Rational solutions of the Boussinesq equation and applications to rogue waves

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1 Transactions of Mathematics and its Applications (017 1, 1 6 doi: 10.10/imatrm/tn00 Rational solutions of the Boussinesq equation and applications to rogue waves Peter A. Clarkson and Ellen Dowie School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT 7FS, UK Corresponding author: P.A.Clarkson@kent.ac.uk E.Dowie@kent.ac.uk [Received on 4 May 017, revised on July 017] We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are epressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger (NLS equation. Further the rational solutions have an interesting structure as they are comprised of a linear combination of four independent solutions of the bilinear equation. Rational solutions of the Kadomtsev Petviashvili I (KPI equation are derived in two ways, from rational solutions of the NLS equation and from rational solutions of the Boussinesq equation. It is shown that these two families of rational solutions of the KPI equation are fundamentally different and a unifying framework is found which incorporates both families of solutions. Keywords: Boussinesq equation; focusing nonlinear Schrödinger equation; Kadomtsev Petviashvili equation, rational solutions; rogue waves. 1. Introduction Rogue waves, sometimes knows as freak waves or monster waves, are waves appearing as etremely large, localized waves in the ocean which have been of considerable interest recently (see, for eample Dysthe et al. (008, Kharif et al. (00, Osborne (010 and Pelinovsky & Kharif (016. The average height of rogue waves is at least twice the height of the surrounding waves, are very unpredictable and so they can be quite unepected and mysterious. A feature of rogue waves is that they come from nowhere and disappear with no trace (Akhmediev et al., 00b,c. In recent years, the concept of rogue waves has been etended beyond oceanic waves: to pulses emerging from optical fibres (Solli et al., 007; Dudley et al., 00, 014; Kibler et al., 010; waves in Bose Einstein condensates (Bludov et al., 00; in superfluids (Ganshin et al., 008; in optical cavities (Montina et al., 00; in the atmosphere (Stenflo & Marklund, 010; and in finance (Yan, 010, 011; for a comprehensive review of the different physical contets rogue waves arise, see Onorato et al. (01. The most commonly used mathematical model for rogue waves involves rational solutions of the focusing nonlinear Schrödinger (NLS equation iψ t + ψ + 1 ψ ψ = 0, (1.1 where subscripts denote partial derivatives, with ψ the wave envelope, t the temporal variable and the spatial variable in the frame moving with the wave, see Section. Downloaded from by guest on 0 December 018 The authors 017. Published by Oford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contactjournals.permissions@oup.com

2 P. A. CLARKSON AND E. DOWIE In this article, we are concerned with rational solutions of the Boussinesq equation u tt + u (u 1 u = 0, (1. which are algebraically decaying and have a similar appearance to rogue-wave solutions of the NLS equation (1.1. Equation (1. was introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water (Boussinesq, 1871, 187; see, also Ursell, 15; Whitham, 174. The Boussinesq equation (1. is also a soliton equation solvable by inverse scattering (Zakharov, 174; Ablowitz & Haberman, 175; Ablowitz & Segur, 181; Deift et al., 18; Ablowitz & Clarkson, 11 which arises in several other physical applications including one-dimensional nonlinear lattice-waves (Zabusky, 167; Toda, 175; vibrations in a nonlinear string (Zakharov, 174 and ion sound waves in a plasma (Scott, 175; Infeld & Rowlands, 10. We remark that equation (1. is sometimes referred to as the bad Boussinesq equation, i.e., when the ratio of the u tt and u terms is negative. If the sign of the u term is reversed in (1., then the equation is sometimes called the good Boussinesq equation. The coefficients of the u and (u terms can be changed by scaling and translation of the dependent variable u. For eample, letting u u + 1in(1. gives u tt u (u 1 u = 0, (1. which is the non-dimensionalized form of the equation originally written down by Boussinesq (1871, 187. There has been considerable interest in partial differential equations solvable by inverse scattering, the soliton equations, since the discovery by (Gardner et al., 167 of the method for solving the initial value problem for the Korteweg de Vries (KdV equation u t + 6uu + u = 0. (1.4 During the past 40 years or so there has been much interest in rational solutions of the soliton equations. For some soliton equations, solitons are given by rational solutions, e.g., for the Benjamin Ono equation (Matsuno, 17; Satsuma & Ishimori, 17. Further applications of rational solutions to soliton equations include: in the description of vorte dynamics (Aref et al., 00; Aref, 007a,b; vorte solutions of the comple sine-gordon equation (Barashenkov & Pelinovsky, 18; Olver & Barashenkov, 005; and in the transition behaviour for the semi-classical sine-gordon equation (Buckingham & Miller, 01. In Section, we discuss rational solutions of the focusing NLS equation (1.1, including some generalized rational solutions which involve two arbitrary parameters. In Section, we discuss rational solutions of the Boussinesq equation (1., also including some generalized rational solutions which involve two arbitrary parameters. Further the generalized rational solutions have an interesting structure as they are comprised of a linear combination of four independent solutions of an associated bilinear equation. In Section 4, we discuss rational solutions of the Kadomtsev Petviashvili I (KPI equation (v τ + 6vv ξ + v ξξξ ξ = v ηη, (1.5 which are derived in two ways, first from rational solutions of the focusing NLS equation (1.1 and second from rational solutions of the Boussinesq equation (1.. In the simplest nontrivial case, it is shown that these two types of rational solutions are different. Further we derive a more general rational solution which has those related to the focusing NLS and Boussinesq equations as special cases and so provides a unifying framework. In Section 5, we discuss our results. Downloaded from by guest on 0 December 018

