Periodic solitons for the elliptic-elliptic focussing Davey-Stewartson equations
PPeriodic solitons for the elliptic-ellipticfocussing Davey-Stewartson equations
Mark D. Groves a , b , Shu-Ming Sun c , Erik Wahl´en d a FR 6.1 - Mathematik, Universit¨at des Saarlandes, Postfach 151150, 66041 Saarbr¨ucken, Germany b Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK c Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA d Department of Mathematics, Lund University, 22100 Lund, Sweden
Received *****; accepted after revision +++++Presented by *****
Abstract
We consider the elliptic-elliptic, focussing Davey-Stewartson equations, which have an explicit bright line solitonsolution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinalspatial direction and are periodic in the transverse spatial direction, is established using dynamical systemsarguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transversedirection.
R´esum´eSolitons p´eriodiques du syst`eme de Davey-Stewartson elliptique-elliptique focalisant.
Nous consid´eronsles ´equations de Davey-Stewartson focalisantes dans le cas elliptique-elliptique, lorsqu’elles poss`edent une solu-tion unidimensionnelle de type soliton. En utilisant des m´ethodes de la th´eorie des syst`emes dynamiques, nousmontrons l’existence d’une famille de solutions bidimensionnelles qui ont le profil d’un soliton dans la directionspatiale longitudinale et sont p´eriodiques dans la direction spatiale transverse. Nous montrons ´egalement que lesoliton unidimensionnel est lin´eairement instable vis-`a-vis des perturbations transverses.
1. Introduction
The Davey-Stewartson equationsi A t + εA xx + A yy + ( γ | A | + γ φ x ) A = 0 , (1) γ φ xx + φ yy − γ | A | x = 0 , (2)where ε = ± γ , γ , γ ∈ R \{ } with γ + γ = ±
2, arise in the modelling of wave packets on the surfaceof a three-dimensional body of water; the variables A = A ( x, y, t ) and φ = φ ( x, y, t ) are the complex waveamplitude and real mean flow and the signs of the parameters depend upon the particular physicalregime under consideration (see Ablowitz & Segur [2, § γ + γ = 2and γ + γ = − focussing and defocussing , and the system is classified ashyperbolic-hyperbolic, hyperbolic-elliptic, elliptic-hyperbolic or elliptic-elliptic according to the signs of ε and γ . Certain special cases of the mixed-type systems are often referred to as DS-I and DS-II and areknown to be completely integrable (see Ablowitz & Clarkson [1, p. 60]). Note that solutions of (1), (2)which are spatially homogeneous in the y -direction satisfy the cubic nonlinear Schr¨odinger equationi A t + A xx + ( γ + γ ) | A | A = 0 (3)(where φ is recovered from (2)). Email addresses: [email protected] (Mark D. Groves), [email protected] (Shu-Ming Sun), [email protected] (Erik Wahl´en).
Preprint submitted to the Acad´emie des sciences March 27, 2018 a r X i v : . [ m a t h . A P ] S e p olutions of (3) which converge to an equilibrium as x → ±∞ and are 2 π -periodic in t are referredto as line solitons . In the defocussing case the equation admits a ‘dark’ line soliton which decays to anontrivial equilibrium, while in the focussing case it has the ‘bright’ line soliton A (cid:63) ( x, t ) = e i t sech( x ) (4)which satisfies A (cid:63) ( x, t ) → x → ±∞ . In this note we examine periodic solitons which decay as x → ±∞ and are periodic in y and t , and in particular consider how they emerge from line solitonsin a dimension-breaking bifurcation . Explicit formulae for dark periodic solitons have been obtained forthe integrable versions of the equations by Watanabe & Tajiri [9] and Arai, Takeuchi & Tajiri [3]; herewe establish the existence of bright periodic solitons to the elliptic-elliptic, focussing equations ( ε = 1, γ + γ = 2, γ >
0) under the additional condition γ > Theorem 1.1
Suppose that ε = 1 , γ + γ = 2 and γ , γ > . There exist an open neighbourhood N ofthe origin in R , a positive real number ω and a family of periodic solitons { e i t u s ( x, y ) , φ s ( x, y ) } s ∈N to(1), (2) which emerges from the bright line soliton in a dimension-breaking bifurcation. Here u s ( x, y ) = sech( x ) + u (cid:48) s ( x, y ) , φ s ( x, y ) = tanh( x ) + φ (cid:48) s ( x, y ) , in which u (cid:48) s ( · , · ) , φ (cid:48) s ( · , · ) are real, have amplitude O ( | s | ) and are even in both arguments and periodic intheir second with frequency ω + O ( | s | ) . We also present a corollary to this result which asserts that the bright line soliton is transversely linearlyunstable and thus confirms the prediction made by Ablowitz & Segur [2, § Theorem 1.2
Suppose that ε = 1 , γ + γ = 2 and γ , γ > . For each sufficiently small positive valueof λ the linearisation of (1), (2) at A (cid:63) ( x, t ) = e i t sech( x ) , φ (cid:63) ( x, y ) = tanh( x ) has a solution of the form e λt +i t ( A ( x, y ) , φ ( x, y )) , where ( A ( x, y ) , φ ( x, y )) is periodic in y and satisfies ( A ( x, y ) , φ ( x, y )) → (0 , as x → ±∞ . In the remainder of this article we suppose that ε = 1, γ + γ = 2 and γ , γ >
0. Equations (1), (2)with these coefficients arise when modelling water waves with weak surface tension. The existence andtransverse linear instability of periodic solitons for the water-wave problem in this physical regime hasrecently been established by Groves, Sun & Wahl´en [6].
