On elementary four-wave interactions in dispersive media

Abstract The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.


Introduction
The objective of the present study is to provide an account of the solutions of the system: where b i (t), i = 1, . . ., 4, are complex valued functions of time t, where an asterisk stands for complex conjugate and where the real-valued coefficients ω i , T ij = T ji and T are independent of time.System (1.1) is completed by initial conditions: b i (0) = √ q i e iϕ i , q i ≥ 0, ϕ i ∈ [0, 2π]. (1.2) The positive q i are the initial wave actions.Equations similar to (1.1) were first derived by Armstrong et al. (1962) in nonlinear optics (time being replaced by a spatial direction) and by Benney (1962) in hydrodynamics.Physically, (1.1) governs the evolution of an isolated quartet of weakly nonlinear waves with wave vectors (k 1 , k 2 , k 3 , k 4 ) satisfying (1.3) propagating with linear frequency ω i = ω(k i ) > 0 in a conservative dispersive medium with non-decay dispersion law, i.e.where three-wave quadratic interactions are excluded.
The coupling coefficients T ij and T depend on the wave vectors and on the physical properties of the medium under consideration.Defining the frequency mismatch (1.4) the interaction (1.3) is said to be resonant if ω = 0. Systems similar to (1.1) were also derived by Bretherton (1964) and Inoue (1975) from scalar model equations, by Boyd & Turner (1978) in plasma physics and by Chen & Snyder (1989) in nonlinear optics.Stiassnie & Shemer (2005) deduced (1.1) from the Zakharov equation (Zakharov 1966(Zakharov , 1968;;Krasitskii 1990Krasitskii , 1994) that governs the evolution of discrete or continuous spectra of weakly nonlinear gravity waves, and which is generally used as the starting point for weak turbulence statistical theory (Yuen & Lake 1982;Zakharov, L'vov & Falkovich 1992;Zakharov 1999;Janssen 2004;Nazarenko & Lukaschuk 2016), even though Hasselmann (1962) in his pioneering work used primitive equations.
Complementary to (1.1), an isolated wave triad (k 1 , k 2 , k 3 ) satisfying (1.5) also interacts nonlinearly at third order and is governed by as first derived by Benney (1962) for resonant gravity-wave interactions: and deduced by Shemer & Stiassnie (1985) from the Zakharov equation.In nonlinear optics, a spatial analogue of (1.6) has been derived by Cappellini & Trillo (1991), who also pointed out that the three-wave system (1.6) cannot be deduced from the four-wave system (1.1) by simple algebraic relations between the complex amplitudes b i (t) and c i (t).System (1.1) may be solved following a procedure introduced by Armstrong et al. (1962) for wave triads in quadratic interaction and extended to quartets in cubic interaction by Bretherton (1964): it consists of reducing (1.1) to a single scalar equation governing the evolution of an auxiliary variable, say q(t).Bretherton proved that q(t) is 'a periodic function of [t], with period depending on the initial conditions but which is only in exceptional cases infinite', but did not give explicit solutions.This was achieved by Inoue (1975), Boyd & Turner (1978), Turner (1980), Chen & Snyder (1989) and Stiassnie & Shemer (2005) using elliptic functions.Some of the 'exceptional cases' mentioned by Bretherton correspond to solutions now called 'pump' and 'breathers'; some of these were given by Inoue (1975) and Turner (1980), but some others were missing.
Following a similar procedure, periodic solutions of (1.6) involving elliptic functions were found by Shemer & Stiassnie (1985) and Cappellini & Trillo (1991).These latter authors, who also found breather solutions, proved that the problem may be recast to an elegant one-degree integrable Hamiltonian system; see also Trillo & Wabnitz (1991).This approach allows one to plot phase portraits representing level-lines of the Hamiltonian from which interesting qualitative and quantitative results can be deduced: isolated points surrounded by closed orbits in phase space correspond respectively to stable equilibria and to periodic solutions, while saddle points connected by separatrices correspond respectively to unstable equilibria and to non-periodic solutions (Cappellini & Trillo 1991;Trillo & Wabnitz 1991;Andrade & Stuhlmeier 2023b).A similar Hamiltonian formulation is used in the present study, but phase portraits are not necessary for our purpose.