3 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION. Rational solutions of the focusing NLS equation The NLS equation iψ t + ψ + 1 σ ψ ψ = 0, σ =±1, (.1 is one of the most important nonlinear partial differential equations. Zakharov & Shabat (17 developed the inverse scattering method of solution for it. Prior to the discovery that the NLS equation (.1 was solvable by inverse scattering, it had been considered by researchers in water waves (Benney & Newell, 167; Zakharov, 168; Benney & Roskes, 16. Hasegawa & Tappert (17a,b discussed the relevance of the NLS equation (.1 in optical fibres and their associated solitary wave solutions. Hasegawa and Tappert showed that optical fibres could sustain envelope solitons both bright and dark solitons. Bright solitons, which decay as, arise with anomalous (positive dispersion for (.1 with σ = 1, the focusing NLS equation. Dark solitons, which do not decay as, arise with normal (negative dispersion for (.1 with σ = 1, the de-focusing NLS equation. Rational solutions of the focusing NLS equation (1.1 have the general form ψ n (, t = { 1 4 G } n(, t + ith n (, t ep ( 1 D n (, t it, (. where G n (, t and H n (, t are polynomials of degree 1 (n + (n 1 in both and t, with total degree 1 (n + (n 1, and D n(, t is a polynomial of degree 1 n(n + 1 in both and t, with total degree 1 n(n + 1 and has no real zeros. The polynomials D n(, t, G n (, t and H n (, t satisfy the Hirota equations with D and D t the Hirota operators 4(tD t + 1H n D n + D D n D n 4D D n G n = 0, D t G n D n + td H n D n = 0, D D n D n = 8G n + 8t H n 4D ng n, [ ( D l Dm t F(, t F(, t = l ( t m F(, tf(, t ]. (. t =,t =t The first two rational solutions of the focusing NLS equation (1.1 have the form ψ 1 (, t = ψ (, t = { 1 } 4(1 + it ep ( 1 + t + 1 { 1 1 G (, t + ith (, t D (, t it, (.4 } ep ( 1, (.5 Downloaded from by guest on 0 December 018

4 P. A. CLARKSON AND E. DOWIE 4 where G (, t = 4 + 6(t t t, H (, t = 4 + (t + (t + 5(t, (.6a (.6b D (, t = 6 + (t (t + t 6 + 7t 4 + t + (.6c (see Akhmediev et al. (00a and Ankiewicz et al. (010 for further details. The solution ψ 1 (, t given by (.4 is known as the Peregrine solution (Peregrine, 18. Further ψ n (, t = ln D n(, t. Dubard et al. (010 show that the rational solutions of the focusing NLS equation (1.1 can be generalized to include some arbitrary parameters. The first of these generalized solutions has the form { } ψ (, t; α, β = 1 1 Ĝ(, t; α, β + iĥ (, t; α, β ep ( 1 D (, t; α, β it, (.7 where Ĝ (, t; α, β = G (, t αt + β, Ĥ (, t; α, β = th (, t + α( t βt, D (, t; α, β = D (, t + αt( t β( t + α + β, (.8a (.8b (.8c with α and β arbitrary constants (see also Dubard & Matveev (011, 01 and Kedziora et al. (011, 01, 01. These generalized solutions have now been epressed in terms of Wronskians (see Gaillard (011, 01, 01a,b,c,d,e,f, 014a,b,c,d, 015a,b,c, Guo et al. (01, Ohta & Yang (01 and Gaillard & Gastineau (015a,b, 016. We note that the polynomial D (, t; α, β has the form D (, t; α, β = D (, t + αtp 1 (, t + βq 1 (, t + α + β, (. where P 1 (, t and Q 1 (, t are linear functions of and t. In Fig. 1, plots of the generalized rational solution ψ (, t; α, β given by (.7 of the focusing NLS equation for various values of the parameters α and β. The solution has a single peak when α = β = 0, which splits into three peaks as α and β increase; this solution is called a rogue wave triplet (Ankiewicz et al., 011; Kedziora et al., 011 and the three sisters (Gaillard, The Boussinesq equation.1. Introduction Downloaded from by guest on 0 December 018 Clarkson & Kruskal (18 showed that Boussinesq equation (1. has symmetry reductions to the first, second and fourth Painlevé equations (P I,P II,P IV w = 6w + z, (.1

