]. Therefore,we adopt the following characteristic coordinates: ξ = 12 ( x + t ) , η = 12 ( x − t ) . (3.1)The sine–Gordon equation (1.3) can be written in a simpler form: u ξη = sin( u ) , (3.2)where u = u ( ξ, η ). The travelling periodic wave is now given by u ( ξ, η ) = ˆ f ( ξ − η ), whereˆ f ( ξ ) : R → R satisfies the second-order equation (2.2), where the prime represents the derivativewith respect to ˆ x = ξ − η = t . Note that t and ˆ x are equivalent due to the Lorenz transformation(2.3).Lax equations for the sine-Gordon equation in characteristic coordinates (3.2) are given bythe following system: ∂∂ξ (cid:20) pq (cid:21) = 12 (cid:20) λ − u ξ u ξ − λ (cid:21) (cid:20) pq (cid:21) (3.3)and ∂∂η (cid:20) pq (cid:21) = 12 λ (cid:20) cos( u ) sin( u )sin( u ) − cos( u ) (cid:21) (cid:20) pq (cid:21) , (3.4)where λ ∈ C is the spectral parameter and χ := ( p, q ) T is an eigenfunction written in variables( ξ, η ). Validity of the sine–Gordon equation (3.2) as the compatibility condition χ ξη = χ ηξ canbe checked by direct differentiation [1]. The first equation (3.3) is referred to as the AKNSspectral problem with the potential w := − u ξ .When w = − ˆ f (cid:48) (ˆ x ) is a travelling periodic wave with the fundamental period L , the AKNSspectral problem determines the Lax spectrum in L ( R ) as the set of all admissible values of λ for which χ ∈ L ∞ ( R ). By Floquet theorem, bounded solutions of the linear equation (3.3) canbe represented in the form χ ( ξ, η ) = φ ( ξ − η ) e iµ ( ξ − η )+Ω η , (3.5)8here φ is L -periodic, µ is defined in the fundamental region [ − πL , πL ], and Ω is a new spectralparameter arising in the separation of variables in the second Lax equation (3.4) [12]. Theadmissible values of λ in C are defined by periodic solutions of the following eigenvalue problem: (cid:20) dd ˆ x + 2 iµ ˆ f (cid:48) (ˆ x )ˆ f (cid:48) (ˆ x ) − dd ˆ x − iµ (cid:21) φ = λφ, (3.6)where ˆ x := ξ − η and µ ∈ [ − πL , πL ]. The spectral parameter Ω determines an eigenvalue of thespectral stability problem for the travelling periodic wave evolving with respect to the coordintae η (see Theorem 5.1 in [12] for spectral stability of the travelling periodic wave evolving withrespect to the time variable t ). Compared to [12], we will not explore the spectral stabilityof travelling periodic waves but will construct solutions to the Lax equations (3.3) and (3.4)which correspond to Ω = 0. Such eigenfunctions χ are bounded in both ξ and η , hence in thespace-time coordinates ( x, t ). The purpose of the algebraic method is to relate solutions of the nonlinear integrable equa-tion and solutions of the associated linear Lax equations in order to obtain an explicit expressionfor the particular eigenvalues of the Lax spectrum. These eigenvalues correspond to boundedeigenfunctions in the space-time coordinates. Our presentation of the algebraic method followsclosely to [7, 8] devoted to the mKdV equation because the AKNS spectral problem (3.3) isidentical with the potential w (ˆ x ) := − ˆ f (cid:48) (ˆ x ), where ˆ x = ξ − η = t . As previously mentioned, wewill drop hats for simplicity of notations.Assume that ( p , q ) is a solution to the AKNS spectral problem (3.3) for a fixed value of λ = λ . Assume that the solution u = u ( ξ, η ) to the sine–Gordon equation (3.2) is related tothe squared eigenfunctions by − u ξ = p + q . (4.1)The linear equation (3.3) with the constraint (4.1) becomes a nonlinear Hamiltonian systemwith Hamiltonian H ( p , q ) = λ p q + 14 ( p + q ) , (4.2)so that ∂∂ξ (cid:20) p q (cid:21) = 12 (cid:20) − (cid:21) (cid:20) ∂H∂p ∂H∂q (cid:21) . (4.3)Let us denote the constant value of H ( p , q ) at the solutions of (4.3) by H so that λ p q = H −