Steady equilibria of (1.1) or of (1.6) are of fundamental importance.By steady equilibria, we mean solutions with steady amplitudes Benney (1962) noticed that the simplest of these is the finite-amplitude travelling wave discovered by Stokes in 1847.The existence of finite-amplitude bichromatic wavetrains (k 1 , k 2 ) was first proved by Phillips (1960), Longuet-Higgins & Phillips (1962) and Benney (1962).These steady bichromatic waves were one of the essential ingredients for the development of the theory of weakly nonlinear wave interactions (see also the historical survey by Phillips (1981)): in a few words, we recall that Phillips (1960) and Longuet-Higgins (1962) proved that the propagation of a finite-amplitude bichromatic gravity wavetrain, say (k 1 , k 3 ), leads initially by cubic (or 'tertiary') nonlinear resonant interaction to the spontaneous linear growth of a third wave k 2 satisfying (1.5) and (1.7).As noticed by Phillips (1967), this behaviour may be deduced from (1.6) considering |c 2 | |c 1 |, |c 3 |.The previous mechanism has to be distinguished from quadratic (or 'secondary') nonlinear interactions inside wave triads for which it has been proved by Galeev & Karpman (1963) and Hasselmann (1967) that a finite-amplitude wave k 3 is exponentially unstable to a couple of infinitesimal disturbances (k 1 , k 2 ) if (see also Craik 1985, p. 131) (1.8) Sometimes called 'decay instability', this mechanism cannot, however, operate in gravity waves, as proved by Phillips (1960) andHasselmann (1962).Back to cubic resonant interactions between three waves satisfying (1.5), (1.6) and (1.7), the continuous energy transfer from (k 1 , k 3 ) to k 2 discovered by Phillips excludes therefore the possibility of steady states at resonance.However, the existence of non-resonant steady states in a system of three waves (k 1 , k 2 , k 3 ) with finite constant amplitudes |b i | has been established near resonance by Cappellini & Trillo (1991), Liao, Xu & Stiassnie (2016) and Andrade & Stuhlmeier (2023b).
In the case of 'non-degenerate' quartets satisfying (1.1) and (1.3), the existence of steady equilibria has been established at resonance in numerical simulations by Liu & Liao (2014), observed experimentally by Liu et al. (2015) and identified off-resonance by Andrade & Stuhlmeier (2023a) using a Hamiltonian approach.The bidirectional standing wave of Okamura (1985) is also a particular case of such steady quartets.It was clear from these studies that the existence of steady equilibria is conditioned by certain constraints between the finite amplitudes |b i |, but a general explicit formulation of these compatibility conditions was missing.
The present study focuses only on four-wave interactions (1.3) inside a single quartet governed by (1.1).Motivated by the recent work of Andrade & Stuhlmeier (2023a) who established some interesting links between various aspects mentioned above, but who restricted their analysis to quartets with symmetric initial conditions |b 1 (0)| = |b 2 (0)| and |b 3 (0)| = |b 4 (0)|, the present study aims at answering the questions that remain open, putting the various pieces of the puzzle together and finding the missing ones.In some sense, our work may be viewed as the extension to non-degenerate quartets (1.3) of the analyses of Shemer & Stiassnie (1985), Cappellini & Trillo (1991) and Andrade & Stuhlmeier (2023b) for the 'degenerate' case (1.5).
The paper is organized in two parts: the first one is generic to dispersive media ( § § 2-4), the second specific to deep-water gravity waves ( § § 5-7).More precisely: reduction to a single equation is carried out in § 2; steady equilibria and their stability are investigated in § 3; unsteady solutions are presented in § 4; models of steady and periodic solutions on deep water are described in § 5; examples of pump and breathers are elaborated in § § 6 and 7. Results are summarized in § 8 and complements are given in the appendices.