5 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 5 Fig. 1. Plots of the generalized rational solution ψ (, t; α, β given by (.7 of the focusing NLS equation for various values of the parameters α and β. w = w + zw + α, (. w = (w w + w + 4zw + (z αw + β w, (. with = d/dz, and α, β arbitrary constants. Vorob ev (165 and Yablonskii (15 epressed the rational solutions of P II (. in terms of polynomials, now known as the Yablonskii Vorob ev polynomials (see also Clarkson & Mansfield (00. Okamoto (186 derived analogous polynomials, now known as the Okamoto polynomials, related to some of the rational solutions of P IV (.. Subsequently Okamoto s results were generalized by Noumi & Yamada (1 who showed that all rational solutions of P IV can be epressed in terms of logarithmic derivatives of two sets of special polynomials, called the generalized Hermite polynomials and the generalized Okamoto polynomials (see also Clarkson (00, 006. Consequently rational solutions of (1. can be obtained in terms the Yablonskii Vorob ev, generalized Hermite and generalized Okamoto polynomials (see Clarkson (008. Some of the rational solutions that are epressed in terms of the generalized Okamoto polynomials are generalized to give the rational solutions of the Boussinesq equation (1. (see Clarkson (008, Galkin et al. (15 and Pelinovsky (18, which are analogs of the rational solutions of the KdV equation (1.4 (Airault et al., 177; Choodnovsky & Choodnovsky, 177; Ablowitz & Satsuma, 178; Adler & Moser, 178. However none Downloaded from by guest on 0 December 018

6 P. A. CLARKSON AND E. DOWIE 6 of these rational solutions of the Boussinesq equation (1. are bounded for all real and t, so are unlikely to have any physical significance. It is known that there are additional rational solutions of the Boussinesq equation (1. which do not arise from the above construction. For eample, Ablowitz & Satsuma (178 derived the rational solution u(, t = ln(1 + + t = 4(1 + t (1 + + t, (.4 by taking a long-wave limit of the two-soliton solution (see also Tajiri & Murakami (11 and Tajiri & Watanabe (17. This solution is bounded for real and t, and tends to zero algebraically as, t. If in the Boussinesq equation (1., we make the transformation u(, t = ln F(, t, (.5 then we obtain the bilinear equation FF tt F t + FF F ( 1 FF 4F F + F = 0, (.6 first derived by Hirota (17, which can be written in the form where D and D t are Hirota operators (.... Rational solutions of the Boussinesq equation (D t + D 1 D4 F F = 0, (.7 Since the Boussinesq equation (1. has the rational solution (.4 then we seek solutions in the form u n (, t = ln F n(, t, n 1, (.8 where F n (, t is a polynomial of degree 1 n(n + 1 in and t, with total degree 1 n(n + 1, of the form F n (, t = n(n+1/ m=0 m a j,m j t (m j, (. with a j,m constants which are determined by equating powers of and t. Using this procedure we obtain the following polynomials F 1 (, t = + t + 1, F (, t = 6 + ( t + 5 F (, t = 1 + ( 6t + 8 j=0 4 + ( t 4 + 0t 15 + t ( 15t 4 + 0t ( 15t t t t t t ( 0t (.10a, (.10b 6 t t Downloaded from by guest on 0 December 018