2. Spatial dynamics
The equations for solutions of (1), (2) for which A ( x, y, t ) = e i t (cid:0) u ( x, y, t ) + i u ( x, y, t ) (cid:1) (and u , u arereal-valued) can be formulated as the evolutionary system u y = v , (5) v y = u t − u xx + u − ( γ u + γ u + γ φ x ) u , (6) u y = v , (7) v y = − u t − u xx + u − ( γ u + γ u + γ φ x ) u , (8) φ y = ψ, (9) ψ y = − γ φ xx + γ ( u + u ) x , (10)where the spatial direction y plays the role of time. To identify an appropriate functional-analytic settingfor these equations, let us first specialise to stationary solutions, so that2 y = v , (11) v y = − u xx + u − ( γ u + γ u + γ φ x ) u , (12) u y = v , (13) v y = − u xx + u − ( γ u + γ u + γ φ x ) u , (14) φ y = ψ, (15) ψ y = − γ φ xx + γ ( u + u ) x . (16)Equations (11)–(16) constitute a semilinear evolutionary system in the phase space X = H ( R ) × L ( R ) × H ( R ) × L ( R ) × H ( R ) × L ( R ); the domain of the linear part of the vector field definedby their right-hand side is D = H ( R ) × H ( R ) × H ( R ) × H ( R ) × H ( R ) × H ( R ). This evolution-ary system is reversible, that is invariant under y (cid:55)→ − y , ( u , v , u , v , φ, ψ ) (cid:55)→ S ( u , v , u , v , φ, ψ ),where the reverser S : X → X is defined by S ( u , v , u , v , φ, ψ ) = ( u , − v , u , − v , φ, − ψ ). It isalso invariant under the reflection R : X → X given by R ( u ( x ) , v ( x ) , u ( x ) , v ( x ) , φ ( x ) , ψ ( x )) =( u ( − x ) , v ( − x ) , u ( − x ) , v ( − x ) , − φ ( − x ) , − ψ ( − x )), and one may seek solutions which are invariant un-der this symmetry by replacing X and D by respectively X r := X ∩ Fix R = H ( R ) × L ( R ) × H ( R ) × L ( R ) × H ( R ) × L ( R )and D r := D ∩ Fix R = H ( R ) × H ( R ) × H ( R ) × H ( R ) × H ( R ) × H ( R ) , where H n e ( R ) = { w ∈ H n ( R ) : w ( x ) = w ( − x ) for all x ∈ R } ,H n o ( R ) = { w ∈ H n ( R ) : w ( x ) = − w ( − x ) for all x ∈ R } . It is also possible to replace D r by the extended function space D (cid:63) := H ( R ) × H ( R ) × H ( R ) × H ( R ) × H (cid:63), o ( R ) × H ( R ) , where H (cid:63), o ( R ) = { w ∈ L ( R ) : w x ∈ H ( R ) , w ( x ) = − w ( − x ) for all x ∈ R } (a Banach space with norm (cid:107) w (cid:107) (cid:63), := (cid:107) w x (cid:107) ). This feature allows one to consider solutions to (11)–(16)whose φ -component is not evanescent; in particular solutions corresponding to line solitons fall into thiscategory (see below).Each point in phase space corresponds to a function on the real line which decays as x → ∞ , andthe dynamics of equations (11)–(16) in y describes the behaviour of their solutions in the y -direction. Inparticular, equilibria correspond to line solitons (the equilibrium( u (cid:63) ( x ) , v (cid:63) ( x ) , u (cid:63) ( x ) , v (cid:63) ( x ) , φ (cid:63) ( x ) , ψ (cid:63) ( x )) = (sech( x ) , , , , tanh( x ) , u , v , u , v , φ, ψ ) = ( u (cid:63) , v (cid:63) , φ (cid:63) , u (cid:63) , v (cid:63) , φ (cid:63) , ψ (cid:63) ) + ( u (cid:48) , v (cid:48) , u (cid:48) , v (cid:48) , φ (cid:48) , ψ (cid:48) ) (17)and seeking small-amplitude periodic solutions of the resulting evolutionary system w y = Lw + N ( w ) (18)for w = ( u (cid:48) , v (cid:48) , u (cid:48) , v (cid:48) , φ (cid:48) , ψ (cid:48) ), where 3 yx y Figure 1. A family of periodic solutions surrounding a nontrivial equilibrium solution to (11)–(16) in its phase space (left)corresponds to a dimension-breaking