Linear stability
Analogy with Newtonian dynamics allows us to formulate a simple criterion for linear stability of steady equilibria.Indeed, (3.1) yields, after linearization around q = 0, where from (2.20) with (2.21) and (2.22), we have γ 2 = c.Therefore, the null solution is linearly stable if c < 0 and unstable otherwise; growth is exponential if c > 0 or algebraic if c = 0. Consider first the linear stability of the bichromatic solution (3.3) for which q 3 = q 4 = 0.In that case, c = 16T 2 q 1 q 2 − B 2 , where B is defined in (2.9).Therefore, the bichromatic wavetrain (3.3) is exponentially unstable if (3.8) Recalling that q i = |b i | 2 and that B i are defined in (2.9), we recover the criterion derived in Leblanc (2009, equation (15)) and recovered by Andrade & Stuhlmeier (2023a).We also recall that (3.8) characterizes respectively 'type B' and 'class-Ia' instabilities following the respective classifications of Okamura (1984) for one-dimensional standing waves and of Ioualalen & Kharif (1994) for steady bichromatic wavetrains.
Turning now to the stability of steady wave quartets with non-zero initial wave actions (q 1 , q 2 , q 3 , q 4 ) satisfying (3.5) and cos p 0 = ±1, the linear stability criterion becomes, after replacing B by the right-hand side of (3.5) and C by (2.19) onto the expression of c  in (2.21):   THEOREM 3.2. The steady wave quartet (3.6) satisfying (3.5 algebraically unstable if equality holds, otherwise linearly stable.

The particular case of equal wave actions
Consider the particular case of the steady wave quartet with equal wave actions q 1 = q 2 = q 3 = q 4 ≡ q 0 > 0 and initial phase mismatch p 0 = 0 or π.We recall that the existence of such steady interactions is conditioned by (3.5) which reduces in the present case to where e. from (2.9): At resonance, (3.10) implies B = 0, while off resonance we get q 0 = − ω/ B. Since q 0 > 0, ω and B must have opposite signs for the quartet to exist.The corresponding instability criterion may be easily deduced from (3.9).Therefore: CRITERION 3.1.Steady wave quartets (3.6) with q 1 = q 2 = q 3 = q 4 ≡ q 0 exist if q 0 = − ω/ B > 0. At resonance, they exist for any q 0 > 0 if B = 0.In both cases, they are exponentially unstable if (3.12)

Lagrange theorem and Lyapunov stability
In his treatise on analytical mechanics published in 1788, Lagrange presented his famous principle on the stability of equilibrium positions, which was rigorously proved by Lejeune-Dirichlet in 1846 and generalized by Lyapunov in 1892 (see Loria & Panteley 2017).Lagrange theorem may be stated as (Gantmacher 1975, pp. 166-173;Arnold 1989, p. 99): LAGRANGE THEOREM.If U(0) = 0 is a strict local minimum of the potential U(q), then the null solution of (2.18) is Lyapunov stable.
For our purpose, stability in the sense of Lyapunov is defined by: LYAPUNOV STABILITY.The null solution is Lyapunov stable if for each ε > 0 there exists In the present case, U(q) = − 1 2 (aq 4 + bq 3 + cq 2 + dq + e).But, as stated previously, q = 0 is a steady equilibrium if U(0) = 0 and U (0) = 0, or equivalently d = e = 0. Therefore, at equilibrium, U(q) = − 1 2 q 2 (aq 2 + bq + c).But since in that case U (0) = −c, then U(q) admits a strict local minimum at q = 0 if c < 0. Therefore in our case, a linearly stable equilibrium is also Lyapunov stable.This means that if c < 0, any sufficiently small disturbance will remain bounded.
Of course it would be necessary to be precise about what is meant by 'sufficiently small' but we shall not pursue that direction for at least two reasons: firstly, because my study is restricted to the interactions inside a single quartet while interactions with other waves may be destabilizing (see e.g.Okamura 1985); secondly, because higher-order interactions are not taken into account (see e.g.Andrade & Stuhlmeier 2023a).As a consequence, we have to keep in mind that stability is only indicative because limited to the discrete four-wave interactions considered in the present study.
By contrast, linear instability criteria are meaningful in the nonlinear regime as it is known that the existence of an exponentially growing solution of the linearized equation (3.7) implies instability of the null solution in the nonlinear equation (3.1) (see Verhulst 1996, p. 88).Furthermore, if other interactions were taken into account, various instability mechanisms would compete without mutual cancellation.
Therefore, the various instability criteria presented in the present study have to be considered as sufficient conditions for instability.