14 ( f (cid:48) ) . (4.4)Recall that u ( ξ, η ) = f ( x ) solves the second-order equation f (cid:48)(cid:48) + sin( f ) = 0 , (4.5)9erivative of which yields f (cid:48)(cid:48)(cid:48) + cos( f ) f (cid:48) = 0 . (4.6)Comparing (4.6) with (2.4) and eliminating cos( f ) produces the third-order equation f (cid:48)(cid:48)(cid:48) = f (cid:48) ( E − −
12 ( f (cid:48) ) , (4.7)where E is constant.Differentiating the constraint (4.1) twice and using (4.3) gives − f (cid:48)(cid:48) = λ ( p − q ) (4.8)and f (cid:48)(cid:48)(cid:48) = λ f (cid:48) + 2 λ f (cid:48) p q . (4.9)Substituting (4.4) for λ p q into (4.9) yields f (cid:48)(cid:48)(cid:48) = λ f (cid:48) + 2 H f (cid:48) −
12 ( f (cid:48) ) . (4.10)Comparing (4.10) with (4.7) gives the following relation: E = λ + 2 H + 1 . (4.11)In order to determine the explicit formula for λ in terms of E , we shall integrate thenonlinear system (4.3) by using the Lax equation:2 ∂∂ξ W ( λ ) = Q ( λ ) W ( λ ) − W ( λ ) Q ( λ ) , (4.12)where Q ( λ ) = (cid:20) λ p + q − p − q − λ (cid:21) , W ( λ ) = (cid:20) W ( λ ) W ( λ ) W ( − λ ) − W ( − λ ) (cid:21) (4.13)with W ( λ ) = 1 − p q λ − λ + p q λ + λ , W ( λ ) = p λ − λ + q λ + λ . (4.14)Substituting (4.1), (4.4), and (4.8) into (4.14) yields the following expressions: W ( λ ) = 1 − H − ( f (cid:48) ) λ − λ ) , W ( λ ) = − λf (cid:48) − f (cid:48)(cid:48) λ − λ . (4.15)The determinant of W ( λ ) is computed from (4.14) asdet[ W ( λ )] = − [ W ( λ )] − W ( λ ) W ( − λ )= − λ p q + ( p + q ) λ − λ = − H λ − λ , W ( λ )] only admits simple poles at λ = ± λ . On theother hand, the determinant of W ( λ ) is computed from (4.15) as det [ W ( λ )] = − H − ( f (cid:48) ) λ − λ + ( λ + 2 H )( f (cid:48) ) − ( f (cid:48)(cid:48) ) − H − ( f (cid:48) ) ( λ − λ ) = − H λ − λ + 4( λ + 2 H )( f (cid:48) ) − f (cid:48)(cid:48) ) − H − ( f (cid:48) ) λ − λ ) . Comparison of these two equivalent expressions yields the constraint:( λ + 2 H )( f (cid:48) ) − ( f (cid:48)(cid:48) ) − H −
14 ( f (cid:48) ) = 0 . (4.16)By the fundamental trigonometric identity, we obtain from (2.4) and (4.5):1 = sin ( f ) + cos ( f ) = ( f (cid:48)(cid:48) ) + 14 ( f (cid:48) ) + (1 − E )( f (cid:48) ) + (1 − E ) . (4.17)Comparing (4.16) and (4.17) yields the relation4 H = E ( E − , (4.18)in addition to (4.11). Expressing H from (4.18) and substituting into (4.11) yield admissiblevalues of λ by λ = E − ∓ (cid:112) E ( E − , (4.19)where the plus and minus sign correspond to the two roots in2 H = ± (cid:112) E ( E − . (4.20)For the rotational waves (2.9), we have E = 2 /k , so that one can extract the square rootfrom (4.19) and obtain two real pairs of admissible values ± λ with λ = 1 ∓ √ − k k , (4.21)where the plus and minus signs correspond to the signs in H = ± √ − k k . (4.22)For the libratitional waves (2.8), we have E = 2 k so that one can again extract the squareroot from (4.19) and obtain a complex quadruplet of admissible values {± λ , ± ¯ λ } with λ = k + i √ − k , (4.23)where the unique λ is located in the first quadrant of the complex plane. This eigenvaluecorresponds to the choice in H = − ik √ − k . (4.24)11 Real Part -1-0.8-0.6-0.4-0.200.20.40.60.81 I m ag i na r y P a r t -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Real Part -1-0.8-0.6-0.4-0.200.20.40.60.81 I m ag i na r y P a r t -1.5 -1 -0.5 0 0.5 1 1.5 Real Part -1.5-1-0.500.511.5 I m ag i na r y P a r t -1.5 -1 -0.5 0 0.5 1 1.5 Real Part -1.5-1-0.500.511.5 I m ag i na r y P a r t Figure 5: The Lax spectrum of (3.3) associated with the rotational (top) and librational (bot-tom) waves for k = 0 .
