Breather and rational solutions of modified Korteweg-de Vries equation

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1 Breather and rational solutions of modified Korteweg-de Vries equation A. Chowdury, A. Ankiewicz and N. Akhmediev Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2600, Australia We present closed form breather solutions of the integrable modified Korteweg-de Vries equation (mkdv). By using a Darboux transformation, we derive first- and second-order doubly-periodic lattice-like solutions. We explicitly derive first-and second-order rational solutions as limiting cases of the breather solutions. We have also found the degenerate solution which corresponds to the equal eigenvalue case. Among the second-order solutions, we single out the doubly-localized high peak solution on a constant background with an infinitely extended trough. This solution plays the role of a rogue wave of the mkdv equation. PACS numbers: Yv, Tg, Qb I. INTRODUCTION The first observation of a solitary wave on a water surface was reported by the Scottish naval engineer J. Scott Russell back in the nineteenth century [1]. Theoretical analysis of such a wave was presented by British scientist Lord Rayleigh [2] and French scientist Boussinesq [3 6]. In 1895, two Dutch mathematicians, Korteweg and de Vries, formulated a partial differential equation (now known as the KdV equation) whose closed form solution precisely modelled the propagation of shallow water waves [7]. Numerical analysis of this equation by Zabuski and Kruskal [8] showed that a localized initial condition results in the excitation of solitons. Gardner et al. [9] were the first to solve an initial value problem involving the KdV equation using the inverse scattering theory with the one-dimensional Schrödinger operator. The mkdv equation and its further modifications are found to describe pulses consisting of a few optical cycles [10 12] and in modelling supercontinuum generation in optical fibres [13]. The mkdv equation has many other applications in various fields such as soliton propagation in lattices [14], nonlinear Alfvén waves propagating in plasma [15] and meandering ocean currents [16]. It can also be applied to the dynamics of traffic flow [17 19]. Furthermore, the mkdv equation is related to Schottky barrier transmission lines [20], ion acoustic soliton experiments in plasmas [21] and fluid mechanics [22]. For convenience, we write the focussing mkdv equation in the following way: ψ x α ( 3 ψ t 3 ) ψ + 6ψ2 = 0, (1) t ψ = ψ(x, t) is a real function with evolution variable x and transverse variable t. We explicitly retain an arbitrary real parameter, α, in the equation in order to have consistent notation with our previous results [23, 24] and because it is useful at times. It can be removed by rescaling the x-variable: αx x. The equation containing the Hirota operator alone [25, 26] differs from Eq.(1) in that ψ 2 in the third term is replaced by ψ 2. (This Hirota operator equation is sometimes called the complex modified Korteweg-de Vries equation [27, 28], but this label is misleading, since it is in the NLS family, not the KdV family). A real solution of the Hirota equation will also solve the mkdv. Some solutions, mainly with zero background have been given for α set to -1 in [29, 30]. If ψ is a solution of the defocussing mkdv, ψ x +ψ ttt 6ψ 2 ψ t = 0, then the Miura transformation [31], φ = ψ t +ψ 2, is a solution of the KdV, φ x +φ ttt 6φφ t = 0. This is not the case we deal with here. Both equations are integrable and have infinitely many conserved quantities [32]. The mkdv admits a scaling transformation. Namely, if ψ = ψ(x, t) is a solution of Eq.(1), then ψ = qψ(q 3 x, qt) (2) is also a solution of eq.