bifurcation of a branch of periodic solitons from a line soliton (right, plot of u ( x, y )). L u v u v φψ = v − u xx + u − (3 γ + γ ) sech ( x ) u − γ sech( x ) φ x v − u xx + u − ( x ) u ψ − γ φ xx + 2 γ (sech( x ) u ) x ,N u v u v φψ = − γ sech( x ) u − γ sech( x ) u − γ u φ x − γ u − γ u u − γ sech( x ) u u − γ u φ x − γ u − γ u u γ ( u + u ) x and we have dropped the primes for notational simplicity. Note that (18) has the invariant subspace˜ X = { ( u , v ) = (0 , } , and we define ˜ X r = X r ∩ ˜ X , ˜ D r = D r ∩ ˜ X and ˜ D (cid:63) = D (cid:63) ∩ ˜ X .Returning to (5)–(10), observe that these equations constitute a reversible evolutionary equation withphase space H (( − t , t ) , X ); the domain of its vector field is H (( − t , t ) , X ) ∩ H (( − t , t ) , D ) and itsreverser is given by the pointwise extension of S : X → X to H (( − t , t ) , X ). Seeking solutions of theform (17), we find that w y = T w t + Lw + N ( w ) , where T ( u , v , u , v , φ, ψ ) = (0 , u , , − u , ,
0) and we have again dropped the primes. In Section 5we demonstrate that the solution ( u (cid:63) , v (cid:63) , u (cid:63) , v (cid:63) , φ (cid:63) , ψ (cid:63) ) of (5)–(10) is transversely linearly unstable byconstructing a solution of the linear equation w y = T w t + Lw (19)of the form e λt u λ ( y ), where u λ ∈ C ( R , X ) ∩ C b ( R , D ) is periodic, for each sufficiently small positivevalue of λ . 4 . Spectral theory In this section we determine the purely imaginary spectrum of the linear operator L : D ⊆ X → X . Tothis end we study the resolvent equations ( L − i kI ) w = w † (20)for L , where w = ( u , v , u , v , φ, ψ ), w † = ( u † , v † , u † , v † , φ † , ψ † ) and k ∈ R \ { } ; since L is real andanticommutes with the reverser S it suffices to examine non-negative values of k , real values of u , u , φ , v † , v † , ψ † and purely imaginary values of u † , u † , φ † , v , v , ψ . Observe that (20) is equivalent to thedecoupled equations( A + k I ) u φ = v † + i ku † ψ † + i kφ † , ( A + k I ) u = v † + i ku † , where A : H ( R ) × H ( R ) ⊆ L ( R ) × L ( R ) → L ( R ) × L ( R ) and A : H ( R ) ⊆ L ( R ) → L ( R ) aredefined by A u φ = − u xx + u − (3 γ + γ ) sech ( x ) u − γ sech( x ) φ x − γ φ xx + 2 γ (sech( x ) u ) x , A u = − u xx + u − ( x ) u ;the values of v , v and ψ are recovered from the formulae v = u † + i ku , v = u † + i ku , ψ = φ † + i kφ. It follows that L − i kI is (semi-)Fredholm if A + k I and A + k I are (semi-)Fredholm and the dimensionof the (generalised) kernel of L − i kI is the sum of those of A + k I and A + k I .Lemmata 3.1 and 3.2 below record the spectra of A and A ; part (i) of the following proposition (seeDrazin [4, Chapter 4.11]) is used in the proof of the former while the latter follows directly from part (ii). Proposition 3.1 (i) The spectrum of the self-adjoint operator − ∂ x − ( x ) : H ( R ) ⊆ L ( R ) → L ( R ) consists ofessential spectrum [1 , ∞ ) and two simple eigenvalues at − and (with corresponding eigenvectors sech ( x ) and sech (cid:48) ( x ) ).(ii) The spectrum of the self-adjoint operator − ∂ x − ( x ) : H ( R ) ⊆ L ( R ) → L ( R ) consists ofessential spectrum [1 , ∞ ) and a simple eigenvalue at (with corresponding eigenvector sech( x ) ). Lemma 3.1
The spectrum of the operator A consists of essential spectrum [0 , ∞ ) and an algebraicallysimple negative eigenvalue − ω whose eigenspace lies in L ( R ) × L ( R ) . Proof.