7 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 7 Fig.. Plots of the rational solutions u n (, t, for n = 1,,..., 6, of the Boussinesq equation. + ( 6t t t t t t t t t t t , (.10c and the polynomials F 4 (, t and F 5 (, t are given in the Appendi. We note that these polynomials have the following form F n (, t = ( + t n(n+1/ + G n (, t, where G n (, t is a polynomial of degree 1 (n+(n 1 in both and t. We remark that the polynomials F n (, t which arise in the rational solutions of the focusing NLS equation (1.1 have a similar structure, see for eample (.6c, though the coefficients in the polynomials G n (, t are different. The polynomials F j (, t, for j =,, 4, in scaled variables, are given by Pelinovsky & Stepanyants (1 see their equations (6 (8. However, while they state that the polynomials are associated with solutions of their equation (, which is a scaled variant of the Boussinesq equation (1., Pelinovsky & Stepanyants (1 don not mention, or reference, the Boussinesq equation. In Fig., plots of the rational solutions u n (, t, for n = 1,,..., 6, of the Boussinesq equation. These show that the maima of the solutions all lie on the line t = 0, with n local maima for the rational solution u n (, t. In Fig., plots of the comple roots of F n (, t, for, 4, 5, for t = 0 and t = n, i.e., t = for n =, t = 1 for n = 4 and t = 15 for n = 5, are given. Each plot shows the comple -plane with roots in of F n (, t are shown at two different values of t. These show a triangular structure for both t = 0 and t = n, though with a different orientation. For t = 0 the roots of the polynomials approimately form Downloaded from by guest on 0 December 018

8 P. A. CLARKSON AND E. DOWIE 8 Fig.. Plots of the comple roots of the polynomials F n (, t, for, 4, 5, for t = 0 (red and t = n (blue, i.e., t = for n =, t = 1 for n = 4 and t = 15 for n = 5. Each plot shows the comple -plane with roots in of F n (, t are shown at two different values of t. Fig. 4. Plots of the loci of the comple roots of F n (, t, for, 4, 5, as t varies, with t = 0 (red and t = n (blue, i.e., t = for n =, t = 1 for n = 4 and t = 15 for n = 5. two isosceles triangles with curved sides. For t = n the roots of the polynomials again approimately form two isosceles triangles, though the values of the roots show that they actually also lie on curves rather than straight lines. An analogous situation arises for the Yablonskii Vorob ev polynomials (Clarkson & Mansfield, 00 and generalized Okamoto polynomials (Clarkson, 00. In Fig. 4, plots of the loci of the comple roots of F n (, t, for, 4, 5, as t varies, between the triangular structures for t = 0 and t = n are given. These show that as t increases the roots move away from the real ais. In Fig. 5, plots of the loci of the comple roots of F 6 (, t with the solution u 6 (, t superimposed, as t varies are given. The scale on the vertical ais relates to the comple -plane for the roots of F 6 (, t. These show that as the roots move away from the real ais, the solution decays to zero. Downloaded from by guest on 0 December 018

9 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION Fig. 5. Plots of the loci of the comple roots of F 6 (, t with the solution u 6 (, t superimposed (blue, as t varies. The scale on the vertical ais relates to the comple -plane for the roots of F 6 (, t... Generalized rational solutions of the Boussinesq equation Since the focusing NLS equation (1.1 has generalized rational solutions, see (.7, then a natural question is whether the Boussinesq equation (1. also has generalized rational solutions. To investigate this, we are concerned with the following theorem. Theorem.1 The Boussinesq equation (1. has generalized rational solutions in the form for n 1, with ũ n (, t; α, β = ln F n (, t; α, β, (.11 F n+1 (, t; α, β = F n+1 (, t + αtp n (, t + βq n (, t + ( α + β F n 1 (, t, (.1 where F n (, t is given by (.10, P n (, t and Q n (, t are polynomials of degree 1 n(n + 1 in and t, and α and β are arbitrary constants. Downloaded from by guest on 0 December 018 Since the generalized polynomial D (, t; α, β for the focusing NLS equation has the structure given by (., we suppose that the Boussinesq equation (1. has a solution in the form (.11, with F n (, t

10 P. A. CLARKSON AND E. DOWIE 10 given by (.10 and the polynomials P n (, t and Q n (, t, which are of degree 1 n(n + 1 in and t, have the form P n (, t = n(n+1/ m=0 m b j,m j t (m j, Q n (, t = j=0 n(n+1/ m=0 m c j,m j t (m j, (.1 where the coefficients b j,m and c j,m are to be determined. Substituting (.1 into the bilinear equation (.6 with F 1 (, t, F (, t, F (, t and F 4 (, t given by (.10, P n (, t and Q n (, t in the form (.1, then by equating powers of and t we find that P 1 (, t = t + 5, Q 1 (, t = t 1, P (, t = 5 6 ( 5t 5 4 ( t Q (, t = 6 ( t 1 t ( 5t t + 45 j=0 + t 6 7 t4 45 (.14a (.14b t , (.14c t t t , (.14d 81 with α and β arbitrary constants; the polynomials P (, t, Q (, t, P 4 (, t and Q 4 (, t are given in the Appendi. The first two generalized rational solutions are where and ũ (, t; α, β = ln F (, t; α, β, (.15 ũ (, t; α, β = ln F (, t; α, β, (.16 F (, t; α, β = F (, t + αtp 1 (, t + βq 1 (, t + α + β = 6 + ( t ( t 4 + 0t 15 + t t t αt ( t β ( t 1 + α + β (.17 F (, t; α, β = F (, t + αtp (, t + βq (, t + (α + β F 1 (, t = 1 + ( 6t ( 15t 4 + 0t ( 0t t t ( 15t t t t ( 6t t t t t t t t t t t αt { 5 6 ( 5t 5 4 ( t t t 6 7 t4 45 t } Downloaded from by guest on 0 December 018