(1) for arbitrary real q. For example, for the basic soliton solution ψ = sech (t + α x), the extended one-parameter family of solutions can be written as ψ = q sech ( qt + α q 3 x ), q is an arbitrary real number. This scaling applies to all mkdv solutions including those given in this work. We will provide solutions with q = 1 as unit-background solutions. The mkdv can also be written in the form of the continuity equation [33]: ψ x = [ ( α ψtt + 2ψ 3)]. t In contrast to KdV, the mkdv equation is not invariant under Galilean transformation. Explicit solutions of the mkdv have been derived using various methods. For instance, Hirota derived the exact envelope soliton for the mkdv equation in [34] and for multiple collision of solitons with different amplitudes in [35]. The inverse scattering technique (IST) has been developed to solve a range of nonlinear evolution differential equations, including the mkdv equation in [36]. The IST for the mkdv equation was given by Tanaka [37]. Wadati used the IST to obtain the exact N-soliton solution for the mkdv [38]. Direct methods can also be used to derive certain solutions. Some classes of periodic solutions of mkdv have been derived in [39]. In this work, using a Darboux transformation with seeding

2 2 solution ψ = 1, we derive exact breather and rational solutions. These correspond to imaginary eigenvalues of the IST. The Lax-pair for the mkdv equation was given by Wadati [38]. We note that Wadati used the opposite sign for α. Our notation below is adapted to Eq.(1). Now, two linear equations R t = U R, R = V R, (3) x are such that the zero-curvature condition: U x V t + [U, V ] = 0, (4) The solution of Eq.(7) is real and contains a trigonometric function only. Here, κ is the frequency of the periodic function. An asymptotic reduction from the modified Kortewegde Vries equation produces the NLS equation [42, 43]. So modulation instability can occur for some parameter values, and an almost periodic wave-train can provide short term pulses of relatively high energy. Fig.1 shows a periodic pattern with the frequency κ = 2 1 b 2 < 2. The eigenvalue is λ = ib b is real. Then the breather period is T = π/ 1 b 2 along the t axis. Breathers exist for 0 < b < 1 and consequently their frequencies remain within 0 < κ < 2. The longest period of breather oscillations occurs in the limit κ 0. will reproduce Eq.(1). Here U and V are 2 2 matrices with U given by [ ] λ ψ(x, t) U = i, (5) ψ(x, t) λ while V is a matrix polynomial in eigenvalue λ. For the mkdv, it is a simple cubic polynomial that can be written in general form V = α 3 j=0 λj V j, submatrices V j are [ ] Aj Bj V j = i, (6) B j A j with A 0 = i (ψt ψ ψ t ψ ), B 0 = 2ψ 2 ψ + ψ tt, A 1 = 2ψ 2, B 1 = 2iψ t, A 2 = 0, B 2 = 4ψ, A 3 = 4, B 3 = 0. It is easy to check that substitution of the matrices U and V into Eq.(4) leads directly to Eq.(1) II. PERIODIC FIRST-ORDER BREATHER AND RATIONAL SOLUTIONS Using the the seeding solution ψ = 1, the imaginary eigenvalue λ = ib and the same basic steps as in [40], we obtain the periodic breather of the mkdv: ψ 1 = 1 + κ κ 2 cos[κ(t + vx)], (7) κ = λ 2 and v = α ( 6 κ 2). Thus we need κ < 2. This solution is shown in Fig.1. In contrast to the Akhmediev breather of the nonlinear Schrödinger equation (NLSE) [41, 44], which is localized in x and periodic in t, the mkdv breather is periodic in t but maintains constant amplitude of oscillations 4 κ 2 every. The oscillations move with velocity v in the (x, t)-plane. FIG. 1: (Color online) A first order mkdv breather solution Eq.(7) with λ = i (or κ = ) and α = Here, the amplitude remains constant. Similar to the case of an Akhmediev breather that becomes a rogue wave [46, 47] in the limit κ 0, the mkdv breather of Eq.(7) in the same limit, κ 0, becomes a rational soliton: ψ 1 = (t + 6xα) 2, (8) This corresponds to the Peregrine solution of the NLSE [48, 49]. Adding to the similarity, the highest amplitude of the solution is 3, just as occurs with the Peregrine solution. However, instead of being doubly localized, the solution of eq.(8) looks more like a soliton on a constant background, ψ = 1, moving with velocity v in the (x, t)-plane. Both solutions, Eq.(7) and Eq.(8) lack phase evolution which is an essential part of any NLSE solution. This is in accordance with the fact that the mkdv directly describes the wave profile rather than a wave envelope function.