General properties
We have seen in § 2 that solutions of (2.18) with (2.5) are bounded.Furthermore, following Inoue (1975), we get from (2.18) the following inequalities: Therefore we can conclude that, by continuity, f admits at least two real roots, say ξ − and ξ + , verifying ξ − ≤ 0 ≤ ξ + ; f is therefore a polynomial function of degree at least equal to two: either quartic, cubic or quadratic.
Excluding the case ξ − = ξ + = 0 corresponding to root 0 with multiplicity at least equal to 2 corresponding to a steady solution since d = 0 and e = 0 as explained in § 3.1, we restrict from now on our discussion to the cases where either ξ − ≤ 0 < ξ + or ξ − < 0 ≤ ξ + .Without lost of generality, suppose that ξ − and ξ + are the closest roots from 0. Equation (2.18) shows that unsteady solutions exist if f (q) ≥ 0; since q(0) = 0 and f (0) ≥ 0, Jeffreys & Jeffreys (1956, pp. 667-668), Bretherton (1964) and Pars (1965, pp. 4-6) showed that ξ − ≤ q(t) ≤ ξ + , ∀t ≥ 0, and that the particle 'velocity' vanishes on the boundaries of this interval: q(ξ ± ) = 0.If ξ − and ξ + are both simple roots, they are turning points, i.e. the sign of q(t) changes on turning points and the direction of the particle is reversed; therefore q(t) is periodic.If the multiplicity of either ξ − or ξ + , say ξ − , is strictly greater than one, then lim t→±∞ q(t) = ξ − and the solution is a breather.Finally if ξ − and ξ + are both double roots, then either lim t→±∞ q(t) = ξ ± or lim t→±∞ q(t) = ξ ∓ and the solution is a pump.Finally, from (2.6), we have also Equation (2.18) with (2.5) may be integrated (see e.g.Craik 1985, p. 138): from which q ≡ q(t) may formally be obtained by inversion.The sign indeterminacy above shows that for each fixed values of the parameters, (2.18) admits two solutions, say The solution to choose is determined because of (2.7) from which we get q(0) = −4T √ q 1 q 2 q 3 q 4 sin p 0 .(4.4) Thus, if −4T √ q 1 q 2 q 3 q 4 sin p 0 ≥ 0, then Finally, if q(t) is periodic, the period τ is In the quartic case, f in (2.20) with a = 0 may be factorized as where ξ 1 , ξ 2 , ξ 3 and ξ 4 are the four roots of f .(The labelling of the roots ξ i is independent of the labelling of the initial wave actions q i .)Since f is real-valued, roots are either real or complex conjugate by pairs.The roots of a quartic polynomial may formally be obtained by quadrature with Ferrari's method (published by Cardan in 1545) but formulae, which are too lengthy to be reported here, are implemented in computer algebra systems.Simple expressions are given in Appendix A in the case q 1 = q 2 = q 3 = q 4 .The nature of the solutions of the Bretherton equation (2.18) depends on the sign of a and on the nature of the roots, as illustrated in figure 1 (see also Turner 1980).If a = 0, the potential is either cubic or quadratic and the solutions are postponed to Appendix B. 4.2.The case a > 0 4.2.1.Periodic solution If f admits four distinct real roots ξ 1 , . . ., ξ 4 such that (figure 1a) then (2.18) with (2.5) has the pair of periodic solutions {Q where sn and sn −1 are Jacobi elliptic functions defined here by (see e.g.Byrd & Friedman (1971, p. 18)) where K is the complete elliptic integral of the first kind.(Different conventions exist for the arguments of elliptic functions; we follow here the notations of Byrd & Friedman Figure 1.The possible configurations for bounded unsteady solutions of (2.18) satisfying (2.5).Motion occurs in the potential well defined by q ∈ [ξ − , ξ + ], where ξ ± such that ξ − ≤ 0 < ξ + or ξ − < 0 ≤ ξ + are the nearest roots around zero between which U(q) = − 1 2 f (q) ≤ 0. Quartic potential (4.6) with a > 0: (a) periodic solution Cubic potential (B1) with b > 0: (h) periodic solution (B2); (i) breather solution (B4).Any other possibility may be deduced by symmetry with respect to the vertical axis.The case of quadratic potential with periodic solution (B6) has been omitted.(1971).By contrast, in Wolfram Mathematica the entry for sn(x, k) as defined in (4.9) is JacobiSN[x,m], where m = k 2 ; similarly for K(k) defined in (4.10) for which the entry is Note that at the initial time where at fixed modulus k: cn(x) = 1 − sn 2 (x) and dn(x) = 1 − k 2 sn 2 (x) (Byrd & Friedman 1971, p. 19).Therefore, at fixed k: Then Formula (254.00) in Byrd & Friedman (1971, p. 112) has been used to get (4.8),plotted in figure 2(a).Similar solutions were found by Inoue (1975), Boyd & Turner (1978), Shemer & Stiassnie (1985), Chen & Snyder (1989), Cappellini & Trillo (1991) and Stiassnie & Shemer (2005).