First note that A is a compact perturbation of the constant-coefficient operator H ( R ) × H ( R ) ⊆ L ( R ) × L ( R ) → L ( R ) × L ( R ) defined by( u , φ ) (cid:55)→ ( − u xx + u , − γ φ xx ) , whose essential spectrum is clearly [0 , ∞ ); it follows that σ ess ( A ) = [0 , ∞ ) (see Kato, [8, ChapterIV, Theorem 5.26]). Because A is self-adjoint with respect to the inner product (cid:104) ( u , φ ) , ( u , φ ) (cid:105) = (cid:104) u , u (cid:105) + γ γ − (cid:104) φ , φ (cid:105) for L ( R ) × L ( R ) the remainder of its spectrum consists of negative realeigenvalues with finite multiplicity.One finds by an explicit calculation that (cid:104)A ( u , φ ) , ( u , φ ) (cid:105) = (cid:104) u − u xx − ( x ) u , u (cid:105) + γ (cid:90) R ( φ x − x ) u ) d x, u , φ ) ∈ H ( R ) × H ( R ) with (cid:104) ( u , φ ) , (sech ( x ) , (cid:105) = 0(see Proposition 3.1(ii)). It follows that any subspace of H ( R ) × H ( R ) upon which A is strictly negativedefinite is one-dimensional. The calculationlim R →∞ (cid:104)A (sech( x ) , φ R ( x ) tanh( x )) , (sech( x ) , φ R ( x ) tanh( x )) (cid:105) = − , where φ ( R ) = χ ( x/R ) and χ ∈ C ∞ ( R ) is a smooth cut-off function equal to unity in [ − , σ ( A ) <
0, so that the spectral subspace of H ( R ) × H ( R ) corresponding to the part of the spectrumof A in ( −∞ , − ε ) is nontrivial and hence one-dimensional for every sufficiently small value of ε >
0. Weconclude that A has precisely one simple negative eigenvalue − ω .Finally, the same argument shows that A | L ( R ) × L ( R ) also has precisely one simple negative eigen-value. It follows that this eigenvalue is − ω , whose eigenspace therefore lies in L ( R ) × L ( R ). (cid:50) Lemma 3.2
The spectrum of the operator A consists of essential spectrum [1 , ∞ ) and an algebraicallysimple negative eigenvalue at whose eigenspace lies in L ( R ) . Corollary 3.3
The purely imaginary number i k belongs to the resolvent set of L for k ∈ R \ { , ± ω } and ± i ω are algebraically simple purely imaginary eigenvalues of L whose eigenspace lies in ˜ X r .
4. Application of the Lyapunov-Iooss theorem
Our existence theory for periodic solitons is based upon an application of the following version of theLyapunov centre theorem for reversible systems (see Iooss [7]) which allows for a violation of the classicalnonresonance condition at the origin due to the presence of essential spectrum there (a feature typical ofspatial dynamics formulations for problems in unbounded domains) provided that the ‘Iooss condition atthe origin’ (hypothesis (viii)) is satisfied.