3 3 FIG. 2: (Color online) A first order mkdv rational solution Eq.(8) with α = 41. III. SECOND ORDER BREATHER SOLUTION The second step in the Darboux transformation scheme [40] provides us with the second-order breather of the mkdv equation: ψ2 = 1 + N2, D2 (9) N2 = (v12 v22 )[v2 (v1 cos F1 2) (2 cos F2 v2 ) v1 (2 cos F1 v1 ) (v2 cos F2 2)] D2 = 2v1 v2 κ1 κ2 sin F1 sin F2 + [2 cos F1 v1 ][2 cos F2 v2 ] FIG. 3: (Color online) A second order mkdv breather solution Eq.(9) with κ1 = 0.7, κ2 = 1.4 and α = 16. other one. This happens with all second-order solutions below. Figure 4 shows another example of a second order doubly-periodic breather with different values of κ1 and κ2. The velocities of propagation of the two individual breathers, α(6 κ21 ) and α(6 κ22 ), are now closer to each other. Thus, the two periodic structures comprising the lattice are located at smaller angles to each other. Their periods differ significantly. In Fig.4, the maximum amplitude is 4.57 and the minimum is The frequencies of the individual breathers, κ1 and κ2, in the above solutions must remain within the range 0 < κ1, κ2 2. When we have κi 2 for either one, the corresponding breather reduces to a constant amplitude background wave. + (κ21 + κ22 8)[2 v1 cos F1 ] [v2 cos F2 2], and F1 = xακ1 (6 κ21 ) + tκ1, F2 = xακ2 (6 κ22 ) + tκ2, q q v2 = 4 κ22, v1 = 4 κ21, Here, the maximum value is 1 + v1 + v2 and the minimum value is 1 v1 v2. So, the sum of the maximum and minimum values is always 2, independent of α, and the total structure always has height of 2(v1 + v2 ). The solution is shown in Fig.3. This is the nonlinear superposition of the set of two periodic breathers obtained in the previous step crossing each other. Thus, the solution creates the two-dimensional lattice. Each individual wave train has specific values of frequencies, κ1 and κ2, respectively. The maximum amplitude of the superposition is 4.30, while the minimum one is The maxima of the lattice appear at the positions of intersection of troughs of one periodic structure with the maxima of the FIG. 4: (Color online) A second order mkdv breather solution Eq.(9) with κ1 = 0.98, κ2 = 0.8 and α = 61. Either of the frequencies, κ1 or κ2, in the breather can

4 4 be zero. When κ 2 = 0, the solution is: with ψ 2 = 1 κ2 [ v1 cos (R 1 ) + ( 1 + 4p 2 ) ] 1 κ 2 D 1,(10) 3 D 3 = 16p 2 ( 1 v ) 16p 1 v 1 κ 1 sin (R 1 ) + 4 [ 4 4v 1 cos (R 1 ) + v1 2 ] + ( 1 + 4p 2 1) [v1 cos (R 1 ) 2]κ 2 1 R 1 = (t V b x) κ 1, p 1 = t + 6xα V b = α ( 6 + κ 2 1). The solution is significantly simplified and contains a rational component and only one wave frequency. This solution is shown in Fig.5. The background here has only one periodic component. The presence of the rational component is revealed in a single ridge of higher amplitudes crossing the origin. The solution is also periodic along this line, as the ridge crosses the periodic background at a finite angle. The periodicity of the maxima along the ridge is defined by this angle. The maximal amplitude of periodic field variation along this line is 4.2. This leaves us with a real second order degenerate mkdv solution: ψ 2 = 1 + N d D d, (11) N d = 4κ 2 va 2 { κva 2 [ ( t 3α κ 2 2 ) x ] sin R b ( κ 2 8 ) } cos R b 4v a and D d = 8κv 4 a[t 3α(κ 2 2)x] sin R b v 5 a cos(2r b ) 8κ 2 v 2 a cos R b v a { 2κ 6 (t αtx + 468α 2 x 2 ) + 12ακ 8 x(t + 18αx) + κ 4 [16(t + 6αx)(t + 18αx) + 1] 8κ 2 [4(t + 6αx) 2 + 1] 18α 2 κ 10 x 2 16}, with v a = 4 κ 2, V b = α ( κ 2 6 ), R b = (t V b x)κ Here, κ = κ 1 = κ 2. An example of this degenerate solution of the mkdv is presented in Fig.6. The wave pattern of this solution consists of a single periodic ridge of high amplitude peaks on a periodic wave background. The background modulation is much weaker than the modulation along the periodic ridge. The maximum amplitude of the peaks along the ridge is (ψ 2 ) max = 5. The line of peaks crosses the origin. The positions of the peaks are synchronised with the periodic wave structure of the background. FIG. 5: (Color online) A second order mkdv breather solution Eq.(10) with κ 1 = 1.6 while κ 2 0 and α = 1 6. IV. DEGENERATE BREATHER SOLUTION Generally, the inverse scattering technique does not allow the presence of two equal eigenvalues in the solution. The solution becomes undefined when λ 1 λ 2, or, equivalently κ 1 κ 2. Despite having this restriction, it is still possible to derive the solution in this limiting case. To proceed with the derivation, we set κ 2 κ 1 +ɛ. Then, we take a Taylor series expansion of this expression in terms of ɛ and retain only the lowest order terms. FIG. 6: (Color online) A second order degenerate mkdv breather solution, Eq. (11), with κ = 1.2 and α = 1 6.