The truncated quartet model
Deep-water irrotational gravity waves propagating in an inviscid incompressible fluid are governed in spectral space at third order in amplitude by the Zakharov equation (Zakharov 1966(Zakharov , 1968;;Krasitskii 1990Krasitskii , 1994)) where is Dirac delta function and the real-valued function T(k, p, q, r) is Krasitskii's kernel given in Appendix C. At leading order, B(k, t) is related to the free-surface elevation z = η(x, t), x ∈ R 2 , by (we follow Janssen's (2004, p. 132) convention for the Fourier transform, so that expressions given here for T (and T ij ) must be divided by (2π) 2 to recover those given in Krasitskii (1994)) (5.2) According to Zakharov (1966Zakharov ( , 1968)), an exact solution of (5.1) is the Stokes wave: (5.3) If we now consider a linear combination of waves B(k, t) = N i=1 b i (t)δ(k − k i ) with N > 1, the Zakharov equation (5.1) yields a system of ordinary differential equations which is not closed, as noticed by Okamura (1985); it leads indeed to the generation of higher harmonics on time scales of order (|T||b| 2 ) −1 .The mathematical validity of such an ansatz is therefore an open question.(This was pointed to me out by an anonymous reviewer even for N = 2; see also discussion in Badulin et al. (1995).Zakharov (1967) already noticed that the bichromatic wave (3.3) is an approximate solution.) If the terms corresponding to higher harmonics are neglected, one gets a truncated model consisting of a closed system of ordinary differential equations considered in various textbooks and review articles (e.g.Yuen & Lake 1982;Craik 1985;Shemer & Stiassnie 1991;Janssen 2004;Kartashova 2010) and implicitly or explicitly used in a number of articles, including those by Saffman & Yuen (1980), Caponi, Saffman & Yuen (1982), Okamura (1984Okamura ( , 1985)), Shemer & Stiassnie (1985), Hogan et al. (1988), Badulin et al. (1995), Stiassnie & Shemer (2005), Leblanc (2009) and Andrade & Stuhlmeier (2023a,b): (5.4) where and 0 otherwise.In the case of four waves satisfying (1.3), (5.4) reduces to (1.1) where The 'free-surface elevation' of this truncated low-order model corresponding to an 'isolated' quartet would be (5.5) Although the formal validity of such a model with respect to actual solutions of the Zakharov equation (5. 1) is an open question, numerical and experimental results support its usefulness (Liu & Liao 2014;Liu et al. 2015;Liao et al. 2016).If the quartet is resonant, Benney's equations (1.1) obtained with the method of multiple scales are recovered.

Bidirectional standing waves and their stability
A particular case of interaction (1.3) concerns bidirectional standing waves for which (5.6)For simplicity, we consider the case where initially q 1 = q 2 = q 3 = q 4 ≡ q 0 > 0. From criterion 3.1 ( § 3.3), the wave quartet is steady providing that q 0 = − ω/ B > 0, where B is given in (3.11).For deep water, we have (see Appendix C) ) is strictly positive for any κ > 0 (the discontinuity at κ = 1 may be removed since ρ(κ) → 1/6 when κ → 1).Therefore then both ω = 0 and B = 0 so that, from criterion 3.1, q 0 > 0 may be chosen arbitrarily.Now,from (3.6): (5.8)where T may be evaluated explicitly thanks to the expressions given in Appendix C.