11 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 11 Fig. 6. Plots of the generalized rational solution ũ (, t; α, β of the Boussinesq equation for various values of the parameters α and β. + β { 6 ( t 1 4 ( } 5t t t t t (α + β ( + t + 1, (.18 with α and β arbitrary constants. Plots of the solutions ũ (, t; α, β, ũ (, t; α, β and ũ 4 (, t; α, β of the Boussinesq equation for various values of the parameters α and β are given in Figs 6 8, respectively. Contour plots of the solutions ũ (, t;10 4,10 4, ũ (, t;10 7,10 7 and ũ 4 (, t;10 10,10 10 of the Boussinesq equation (1. illustrating this behaviour are given in Fig.. Figure 6 shows that the solution ũ (, t; α, β has two peaks when α = β = 0, then as α and β increase a third peak appears. Numerical evidence suggests that as α and β increase then the three peaks all tend to the same height ma(ũ = 4. For α and β sufficiently large, then ũ (, t; α, β has three lumps which are essentially copies of the lowest-order solution, i.e., u 1 (, t, which equally spaced on a circle; an analogous situation arises for the second generalized rational solution of the NLS equation (Kedziora et al., 01, 01. Figure 7 shows that the solution ũ (, t; α, β has three peaks when α = β = 0, then as α and β increase three more peaks appear, for α and β sufficiently large with one central peak and five in a circle around it, so forming a pentagram. Again, numerical evidence suggests that as α and β increase then the three peaks all tend to the same height ma(ũ = 4. For α and β sufficiently large, the rational solution ũ (, t; α, β has si lumps, again essentially copies of the lowest-order solution u 1 (, t, with five equally spaced on a circle; an analogous situation arises for the third generalized rational solution of the NLS equation (Kedziora et al., 011, 01. Downloaded from by guest on 0 December 018

12 P. A. CLARKSON AND E. DOWIE 1 Fig. 7. Plots of the generalized rational solution ũ (, t; α, β of the Boussinesq equation for various values of the parameters α and β. Figure 8 shows that the solution ũ 4 (, t; α, β has four peaks when α = β = 0, then as α and β increase five more peaks appear, with for α and β sufficiently large with two central peaks and seven in a circle around it, so forming a heptagram. As for ũ (, t; α, β and ũ (, t; α, β, numerical evidence suggests that as α and β increase then the peaks all tend to the same height ma(ũ 4 = 4. An analogous situation arises for the fourth generalized rational solution of the NLS equation (Kedziora et al., 01. Remark. Ohta & Yang (01, Figure 1 show that for focusing NLS equation (1.1, the generalized rational solution ψ (, t; α, β (.7 has a single peak when α = β = 0, and three peaks otherwise. Ohta & Yang (01, Figure also show that the generalized rational solution ψ (, t; α, β has a single peak when unperturbed, and si peaks otherwise. Define the polynomials Θ n ± (, t, for n N,by Θ ± n (, t = P n(, t ± itq n (, t, (.1 with P n (, t and Q n (, t the polynomials in the generalized rational solution (.1. Then for P n (, t and Q n (, t given by (.14, it is easily verified that Θ n ± (, t, for n = 1,,, 4, satisfy the bilinear equation (.6. Hence in the general case we have the following conjecture. Downloaded from by guest on 0 December 018 Conjecture. The polynomials Θ n ± (, t given by (.1 satisfy the bilinear equation (.6. Consequently, from this and Theorem.1 we have the following result.

13 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 1 Fig. 8. Plots of the generalized rational solution ũ 4 (, t; α, β of the Boussinesq equation for various values of the parameters α and β. Fig.. Contour plots of the generalized rational solutions ũ (, t;10 4,10 4, ũ (, t;10 7,10 7 and ũ 4 (, t;10 10,10 10 of the Boussinesq equation. Downloaded from by guest on 0 December 018 Lemma.4 Let Θ ± n (, t be given by (.1, then the polynomial F n+1 (, t; α, β given by (.1 can be written as F n+1 (, t; α, β = F n+1 (, t + (α + iβθ + n (, t + (α iβθ n (, t + ( α + β F n 1 (, t, (.0