5 5 V. SECOND-ORDER RATIONAL SOLUTION We can further simplify the solution of Eq.(10) by taking the other frequency κ 1 0. Trigonometric functions then disappear and we obtain the second-order purely rational solution of mkdv equation: ψ 2 = G 2 D 2, (12) G 2 = 3 8(6αx + t) [ 2(6αx + t) 3 + 3(22αx + t) ] and [ D 2 = 48αx 62208α 4 x 4 (αx + t) + 432α 3 ( 60t 2 13 ) x α 2 t ( 20t 2 9 ) x 2 + 3α ( 240t 4 120t ) x + t ( 48t 4 8t ) ] + 64t t t This solution is shown in Fig.7. The solution is real without any periodicity and is located on a flat background, ψ = 1. The solution is seemingly a result of a collision of a dark and a bright solitons with slightly different velocities. The collision point is at (0, 0) and has a single peak with the highest amplitude 5 at the origin. Remarkably, this value is the same as for the second-order rogue wave of the NLSE [44, 45, 50]. This solution is a proof that the mkdv also has a high-amplitude rational solution localized in two dimensions. The dark soliton part of this solution has a minimum around 2.75, thus making the central part of the solution significantly higher than the bottom of the trough. FIG. 7: (Color online) A second order mkdv rational solution Eq.(12) with α = 1 6. Returning to the simple scaling of Eq.(2), we note that the rational solution can be transformed to arbitrary background a and arbitrary maximal amplitude 5a. Then it is given by ψ 2 = a ( ) G 2 D 2 (13) each x is replaced by a 3 x and each t by a t in the above expressions for G 2 and D 2. Thus, G 2 = 3 8a ( 6a 2 αx + t ) [ 2a 3 ( 6a 2 αx + t ) 3 + 3a ( 22a 2 αx + t )] and similar modification applies for D 2. VI. SECOND ORDER RATIONAL SOLUTION WITH DIFFERENTIAL SHIFT FIG. 8: (Color online) A second order mkdv rational solution, Eq.(14), with shift parameter c a = 2; here α = 1 6. Continuing the comparison with the rogue wave solutions of the NLSE, we recall that, for the NLSE equation, a form of the second-order rogue wave solution appears as a triplet of well-separated first-order rogue waves [51, 52]. This form requires two additional free parameters that move the rogue wave components out from the centre of the structure. We call these parameters differential shifts [23, 53]. They should not be confused with trivial translations along the x and t co-ordinates. In contrast to the NLSE equation, the mkdv rational solution cannot be split into three components, as there are no first-order rational solutions here. We apply the differential shifts, x d and t d, to the solution of Eq.(12) in the same way as in [23, 53]. We set c a = x d + 6α t d as the combined constant. We find: ψ 2 = 1 + N t D t (14) N t = 12{8(t + 6αx)[ 16c a + 2t 3 + 6α(6t )x + 216α 2 tx 2 + 3t + 432α 3 x 3 ] 3},

6 6 and D t = 4{256c 2 a α 3 x 3 (8c a + 20t 3 9t) + 36α 2 x 2 [24t(16c a + 10t 3 5t) + 139] + 128c a t αx(192c a t 2 176c a + 48t 5 8t t) 96c a t + 16t t α 4 (60t 2 13)x t 2 + 3(12α x) 5 (t + αx)} + 9. Now, the new solution, Eq.(14), has a free real parameter, c a. For c a = 0, the solution Eq.(14) coincides with the second-order rational solution of Eq.(12). Two examples of this solution with nonzero values of x d and t d are shown in Figs. 8 and 9. As we can see, the action of these parameters on the rational solution results in a shift of the peak of along the dark soliton trough. This is different from elementary translations of the solution along the x and t axes. In the present case, the translation is eigenvalue-dependent. In our work, we have found new breather solutions of the mkdv equation. A second order nonlinear superposition of them, constructed with the use of the Darboux transformation, leads to doubly-periodic lattice -type structures. Limiting cases of these structures include a high-amplitude oscillating ridge on a single periodic background, soliton-like structures and, most remarkably, a rational solution on a constant background with an infinitely extended dark soliton trough. Breathers and rogue wave solutions of the NLSE are used mainly to explain wave dynamics in deep water and optical pulse dynamics in nonlinear optics. The corresponding classes of solutions of the mkdv equation derived in this work are fundamentally different from