Theoretical predictions have been compared with numerical integration of (1.1) and results are in excellent agreement.A representative example is plotted in figure 4. Of course, the unsteady perturbed solutions that appear on this plot may be expressed with one of the explicit formulas given in § 4. To this aim, we first determine in each case the function f in (2.20).From the data given in the caption of figure 4, we get a < 0 so that f may be factorized as (4.6), where the four roots ξ i , i = 1, . . ., 4, may be explicitly evaluated thanks to the expressions given in Appendix A since in both cases q 1 = q 2 = q 3 = q 4 = q 0 (1 + ε); their numerical values are reported in table 1.Therefore, because of the Manley-Rowe relations (2.4), the unsteady periodic solutions plotted in figure 4 are |b 1 (t)| 2 = q 1 + q(t), where the functions q(t) are given by (4.21) for p 0 = 0 (dashed line) and by (4.27) for p 0 = π (solid line).Their respective periods are given in  4 for ε = 0.01.Their respective exact solutions q(t) are (4.21) for p 0 = 0 and (4.27) for p 0 = π; their crest-trough amplitude is ξ 2 − ξ 1 and their period is τ .table 1.Their graphs cannot be distinguished from their numerical counterparts, as the relative errors between numerical and exact solutions is of the order of 0.1 % at the final time of the computations.Finally, the free-surface elevation of these bifurcated solutions may be written as where A i and q 0 are defined in (5.9) and where the phases p i (t) are integrated numerically from (2.3).Results are plotted in figure 5: they show in the unstable case (figure 5b) five large-scale beats over ten periods τ of the corresponding function q(t).This is due to the fact that the amplitudes |b i (t)| involve the square root of q(t); see (2.4).
(A companion criterion is given in Appendix D for q 1 = q 3 and q 2 = q 4 .)Unstable disturbed solutions can lie on the same resonant surface (figure 6a), or not.For instance, if we consider the following non-resonant (but nearly resonant) quartet: Let us now construct a steady solution for the non-resonant quartet (5.15).We first test criterion 3.1 ( § 3.3), but since in that case ω/ B > 0, a steady solution with q 1 = q 2 = q 3 = q 4 does not exist; we need one more time another strategy.We are, however, lucky with (5.15) because all the B i defined in (2.9) have the same sign: negative.Thus we can set q i = α/B i expecting α < 0 for existence.From (3.5) we get (5.16) Since T > 0 for (5.15), we conclude that the only possibility to have α < 0 is p 0 = 0. Concerning stability, condition (3.9) with q i = α/B i becomes (5.17) Applying this condition for quartet (5.15) and p 0 = 0 gives stability.983 A27-19 Figure 7. Snapshots of the free-surface elevation (6.6) of the X-pump in the region k 0 dp i /dt = −ω 0 + 2k 3 0 q 0 , i = 1, . . ., 4. Therefore, the solution of (1.1) is the 'X-pump': where q(t) = q 0 tanh( √ aq 0 t), Ω 0 = ω 0 − 2k 3 0 q 0 and ω 0 = √ gk 0 .Free surface is and where a and P 0 are given respectively in (6.3) and (6.4). Figure 7 illustrates the fact that the energy of components (k 0 3 , k 0 4 ) is pumped and totally transferred to (k 0 1 , k 0 2 ), that is, from the y standing wave to the x standing wave.