14 P. A. CLARKSON AND E. DOWIE 14 which is a linear combination of four solutions F n+1 (, t, Θ ± n (, t and F n 1(, t of the bilinear equation ( Rational solutions of the KP I equation 4.1. Introduction The KP equation (v τ + 6vv ξ + v ξξξ ξ + σ v ηη = 0, σ =±1, (4.1 which is known as KPI if σ = 1, i.e., (1.5, and KPII if σ = 1, was derived by Kadomtsev & Petviashvili (170 to model ion-acoustic waves of small amplitude propagating in plasmas and is a two-dimensional generalization of the KdV equation (1.4. The KP equation arises in many physical applications including weakly two-dimensional long waves in shallow water (Ablowitz & Segur, 17; Segur & Finkel, 185, where the sign of σ depends upon the relevant magnitudes of gravity and surface tension, in nonlinear optics (Pelinovsky et al., 15, ion-acoustic waves in plasmas (Infeld & Rowlands, 10, two-dimensional matter-wave pulses in Bose Einstein condensates (Tsuchiya et al., 008 and as a model for sound waves in ferromagnetic media (Turitsyn & Falkovitch, 185. The KP equation (4.1 is also a completely integrable soliton equation solvable by inverse scattering and again the sign of σ is critical since if σ = 1, then the inverse scattering problem is formulated in terms of a Riemann Hilbert problem (Manakov, 181; Fokas & Ablowitz, 18, whereas for σ = 1, it is formulated in terms of a ( DBAR problem (Ablowitz et al., 18. The first rational solution of the KPI equation (1.5, is the so-called lump solution v(ξ, η, τ = ξ ln[(ξ τ + η + 1] = 4 (ξ τ η 1 [(ξ τ + η + 1], (4. which was found by Manakov et al. (177. Subsequent studies of rational solutions of the KPI equation (1.5 include Ablowitz et al. (000, Ablowitz & Villarroel (17, Dubard & Matveev (011, 01, Gaillard (016a,b, Johnson & Thompson (178, Ma (015, Pelinovsky (14, 18, Pelinovsky & Stepanyants (1, Satsuma & Ablowitz (17, Singh & Stepanyants (016 and Villarroel & Ablowitz (1. We remark that the KP equation (4.1 is invariant under the Galilean transformation (ξ, η, τ, v (ξ + 6λ, η, τ, v + λ, (4. with λ an arbitrary constant. In fact the rational solutions of the KPI equation (1.5 derived by Dubard & Matveev (011, 01 and Gaillard (016a,b are equivalent under the Galilean transformation ( Rational solutions of KPI related to the focusing NLS equation Dubard & Matveev (011, 01 derive rational solutions of the KPI equation (1.5 from the generalized rational solution ψ (, t; α, β (.7 of the focusing NLS equation (1.1 (see also Dubard et al. (010 and Gaillard (016a,b. Specifically Dubard & Matveev (011, 01 show that Downloaded from by guest on 0 December 018 v(ξ, η, τ = ξ ln ( D (ξ τ, η; α, 48τ = 1 ψ (, t; α, β 1 =ξ τ,t=η,β= 48τ, (4.4

15 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 15 is a solution of the KPI equation (1.5. If we define F nls (ξ, η, τ; α = D (ξ τ, η; α, 48τ, then F nls (ξ, τ; α = ξ 6 18τξ 5 + ( 45τ + η + 1 ξ 4 1 ( 45τ + η 5 τξ + { η ( τ 1 η + 115τ 4 70τ + 7 } ξ { 18τη ( τ + 5 τη τ 5 68τ + 450τ } ξ + η ( τ + 1 η 4 + ( 7τ τ + 11 η + 7τ 6 4τ τ +. (4.5 The polynomial F nls (ξ, τ; α satisfies ( D 4 ξ + D ξd τ Dη F F = 0, (4.6 which is the bilinear form of the KPI equation (1.5, and so v nls (ξ, η, τ; α = ξ is a rational solution of the KPI equation (1.5. ln Fnls 4.. Rational solutions of KPI related to the Boussinesq equation (ξ, η, τ; α, (4.7 The Boussinesq equation (1. is a symmetry reduction of the KPI equation (1.5 and so the generalized rational solutions ũ n (, t; α, βgiven by (.11 of the Boussinesq equation can be used to generate rational solutions of the KPI equation. If in the KPI equation (1.5 we make the travelling wave reduction v(ξ, η, τ = u(, t, = ξ τ, t = η, then u(, t satisfies the Boussinesq equation (1.. Consequently given a solution of the Boussinesq equation (1., then we can derive a solution of the KPI equation (1.5. In particular, if u(, t = ln F(, t, for some known F(, t, is a solution of the Boussinesq equation (1., then v(ξ, η, τ = ln F(ξ τ, η ξ is a solution of the KPI equation (1.5. For eample the choice F(, t = + t + 1 gives the lump solution (4. of KPI. Using the generalized rational solution ũ (, t; α, β(.15 of the Boussinesq equation (1. we obtain the rational solution of the KPI equation (1.5 given by Downloaded from by guest on 0 December 018 v(ξ, η, τ; α, β = ξ ln Fbq (ξ, η, τ; α, β, (4.8