those of the NLSE. The solutions of the mkdv are related to the wave profile directly. These solutions reveal that the phenomena of breathers and rogue waves are not confined to the deep ocean. Being the rational solutions of the mkdv equation, they may appear in electromagnetic waves in quantized films, internal waves for some density stratifications, elastic media [55], etc., as well. Remarkably, in the deep ocean, modulation instability contributes to the formation of rogue waves through NLSE breather dynamics, while, in the shallow water case, we obtain rational solutions and solitons in the zero frequency limit (κ 0) of mkdv breathers. Hence, the deep and shallow water cases provide two different descriptions in hydrodynamics. These new solutions will undoubtedly be useful and will give insight in modelling nonlinear wave processes in various other physical systems the mkdv is relevant. FIG. 9: (Color online) A second order mkdv rational solution, Eq.(14), with shift parameter c a = 2; here α = 1 6. VII. CONCLUSION ACKNOWLEDGEMENTS The authors acknowledge the support of the Australian Research Council (Discovery Project number DP ). N.A. and A.A. acknowledge support from the Volkswagen Stiftung and A. C. acknowledges Endeavour Postgraduate Award support and support from a Ben Williams student grant. [1] J. S. Russell. Report on waves. In 14-th meeting of the British Association for the Advancement of Science, 311, 390 (1844). [2] Lord Rayleigh. On waves. Phil. Mag, 1: (1876). [3] J. Boussinesq. Essai sur la théorie des eaux courantes. Imprimerie nationale, 2 (1877). [4] J. Boussinesq. Théorie de ĺintumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus Acad. Sci (Paris), 72, (1871). [5] J. Boussinesq. Théorie des ondes et des remous qui se propagent le long duncanal rectangulaire horizontal, en communiquant au liquide continu dans ce canal des vitesses sensiblement pareilles de la surface au fond.journal of Mathématic Pure and Appliquées, 17, (1872). [6] J. Boussinesq. Théorie général des mouvements, qui sont propagés dans un canal rectangulaire horizontal. CR

7 7 Acad. Sci. Paris, 73, (1871). [7] D. J. Korteweg and G. De Vries. Xli. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39, (1895). [8] N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett, 15, (1965). [9] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, (1967). [10] H. Leblond and D. Mihalache, Few-optical-cycle solitons: Modified Korteweg-de Vries sine-gordon equation versus other non-slowly-varying-envelope-approximation models, Phys. Rev. A, 79, (2009). [11] H. Leblond and D. Mihalache, Few-optical-cycle dissipative solitons, J. Phys. A: Math. Theor. 43, (2010). [12] H. Leblond and D. Mihalache, Optical solitons in a fewcycle regime: recent theoretical results, Romanian Reports in Physics, Supplement, 63, (2011). [13] H. Leblond, Ph. Grelu and D. Mihalache, Models for supercontinuum generation beyond the slowly-varyingenvelope approximation Phys. Rev. A, 90, (2014). [14] H. Ono, Soliton fission in anharmonic lattices with reflectionless inhomogeneity, J. Phys. Soc. of Japan, 61, (1992). [15] A. H. Khater, O. H. El-Kalaawy and D. K. Callebaut, Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron positron plasma. Physica Scripta, 58, 545 (1998). [16] E. A. Ralph and L. Pratt, Predicting eddy detachment for an equivalent barotropic thin jet, J. Nonlinear Science, 4, (1994). [17] T. S. Komatsu and Shin-ichi Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52, (1995). [18] Z.-P. Li and Y.-C. Liu, Analysis of stability and density waves of traffic flow model in an ITS environment, European Phys. J. B - Condensed Matter and Complex Systems, 53, (2006). [19] H. X. Ge, S. Q. Dai, Y. Xue and L. Y. Dong, Stabilization analysis and modified Korteweg-de Vries equation in a cooperative driving system, Phys. Rev. E, 71, (2005). [20] V. Ziegler, J. Dinkel, C. Setzer and K. E. Lonngren, On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line, Chaos, Solitons & Fractals, 12, (2001). [21] K. E. Lonngren, Ion acoustic soliton experiments in a plasma. Optical and Quantum Electronics, 30, (1998). [22] M. A. Helal, Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos, Solitons & Fractals, 13, (2002). [23] A. Chowdury, D. J. Kedziora, A. Ankiewicz and N. Akhmediev, Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions, Phys. Rev. E, 91, (2015). [24] A. Chowdury, D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms. Phys. Rev. E, 90, (2014). [25] A. Ankiewicz, N. Akhmediev, Higher-order integrable evolution equation and its soliton solutions, Physics Letters A, 378, (2014). [26] A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions, Phys. Rev E 89, (2014). [27] Zhaqilao, N-th-order rogue wave solutions of the complex modified Korteweg-de Vries equation, Physica Scripta, 87, (2013). [28] L. Li, K. Porsezian, J. He, L. Wang and R. Erdelyi, Fewcycle optical rogue waves: Complex modified Korteweg de Vries equation, Phys. Rev. E, 89, (2014). [29] Sun Ying-Ying and Zhang Da-Jun, Rational Solutions with Non-zero Asymptotics of the Modified Korteweg-de Vries Equation, Commun. Theor. Phys. 57 (2012) ; see alsohttp://arxiv.org/pdf/ v2.pdf [30] Zhang, Da-Jun and Zhao, Song-Lin and Sun, Ying-Ying and Zhou, Jing, Solutions to the modified Korteweg de Vries equation, Reviews in Mathematical Physics. 26, (2014) [31] R. M. Miura. Korteweg-de Vries equation and generalizations. i. a remarkable explicit nonlinear transformation, J. Math. Phys., 9, (1968). [32] R. M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Review, 18, (1976). [33] N. F. Smyth and A. L. Worthy, Solitary wave evolution for MKdV equations, Wave Motion, 21, (1995). [34] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14, (1973). [35] R. Hirota, Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Japan, 33, , (1972). [36] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., 31 (2),125 (1973). [37] Sh. Tanaka, Modified Korteweg-de Vries equation and scattering theory, Proc. Japan Academy, 48, (1972). [38] M. Wadati, The exact solution of the modified Kortewegde Vries equation, J. Phys. Soc. Japan, 32, (1972). [39] P. G. Kevrekidis, A. Khare, A. Saxena and G. Herring, On some classes of mkdv periodic solutions. J. Phys. A: Math. Gen. 37, (2004). [40] N. Akhmediev, V. I. Korneev and N. V. Mitskevich, N- modulation signals in a single-mode optical fiber with allowance for nonlinearity, Zh. Eksp. Teor. Fiz., 94, (1988). English translation in: [Sov. Phys. JETP, 67, 89 (1988)], [41] N. Akhmediev and V. I. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation, Teor. Mat. Fiz., 69, (1986). English translation in: [Theor. Math. Phys., 69, 1089 (1986)]. [42] R. Grimshaw, E. Pelinovsky, T. Talipova, Michael Ruderman and Robert Erdelyi, Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg De Vries Equation, Studies in Applied Mathematics,114, , (2005) [43] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, and T. Talipova,Wave group dynamics in weakly nonlinear longwave models, Physica D: Nonlinear Phenomena, 159, 35 37,(2001)

8 8 [44] N. Akhmediev, V. Eleonsky and N. Kulagin, Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions, Sov. Phys. JETP, 62, (1985). [45] N. Akhmediev, A. Ankiewicz and M. Taki, Waves that appear from no and disappear without a trace, Phys. Lett., A 373, (2009). [46] N. Akhmediev, V. M. Eleonskii and N. E. Kulagin, Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. (USSR), 72, (1987). [47] N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E, 80, (2009). [48] D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16, (1976). [49] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics, Nature Phys., 6, (2010). [50] N. Akhmediev, J. M. Soto-Crespo and A. Ankiewicz, Extreme waves that appear from no: on the nature of rogue waves, Phys. Lett. A, 373, (2009). [51] P. Dubard, P. Gaillard, C. Klein, V.B. Matveev, Eur. Phys. J. Special Topics, 185, 247 (2010). [52] A. Ankiewicz, D. J. Kedziora and N. Akhmediev, Rogue wave triplets, Phys. Lett. A, 375, (2011). [53] D. J. Kedziora, A. Ankiewicz, N. Akhmediev, Secondorder nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits, Phys. Rev. E, 85, (2012). [54] D. J. Kedziora, A. Ankiewicz and N. Akhmediev, Circular rogue wave clusters, Phys. Rev. E, 84, (2011). [55] T. R. Marchant, Asymptotic solitons for a higher-order modified Korteweg-de Vries equation, Phys. Rev. E, 66, (2002).