One might expect that this pump solution could be reproduced experimentally; however, both standing waves constituting the complete solution (6.6) are linearly unstable with respect to infinitesimal disturbances constituted by the other couple of waves according to instability condition (3.8) applied for instance to the bichromatic wave (k 0 1 , k 0 2 ) with q 1 = q 2 disturbed by (k 0 3 , k 0 4 ) yielding 16T 2 > 4(1 + 4ν) 2 k 6 0 , true from (6.2).The converse being true, both instabilities may feed each other.This is illustrated in figure 8 in which is represented the X-pump (dashed line) and a perturbed solution where a relative error of 0.1 % has been made on the initial phase mismatch p 0 : the perturbed solution is periodic (solid line).In fact, the roots of (4.6) are, from (A2) with B = 0 and T and A given by (6.2) and (6.3): They are plotted in figure 9(a).They coincide by pairs for p 0 = P 0 and are distinct for any p 0 / ∈ {0, P 0 , π}, two being positive, two negative.The solution is in that case periodic and completely determined by (4.8) together with a in (6.3) and ξ i (6.7); their period is (4.10) (figure 9b), and their crest-trough amplitude at fixed p 0 is: 2 min |ξ i |, i = 1, . . ., 4. These are the 'X-beats' in which energy is periodically exchanged between components (k 0 1 , k 0 2 ) and (k 0 3 , k 0 4 ).Note finally that 0 is a double root if p 0 = 0 or π, corresponding to steady standing waves discussed in § 5.2.q 0 t q(t)/q 0 Figure 8. Solutions of (2.18) for (6.1) with q 1 = q 2 = q 3 = q 4 ≡ q 0 .X-pump with p 0 = P 0 (dashed line); X-beats with p 0 = 1.0001P 0 (solid line).P 0 is given by (6.4).given by (6.7) of the quartic (4.6) for the X-quartet (6.1).The dashed line corresponds to p 0 = P 0 (6.4) of the X-pump solution (6.6).(b) If p 0 / ∈ {0, P 0 , π}, solutions are periodic X-beats given by (4.8) with dimensionless period τ = k 3 0 q 0 τ given by (4.10).

Breathers
7.1.The Ψ -breathers Looking for breathers is a difficult task because two or three roots of the quartic function (4.6) have to coincide in a non-symmetric way, as compared with the X-pump solution (4.19) for which the roots were equal by pairs and opposite.
Together with a = (4T + A)(4T − A), roots (7.5) completely determine the solution (4.14), referred to as the 'Ψ -breathers' represented in figure 10 for various values of ε.Note that B = 0 for ε = 0, so that (7.4) matches with (6.4) obtained for the X-pump solution.Expansion near ε = 0 gives, up to the second order: It is interesting for the Ψ -breathers to reconstruct the free-surface elevation.Since the relative q(t) is given explicitly by (4.14), the relative phase p(t) defined in (2.2) may be deduced from (2.7) and plotted in figure 10.Then, the four individual moduli |b i (t)| are deduced from the Manley-Rowe equation (2.4), while numerical integration of (2.3) is necessary to obtain the individual phases p i (t) = arg b i (t) in order reconstruct the free-surface elevation given by (5.5).This is achieved in figure 11 for the Ψ -breathers defined by (7.1), (7.4) and (7.5) for different values of the parameter ε that quantifies the departure from the X-pump solution (6.6).Is is clear from figure 11 that the sole considerations of q(t) and p(t) represented in figure 10 cannot explain the local behaviour of η quartet (0, t) and that phase mixing must be invoked to explain such destructive/constructive interferences (the π/2 relative phase jump is also worth noting).
Whether this kind of behaviour may be related or not to the occurrence of extreme events is left for future investigation.Finally, if ε = ε 1 or ε = ε 2 , then A = 4T and a = 0, so that f in (2.18) becomes cubic and reads as (B1), where here b = −2A( B)q 0 > 0. It may be deduced from the considerations of Appendix B in that case that if we constrain the positive roots to coincide, i.e. ξ 23 ≡ ξ 2 = ξ 3 = q 0 , then we get the following compatibility conditions: For both ε = ε 1 and ε = ε 2 we have the same numerical value for B/A ≈ −0.8487; both conditions above are fulfilled and the corresponding solution is breather (B4).