16 where F bq (ξ, η, τ; α, β = F (, t; α, β, i.e., P. A. CLARKSON AND E. DOWIE 16 (ξ, η, τ; α, β = ξ 6 18τξ 5 + ( 45τ + η + 5 ξ 4 1 ( 45τ + η + 5 τξ + { η ( } τ + 5 η + 115τ τ 15 ξ { 18η ( τ + 5 } η τ τ + 50 τξ + η ( τ η 4 + ( 7τ 4 + 0τ η F bq + 7τ τ 4 15τ α { ξ η 18ξτη η + ( 7τ β { ξ ξ τ ( η 7τ + 1 ξ 7τ + τη + τ } + α + β. (4. We remark that this polynomial, in scaled coordinates, is given by Gorshkov et al. (1, see their equation (4., though the authors don t mention the Boussinesq equation A more general rational solution If we compare the polynomials F nls (ξ, η, τ; α and Fbq (ξ, η, τ; α, β, respectively, given by (4.5 and (4., then we see that they are fundamentally different. As we shall now demonstrate, they are special cases of a more general polynomial. Consider the polynomial F (ξ, η, τ; μ, α, β, with parameters μ, α and β, given by F (ξ, η, τ; μ, α, β = ξ 6 18τξ 5 + (η + 15τ 6μ + ξ 4 { 6η + 540τ 1(6μ + 6μ 7 } τξ + { η (τ μ + 1η + 115τ 4 54(6μ + 1μ 5τ + μ(μ + (μ μ + } ξ { 18η 4 + 6(τ + 5η τ 4 4(μ + 6μ 1τ η } + 18μ(μ + 1μ μ + 1 } τξ + η 6 + (7τ + 6μ + 1μ + η 4 + { 4τ (6μ + 7τ + (μ 4 + 4μ + 6μ 4μ + 4 } η + 7τ 6 81(μ + 4μ 1τ 4 + (μ 4 + 7μ + 150μ + 1μ + 16τ + (μ μ + + α { ηξ 18τηξ η + [ τ μ(μ + ] η } + β { ξ τξ 6(η τ + μ ξ + τη 7τ + (μ + 1μ + 4τ } + α + β. (4.10 This polynomial has both the polynomials F nls (ξ, η, τ; α and Fbq (ξ, η, τ; α, β as special cases, specifically Furthermore F nls (ξ, η, τ; α = F (ξ, η, τ;1,α,0, F bq (ξ, η, τ; α, β = F (ξ, η, τ; 1, α, β. Downloaded from by guest on 0 December 018 v(ξ, η, τ; μ, α, β = ξ ln F (ξ, η, τ; μ, α, β, (4.11

17 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 17 with F (ξ, η, τ; μ, α, β given by (4.10, is a solution of the KPI equation (1.5, which includes as special cases the solutions (4.7, when μ = 1 and β = 0, and (4.8, when μ = 1, as is easily shown. In Fig. 10, the initial solution v(ξ, η,0;μ,0,0 given by (4.11 is plotted for various choices of the parameter μ. When μ = 1, then this arises from the solution (4.7 derived from the focusing NLS equation (1.1 whilst when μ = 1, then this arises from the solution (4.8 derived from the Boussinesq equation (1.. From Fig. 10, we can see that for μ<μ, the solution v(ξ, η,0;μ,0,0 has two peaks on the line η = 0, which coalesce when μ = μ to form one peak at ξ = η = 0. By considering when ξ v(ξ,0,0;μ,0,0 = 8(μ4 + 1μ + 16μ 6 = 0, ξ=0 (μ μ + then μ is the real positive root of [ ][ ] μ 4 + 1μ + 16μ 6 = μ + (1 1 6μ + 6 μ + ( μ = 0, i.e., μ = For μ>μ, it can be shown that v(0, 0, 0; μ,0,0 = 4μ(μ + μ μ +, increases until it reaches a maimum height of 4( + 5 when μ = 1 (1 + 5, which is the golden mean! 5. Discussion In this article, we have derived a sequence of algebraically decaying rational solutions of the Boussinesq equation (1. which depend on two arbitrary parameters, have an interesting structure and have a similar appearance to rogue-wave solutions in the sense that they have isolated lumps. The associated special polynomial, which has equal weight in and t, satisfies a bilinear equation of Hirota type and comprises of a linear combination of four independent solutions of the bilinear equation, something remarkable for a solutions of a bilinear equation. The derivation of a representation of these special polynomials as determinants is currently under investigation and we do not pursue this further here. We remark that other types of eact solutions of the Boussinesq equation (1. can be derived using the bilinear equation (.6 including breather solutions (Tajiri & Murakami, 18, 11 and rational-soliton solutions (Rao et al., 017. Using our rational solutions of the Boussinesq equation (1., we derived rational solutions of the the KPI equation (1.5 and compared them to those obtained from rational solutions of the focusing NLS equation (1.1 bydubard & Matveev (011, 01. It was shown that the two sets of solutions are fundamentally different and both are special cases of a more general rational solution. We remark that Ablowitz et al. (000 (see also Ablowitz & Villarroel (17 and Villarroel & Ablowitz (1 derived a hierarchy of algebraically decaying rational solutions of the KPI equation (1.5 which have the form Downloaded from by guest on 0 December 018 v m (ξ, η, τ = ξ ln G m(ξ, η, τ, (5.1