85 (left) and k = 0 .
95 (right). Red dots represent eigenvalues (4.21) and(4.23).We approximate numerically the Lax spectrum of the AKNS spectral problem (3.3) byusing the Floquet theorem and converting the spectral problem to the form (3.6). By usingdiscretization of the spatial domain [0 , L ] and the range of the µ values in [ − πL , πL ] we reduce(3.6) to the matrix eigenvalue problem for each µ , this problem is handled using Matlab’s eig()function. The derivative operator dd ˆ x is replaced with the 12 th order finite difference matrix toensure high accuracy of computations. The union of each set of eigenvalues associated for each µ defines the Lax spectrum.Figure 5 shows the numerically constructed Lax spectra for the rotational and librationalwaves using certain values of k . The end points of the spectral bands outside i R correspond tothe eigenvalues (4.21) and (4.23).Lax spectra on Figure 5 correspond to the AKNS spectral problem (3.3) for the sine–Gordonequation in characteristic variables ( ξ, η ). The location of the Lax spectrum in space-timecoordinates ( x, t ) is different because the bounded eigenfunctions in ξ are located at differentvalues of λ ∈ R compared to bounded functions in x = ξ + η . Nevertheless, the eigenvalues(4.21) and (4.23) belong to the Lax spectrum in ( x, t ) because the corresponding eigenfunctions12re bounded and periodic both in x and t . The same eigenvalues are shown by crosses on Fig.7 in [12], from which it is clear that the eigenvalues (4.21) and (4.23) do not appear as the endpoints of the Lax spectrum in the space-time coordinates ( x, t ). Let ( p, q ) be a solution to the linear equations (3.3) and (3.4) for a fixed value of λ and forthe solution u = u ( ξ, η ) of the sine–Gordon equation (3.2). As is shown in [7], the new solutionˆ u = ˆ u ( ξ, η ) to the sine–Gordon equation is given by the one-fold Darboux transformation:ˆ w = w + 4 λpqp + q , (5.1)where w := − u ξ and ˆ w := − ˆ u ξ . If u = f ( ξ − η ) is the rotational wave given by (2.9) with x = ξ − η and λ = λ is given by the algebraic method with the eigenfunction ( p, q ) = ( p , q )satisfying (4.1) and (4.4), the one-fold Darboux transformation (5.1) yieldsˆ w = w + 4 λ p q p + q = w + 4 H − w w = 4 H w . (5.2)Since w = − f (cid:48) is given by (2.9) and H is given by (4.22), we obtain up to the sign changes:ˆ w = ± k − √ − k dn( k − x ; k ) = ± k − dn( k − x + K ( k ); k ) . (5.3)The new solution (5.3) is just a half-period translated and reflected version of the rotationalwave, which is periodic with the period L = 2 kK ( k ).In order to construct a new solution to the sine–Gordon equation on the background of therotational wave (2.9), we are looking for the second, linear independent solution to the linearequations (3.3) and (3.4) with the same value of λ = λ . We will define the second solution inthe same form as is used in [8]:ˆ p = p φ R − q p + q , ˆ q = q φ R + p p + q , (5.4)where the function φ R = φ R ( ξ, η ) satisfies the system of scalar equations: ∂φ R ∂ξ = − λ p q ( p + q ) , λ ∂φ R ∂η = ( p − q ) sin( f ) − p q cos( f )( p + q ) . (5.5)The representation (5.4) is