7.2.Looking for rational breathers Since a < 0 when ε = 2 lies in the range ε 1 < ε < ε 2 , we expect to find rational breathers (4.25).Suppose that the triple root which characterizes this solution is positive, ξ 2 = ξ 3 = ξ 4 ≡ ξ 234 > 0, so that ξ 1 ≤ 0. Now, contrary to the case a > 0 for which the multiple roots must coincide (in absolute value) with the initial wave actions because of (4.1), roots satisfy in the present case the following inequalities: (7.8)However, no rational breather has been found for the resonant trident (7.1).Thus, constraints imposed by the coincidence of three roots of the quartic function f make the search for rational breathers even more difficult than for other breathers.For reasons that will be made clear later, still for the quartet (7.1), we consider the point in parameter space where A = 12T.It is found numerically that it corresponds to ε = ε 0 , where ε 0 ≈ 0.7475.Now, from this point of bifurcation, we consider the following family of non-resonant quartet: k 0 κ y ), (7.9)where the wave vectors k ε i are defined in (7.1) and where (κ x , κ y ) are real numbers.Now, if we impose on the roots in (7.8) the constraints ξ 1 = 0 and ξ 234 = q 0 > 0, and if we assume that A = 12T and cos p 0 = −1, then it may be deduced from (A2) that q 0 must satisfy the following compatibility condition: (7.10)where all parameters are to be evaluated on the non-resonant manifold (7.9).In figure 12 is represented in the (κ x , κ y ) plane the curve (Γ ) defined by A = 12T that bifurcates from the origin O corresponding to κ = 0.This curve passes through points P 1 , P 2 and P 3 and forms a loop which crosses light-grey regions in which condition (7.10) is violated, so that rational breathers of the kind considered here exist on any point of (Γ ), except in these shadowed regions.Truncated values of a = −128T 2 (since A = 12T) and q 0 in (7.10) are reported in table 2. Since ξ 1 = 0 and ξ 234 = q 0 , the corresponding rational breathers are (4.26).The values of the frequency mismatch show that for | ω|/(k 3 0 q 0 ) to stay of order 1, one must stay on curve (Γ ) between O and approximately P 1 to keep physical insight.
To summarize, the nature of a given non-degenerate wave quartet with wave vectors k i satisfying (1.3) and prescribed initial amplitudes b i (0) = 0, i = 1, . . ., 4, may be determined through the following steps: (a) Compute the initial wave actions q i = |b i (0)| 2 , the initial phase mismatch p 0 from (2.12), the linear frequencies ω i = ω(k i ) from the dispersion law of the medium under consideration and the frequency mismatch ω from (1.4).
(the X-beats for instance) in a way similar to the Fermi-Pasta-Ulam recurrence for the modulational instability (Yuen & Lake 1982;Shemer & Stiassnie 1985, 1991;Trillo & Wabnitz 1991;Janssen 2004;Leblanc 2009).This behaviour is also illustrated in the phase portraits in figure 2 of Andrade & Stuhlmeier (2023a): in this figure, the dots correspond to steady solutions and are connected by a heteroclinic orbit in phase space which corresponds to a breather; one sees that any departures from these saddle points or from heteroclinic orbits bifurcate to closed orbits corresponding to periodic large-amplitude solutions.Similar observations were made by Cappellini & Trillo (1991) and Andrade & Stuhlmeier (2023b) for the three-wave cubic interaction.What was surprising to me is that stability or instability of non-degenerate quartets has to be understood as structural since no external disturbance is necessary for a solution to be unstable, in contrast for instance to the decay instability or to the modulational instability of Stokes wave or bichromatic wavetrains, where disturbances assimilated to interacting waves with initially infinitesimal amplitudes have to be superimposed to the initial states.Here, in the unstable case, any small departure from the initial conditions leads to a solution that diverges initially from the steady state.
I conclude this paper with possible related issues: (a) Compute the response of the Zakharov equation ( 5.1) to initial data of the form B(k, 0) = 4 i=1 b i (0)δ(k − k i ) and compare with solutions of the truncated model (1.1), on or off resonance.Alternatively, is the Stokes wave the unique exact solution of (5.1) in the form B(k, t) = N i=1 b i (t)δ(k − k i ), N ≥ 1? (b) Can isolated quartet models such as the bidirectional standing wave (5.9) and the X-pump (6.6), the Ψ -breather or their bifurcated periodic states be reproduced numerically or experimentally, as for steady states (Liu et al. 2015)?(c) Is there a connection between the large variations observed in figure 11 and the occurrence of rogue waves (Onorato et al. 2013)?(d) Would instabilities grow super-exponentially if a linear forcing term were added to (1.1) or (1.6), as predicted from cubic forced nonlinear Schrödinger and Davey-Stewartson equations (Leblanc 2007(Leblanc , 2008)?

Table 1 .
Truncated values of the roots ξ i of the quartic function (4.6) with a = −25.57corresponding to the parameters of the unsteady periodic solutions represented in figure

Table 2 .
. Coordinates and values of a, q 0 and | ω| for P 1 , P 2 and P 3 of figure 12.