18 P. A. CLARKSON AND E. DOWIE 18 Downloaded from by guest on 0 December 018 Fig. 10. The initial solution v(ξ, η,0;μ,0,0 given by (4.11 is plotted for various choices of the parameter μ. When μ = 1 the initial solution corresponds to that arising from the Boussinesq equation (1. and when μ = 1 to the initial solution from the focusing NLS equation (1.1.

19 RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION 1 where G m (ξ, η, τ is a polynomial of degree m in ξ, η and τ. These rational solutions are derived in terms of the eigenfunctions of the non-stationary Schrödinger equation iϕ η + ϕ ξξ + vϕ = 0, (5. with potential v = v(ξ, η, τ, which is used in the solution of KPI (1.5 by inverse scattering; equation (1.5 is obtained from the compatibility of (5. and ϕ τ + 4ϕ ξξξ + 6vϕ ξ + wϕ = 0, w ξ = v. (5. This is a fundamentally different hierarchy of solutions of the KPI equation (1.5 compared to those discussed in Section 4, not least because it involves polynomials of all even degrees, not just of degree n(n + 1, with n N. Acknowledgements P.A.C. thanks Nail Akhmediev, Adrian Ankiewicz and Andrew Bassom for helpful comments and illuminating discussions and also the School of Mathematics & Statistics at the University of Western Australia, Perth, Australia, for their hospitality during his visits when some of this research was done. We also thank the reviewers for their helpful comments. References Ablowitz, M. J., Bar Yaacov, D. & Fokas, A. S. (18 On the inverse scattering transform for the Kadomtsev Petviashvili equation. Stud. Appl. Math. 6, Ablowitz, M. J., Chakravarty, S., Trubatch, A. D. & Villarroel, J. (000 A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev Petviashvili I equations. Phys. Lett. A 67, Ablowitz, M. J. & Clarkson, P. A. (11 Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 14. Cambridge: Cambridge University Press. Ablowitz, M. J. & Haberman, R. (175 Resonantly coupled nonlinear evolution equations. J. Math. Phys. 16, Ablowitz, M. J. & Satsuma, J. (178 Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 1, Ablowitz, M. J. & Segur, H. (17 On the evolution of packets of water waves. J. Fluid Mech., Ablowitz, M. J. & Segur, H. (181 Solitons and the Inverse Scattering Transform. Philadelphia: SIAM. Ablowitz, M. J. & Villarroel, J. (17 Solutions to the time dependent Schrödinger and the Kadomtsev Petviashvili equations. Phys. Rev. Lett. 78, Adler, M. & Moser, J. (178 On a class of polynomials associated with the Korteweg-de Vries equation. Commun. Math. Phys. 61, 1 0. Airault, H. McKean, H. P. & Moser, J. (177 Rational and elliptic solutions of the KdV equation and related many-body problems. Commun. Pure Appl. Math. 0, Akhmediev, N., Ankiewicz, A. & Soto-Crespo, J. M. (00a Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, Akhmediev, N., Ankiewicz, A. & Taki, M. (00b Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 7, Akhmediev, N., Soto-Crespo, J. M. & Ankiewicz, A. (00c Etreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A 7, Ankiewicz, A., Clarkson, P. A. & Akhmediev, N. (010 Rogue waves, rational solutions, the patterns of their zeros and integral relations. J. Phys. A 4, 100. Downloaded from by guest on 0 December 018

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