non-singular for the rotational waves because w = p + q = − f (cid:48) has the sign-definite f (cid:48) in (2.9). As we prove below, the exact expression for φ R is given by φ R ( ξ, η ) = C + 12 ( ξ + η ) − H (cid:90) ξ − η dx ( f (cid:48) ) , (5.6)13here C is an arbitrary constant of integration. Indeed, by using (4.1), (4.4), and (4.8) werewrite (5.5) in the form: ∂φ R ∂ξ = 12 − H ( f (cid:48) ) , λ ∂φ R ∂η = − f (cid:48)(cid:48) sin( f ) + (4 H − ( f (cid:48) ) ) cos( f )2( f (cid:48) ) . (5.7)By using (2.4), (4.5), (4.11), and (4.16), the second equation of system (5.7) is simplified to ∂φ R ∂η = 12 + 2 H ( f (cid:48) ) , (5.8)which implies (5.6) due to the first equation of system (5.7) and f = f ( ξ − η ).If f and ( p , q ) are L -periodic functions in x := ξ − η with period L = 2 kK ( k ), the function φ R and (ˆ p , ˆ q ) are non-periodic. When the second, linearly independent solution ( p, q ) = (ˆ p , ˆ q )is used in the one-fold Darboux transformation (5.1), it generates a new solution with analgebraic structure on the background of the rotational waves. The new solution approachesthe rotational wave along the directions in the ( ξ, η ) plane where | φ R | grows to infinity.We recall (2.9) and (5.3) to rewrite (5.6) in the equivalent form: φ R ( ξ, η ) = C + 12 ( ξ + η ) − H k − k ) (cid:90) k − ( ξ − η )0 dn ( z + K ( k ); k ) dz. (5.9)Also recall the complete elliptic integral of the second kind E ( k ) = (cid:90) K ( k )0 dn ( z ; k ) dk. Over the period L = 2 kK ( k ), the integral in (5.9) is incremented by 2 E ( k ), hence | φ R ( ξ, η ) | →∞ along every direction in the ( ξ, η )-plane with the exception of the direction of the straightline: Ω := (cid:26) ( ξ, η ) ∈ R : ( ξ + η ) − H k E ( k )(1 − k ) K ( k ) ( ξ − η ) = 0 (cid:27) . (5.10)The integration constant C serves as a parameter which translates the straight line Ω in the( ξ, η )-plane within the period of the rotational wave.Let us now take the one-fold Darboux transformation (5.1) with the second linearly inde-pendent solution (5.4) for the admissible eigenvalues λ given by (4.21). By using the relations(4.1), (4.4), and (4.8), we obtainˆ w = w + 4 λ ˆ p ˆ q ˆ p + ˆ q = w + 4 λ [ p q ( φ R w −
1) + φ R w ( p − q )]( p + q )( φ R w + 1)= w + (4 H − w )( φ R w −
1) + 4 φ R w∂ ξ ww ( φ R w + 1) (5.11)14here ˆ w = − ˆ u ξ and w = − u ξ .We show next that the new solution (5.11) describes a kink propagating on the backgroundof the rotational wave. Indeed, the function φ R ( ξ, η ) : R (cid:55)→ R is bounded and periodicin the direction of the line Ω given by (5.10). In every other direction on the ( ξ, η )-plane, | φ R ( ξ, η ) | → ∞ so that the new solution (5.11) satisfies the limit:lim | φ R |→∞ ˆ w = w + 4 H − w w = 4 H w , (5.12)which coincides with (5.2). As follows from (5.3), this limit is a half-period translated andreflected version of the rotational wave. Since the divergence of | φ R ( ξ, η ) | → ∞ is linear in( ξ, η ) as follows from (5.9), the new solution (5.11) approaches the translated and reflectedrotational wave algebraically fast.Figure 6: Localized waves on the rotational wave with k = 0 .
95 generated from the one-foldDarboux transformation using eigenvalues (4.21) with the lower (left) and upper (right) signs.Along the direction Ω, the new solution (5.11) does not approach the rotational wave. Itfollows from (5.11) at the critical point of w = − f (cid:48) , where ∂ ξ w = − f (cid:48)(cid:48) is zero, that the maximumof | ˆ w | happens at the points, where φ R = 0 andˆ w | φ R =0 = w − H − w w = 2 w − H w . (5.13)Compared to the maximum of the rotational wave sup ( ξ,η ) ∈ R | w ( ξ, η ) | = 2 k − , the maximumof the new solution (5.11) is attained at sup ( ξ,η ) ∈ R | ˆ w ( ξ, η ) | = 2 k − M , where M is the magni-fication factor given by M ( k ) = 2 ∓ √ − k . (5.14)The sign choice in (5.14) corresponds to the sign choice in (4.21) and (4.22). The magnificationfactor M determines the maximum of the localized wave propagating on the background of the15otational waves in the direction of the straight line Ω. Position of the localized wave is changedby the parameter C for the integration constant. The localized wave is greater for the lowersign in (4.21) and (4.22). Note that the magnification factor in (5.14) was previously derivedfor similar solutions to the NLS and mKdV equations in [6, 7].Figure 6 illustrates the exact solution (5.11) for k = 0 .
95 and two sign choices in (4.21).The value of C is set to 0 in (5.9). We see numerically that the solution surface | ˆ w ( ξ, η ) | achieves its maximum at ( ξ, η ) = (0 ,
0) and is repeated along the direction of Ω. This is thedirection of propagation of the localized wave on the background of the rotational waves. Thelocalized wave has a bigger magnification for the larger value of λ (right panel) and smallermagnification for the smaller value of λ (left panel).By using the same solution formula (5.11), we have computed sin(ˆ u ) = ˆ u ξη by numericallydifferentiating ˆ w = − ˆ u ξ in η with a forward difference. The corresponding surface plots ofsin(ˆ u ) in ( x, t ) are presented on Figure 3. Note that the kink and antikink propagate intoopposite directions for the different sign choices of λ in (4.21). Indeed, it follows from (4.22)and (5.10) in variables ( x, t ) that the kink and antikink propagate along the straight lines x = ± E ( k ) √ − k K ( k ) t, (5.15)hence the propagation directions are opposite to each other. Since E ( k ) > √ − k K ( k ),the speed of propagation exceeds one, hence these solutions are relevant for the superluminaldynamics of the sine–Gordon equation (1.3). If the new solution ˆ u = ˆ u ( ξ, η ) to the sine–Gordon equation (3.2) is given by the one-foldDarboux transformation (5.11) and u = u ( ξ, η ) is the librational wave, then ˆ u is no longerreal-valued because H and λ are complex-valued in (4.23) and (4.24). The two-fold Darbouxtransformation is required to generate new real-valued solutions on the background of thelibrational waves.Let ( p , q ) and ( p , q ) be solutions to the linear equations (3.3) and (3.4) with fixed values of λ = λ and λ = λ such that λ (cid:54) = ± λ . As is shown in [7], the two-fold Darboux transformationtakes the form:ˆ w = w + 4( λ − λ )[ λ p q ( p + q ) − λ p q ( p + q )]( λ + λ )( p + q )( p + q ) − λ λ [4 p q p q + ( p − q )( p − q )] , (6.1)where w := − u ξ and ˆ w := − ˆ u ξ . We take λ and H as in (4.23) and (4.24), and define λ = ¯ λ with p = ¯ p and q = ¯ q . By using (4.1), (4.4), (4.8), and (4.16), we obtainˆ w = w + 4( λ − ¯ λ )( H − ¯ H ) w ( λ + ¯ λ ) w − − H + w + ( w (cid:48) ) ] = − w. (6.2)The new solution (6.2) is simply a reflected version of the librational wave. Therefore, we arelooking for the second, linearly independent solution to the linear equations (3.3) and (3.4) for16he same value of λ . One representation for the second solution is given by (5.4). However, w = p + q = − f (cid:48) crosses zero for librational waves, hence the representation (5.4) becomessingular at some points. For librational waves, we should define the second solutions in adifferent form used in [7]: ˆ p = φ L − q , ˆ q = φ L + 1 p , (6.3)where the function φ L = φ L ( ξ, η ) satisfies the system of scalar equations: ∂φ L ∂ξ = f (cid:48) ( p − q )2 p q φ L − f (cid:48) ( p + q )2 p q , λ ∂φ L ∂η = ( p + q ) sin( f )2 p q φ L − ( p − q ) sin( f )2 p q . (6.4)The representation (6.3) is non-singular because if either p or q vanish in some points, thenequations (4.1) and (4.8) yield a contradiction with real f and complex λ . As we prove below,the exact expression for φ L is given by φ L ( ξ, η ) = (4 H − ( f (cid:48) ) ) (cid:18) C + η λ + (cid:90) ξ − η λ ( f (cid:48) ) dx (4 H − ( f (cid:48) ) ) (cid:19) , (6.5)where C is an arbitary constant of integration. By substituting (4.1), (4.4), and (4.8) in (6.4),we obtain: ∂φ L ∂ξ = 2 f (cid:48) f (cid:48)(cid:48) ( f (cid:48) ) − H φ L − λ ( f (cid:48) ) ( f (cid:48) ) − H , λ ∂φ L ∂η = − λ f (cid:48) f (cid:48)(cid:48) ( f (cid:48) ) − H φ L + 2( f (cid:48)(cid:48) ) ( f (cid:48) ) − H . (6.6)By using φ L = (4 H − ( f (cid:48) ) )Υ (6.7)with Υ = Υ( ξ, η ), system (6.6) can be simplified to the form: ∂ Υ ∂ξ = 2 λ ( f (cid:48) ) (4 H − ( f (cid:48) ) ) , λ ∂ Υ ∂η = − f (cid:48)(cid:48) ) (4 H − ( f (cid:48) ) ) . (6.8)If follows from (4.16) and (6.8) that ∂ Υ ∂ξ + ∂ Υ ∂η = 12 λ , (6.9)which implies that Υ( ξ, η ) = C + η λ + G ( ξ − η ) (6.10)for some function G ( x ) : R (cid:55)→ C to be determined. Substituting this into (6.8) yields G (cid:48) = 2 λ ( f (cid:48) ) (4 H − ( f (cid:48) ) ) , so that integration and substitution into (6.7) and (6.10) yields (6.5).17he functions f and ( p , q ) are L -periodic functions with period L = 4 K ( k ) for librationalwaves, however, the functions φ L and (ˆ p , ˆ q ) are non-periodic. We shall prove that | φ L ( ξ, η ) | →∞ as | ξ | + | η | → ∞ everywhere in the ( ξ, η )-plane. Indeed, by factoring out λ in the secondterm of equation (6.5) and by using periodicity of 4 H − ( f (cid:48) ) , we have | φ L ( ξ, η ) | → ∞ if andonly if | ˜ φ L ( ξ, η ) | → ∞ , where˜ φ L ( ξ, η ) = η + (cid:90) ξ − η λ ( f (cid:48) ) (( f (cid:48) ) − H ) dx = η + (cid:90) ξ − η k −
1) + 2 ik √ − k ]( f (cid:48) ) (( f (cid:48) ) + 4 ik √ − k ) dx. Taking the imaginary part yieldsIm[ ˜ φ ] = 8 k √ − k (cid:90) ξ − η ( f (cid:48) ) − k − f (cid:48) ) − k (1 − k )(( f (cid:48) ) + 16 k (1 − k )) ( f (cid:48) ) dx = 128 k √ − k (cid:90) ξ − η k cn ( x ; k ) + (1 − k )cn ( x ; k ) + k − f (cid:48) ) + 16 k (1 − k )) ( f (cid:48) ) dx = − k √ − k (cid:90) ξ − η sn ( x ; k )dn ( x ; k )(( f (cid:48) ) + 16 k (1 − k )) ( f (cid:48) ) dx where we have used (2.8) in order to express f (cid:48) ( x ) = 2 k cn( x ; k ) and simplify the ellipticfunctions. The integrand is clearly positive for every k ∈ (0 , φ ] remainsbounded only in the diagonal direction on the ( ξ, η ) plane, however, in this direction Re[ ˜ φ ]grows linearly in η . Hence, | φ L ( ξ, η ) | → ∞ along every direction in the ( ξ, η ) plane.Let us now take the two-fold Darboux transformation (6.1) with the second, linearly inde-pendent solution (6.3) to the linear equations (3.3) and (3.4) for λ = k + i √ − k and λ = ¯ λ .The new solution is written in the form:ˆ w = w + 4( λ − λ )[ λ ˆ p ˆ q (ˆ p + ˆ q ) − λ ˆ p ˆ q (ˆ p + ˆ q )]( λ + λ )(ˆ p + ˆ q )(ˆ p + ˆ q ) − λ λ [4ˆ p ˆ q ˆ p ˆ q + (ˆ p − ˆ q )(ˆ p − ˆ q )] , (6.11)where (ˆ p , ˆ q ) are taken as the complex conjugate to (ˆ p , ˆ q ).We will prove that the new solution (6.11) describes an isolated rogue wave arising on thebackground of the librational wave. Indeed, the function φ L ( ξ, η ) : R (cid:55)→ R is unbounded inevery direction on the ( ξ, η ) plane, so thatlim | φ L |→∞ ˆ w = w + 4( λ − ¯ λ )( H − ¯ H ) w ( λ + ¯ λ ) w − − H + w + ( w (cid:48) ) ] = − w, (6.12)which coincides with (6.2). The divergence of | φ L ( ξ, η ) | → ∞ is again linear in ( ξ, η ) as followsfrom (6.5), hence the new solution (6.11) approaches the reflected librational wave algebraically.It follows from (6.11) at the critical points of w = − f (cid:48) , where ∂ ξ w = − f (cid:48)(cid:48) is zero, that themaximum of | ˆ w | happens at the points, where φ L = 0 andˆ w | φ L =0 = w − λ − ¯ λ )( H − ¯ H ) w ( λ + ¯ λ ) w − − H + w + ( w (cid:48) ) ] = 3 w. (6.13)18igure 7: Rogue waves on the librational wave with k = 0 . k = 0 . ( ξ,η ) ∈ R | w ( ξ, η ) | = 2 k , the maximum ofthe new solution (6.11) is attained at sup ( ξ,η ) ∈ R | ˆ w ( ξ, η ) | = 6 k , hence the rogue wave has triplemagnification compared to the background wave. Note that the rogue wave (6.11) and thetriple magnification factor was previously obtained for the mKdV equation in [7].Figure 7 illustrates the exact solution (6.11) for two particular values of k . The value of C is set to 0 in (6.5). It is clear that the solution surface of | ˆ w ( ξ, η ) | achieves its maximumat ( ξ, η ) = (0 ,
0) where φ L vanishes. The modulus is shown for a better resolution of theoscillations of the librational wave background.Based on the same solution formula (6.11), we have computed sin(ˆ u ) = ˆ u ξη by numericallydifferentiating ˆ w = − ˆ u ξ in η with a forward difference. The corresponding surface plots ofsin(ˆ u ) in ( x, t ) are presented on Figure 2 for different values of k .Finally, we inspect how the magnification of the rogue wave depends on the constant ofintegration C in (6.5) and (6.11). The magnification factor is defined as M := sup ( ξ,η ) ∈ R | ˆ w ( ξ, η ) | sup ( ξ,η ) ∈ R | w ( ξ, η ) | . Figure 8 presents the plot of M versus C for k = 0 .
8. When C = 0, the magnification factor ismaximal at M = 3. It is periodically continued with respect to C and it reaches the minimalvalue below 2. The minimal value of M depends on k . We have presented new solutions to the sine-Gordon equation using an algebraic methodand the Darboux transformations. The new solutions describe localized structures on thebackground of rotational and librational waves. These localized structures are obtained for theparticular eigenvalues of the linear Lax equations which correspond to bounded solutions in19
Figure 8: The magnification factor M of the rogue wave ˆ w given by (6.11) versus the constantof integration C in (6.5) for k = 0 . Acknowledgement.
The authors thank P.D. Miller and B.Y.Lu for sharing their preprint[18] before submission and many relevant discussions. This project was supported in part bythe National Natural Science Foundation of China (No. 11971103).
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