Theorem 4.1 (Iooss-Lyapunov)
Consider the differential equation w τ = L ( w ) + N ( w ) , (21) in which w ( τ ) belongs to a real Banach space X . Suppose that Y , Z are further real Banach spaces withthe properties that(i) Z is continuously embedded in Y and continuously and densely embedded in X ,(ii) L : Z ⊆ X → X is a closed linear operator,(iii) there is an open neighbourhood U of the origin in Y such that L ∈ L ( Y , X ) and N ∈ C , u ( U , X ) (and hence N ∈ C , u ( U ∩ Z , X ) ) with N (0) = 0 , d N [0] = 0 .Suppose further that(iv) equation (21) is reversible: there exists an involution S ∈ L ( X ) ∩ L ( Y ) ∩ L ( Z ) with SLw = − LSw and SN ( w ) = − N ( Sw ) for all w ∈ U ,and that the following spectral hypotheses are satisfied.(v) ± i ω are nonzero simple eigenvalues of L ;(vi) i nω ∈ ρ ( L ) for n ∈ Z \{− , , } ;(vii) (cid:107) ( L − i nω I ) − (cid:107) X →X = o (1) and (cid:107) ( L − i nω I ) − (cid:107) X →Z = O (1) as n → ±∞ ;(viii) For each w † ∈ U the equation Lw = − N ( w † ) has a unique solution w ∈ Y and the mapping w † (cid:55)→ w belongs to C , u ( U , Y ) . nder these hypotheses there exist an open neighbourhood I of the origin in R and acontinuously differentiable branch { ( v ( s ) , ω ( s )) } s ∈ I of reversible, π/ω ( s ) -periodic solutions in C ( R , Y ⊕ X ) ∩ C per ( R , Y ⊕ Z ) to (21) with amplitude O ( | s | ) . Here the direct sum refers to the decom-position of a function into its mode and higher-order Fourier components, the subscript ‘per’ indicatesa π/ω ( s ) -periodic function and ω ( s ) = ω + O ( | s | ) . Theorem 1.1 is proved by applying the Iooss-Lyapunov theorem to (18), taking X = ˜ X r , Y = ˜ D (cid:63) , Z =˜ D r and U = ˜ D (cid:63) (and of course τ = y and S ( u , v , φ, ψ ) = ( u , − v , φ, − ψ )). The spectral hypotheses(v) and (vi) follow from Corollary 3.3, while (vi) and (vii) are verified in respectively Lemma 4.2 and 4.3below. Lemma 4.2
The operator L satisfies the resolvent estimates (cid:107) ( L − i kI ) − (cid:107) ˜ X r → ˜ X r = O ( | k | − ) and (cid:107) ( L − i kI ) − (cid:107) ˜ X r → ˜ D r = O (1) as | k | → ∞ . Proof.
Notice that L ( u , v , φ, ψ ) = ( B ( u , v ) , B ( φ, ψ )) + C ( u , v , φ, ψ ) , where B ( u , v ) = ( v , − u xx + u ), B ( φ, ψ ) = ( ψ, − γ φ xx + γ φ ) and C ( u , v , φ, ψ ) = (0 , − (3 γ + γ ) sech ( x ) u − γ sech( x ) φ x , , − γ φ + 2 γ (sech( x ) u ) x ) . Writing ˜ X r = ˜ X × ˜ X , ˜ D r = ˜ Y × ˜ Y and equipping ˜ X with the usual inner product and ˜ X withthe inner product (cid:104) ( φ , ψ ) , ( φ , ψ ) (cid:105) = (cid:104) φ , φ (cid:105) + γ − (cid:104) ψ , ψ (cid:105) , observe that B j : ˜ Y j ⊂ ˜ X j → ˜ X j isself-adjoint, so that (cid:107) ( B j − i kI ) − (cid:107) ˜ X j → ˜ X j ≤ | k | − for k (cid:54) = 0. Furthermore (cid:107)B j ( · ) (cid:107) ˜ X j = (cid:107) · (cid:107) ˜ Y j , so that (cid:107) ( B j − i kI ) − (cid:107) ˜ X j → ˜ Y j = (cid:107)B j ( B j − i kI ) − (cid:107) ˜ X j → ˜ X j = (cid:107) I + i kI ( B j − i kI ) − (cid:107) ˜ X j → ˜ X j ≤ k (cid:54) = 0. It follows that B = B × B : ˜ D r ⊆ ˜ X r → ˜ X r satisfies the estimates (cid:107) ( B − i kI ) − (cid:107) ˜ X r → ˜ X r ≤ | k | − , (cid:107) ( B − i kI ) − (cid:107) ˜ X r → ˜ D r ≤ k (cid:54) = 0.Finally, note that C : ˜ X r → ˜ X r is bounded, whence (cid:107)C ( B − i kI ) − (cid:107) ˜ X r → ˜ X r = O ( | k | − )as | k | → ∞ . Consequently I − C ( B − i kI ) − : ˜ X r → ˜ X r is invertible for sufficiently large values of | k | with (cid:107) ( I − C ( B − i kI ) − ) − (cid:107) ˜ X r → ˜ X r = O (1) (23)as | k | → ∞ , and the stated result follows from the identity( L − i kI ) − = ( B − i kI ) − (cid:0) I − C ( B − i kI ) − (cid:1) − and the estimates (22), (23). (cid:50) Lemma 4.3
The equation Lw = − N ( w † ) (24) has a unique solution w ∈ ˜ D (cid:63) for each w † ∈ ˜ D (cid:63) and the formula w † (cid:55)→ w defines a smooth mapping ˜ D (cid:63) → ˜ D (cid:63) . Proof.
Equation (24) is equivalent to the equations u − u xx − ( x ) u = f ( w † ) , where f ( w † ) = (3 γ + γ ) sech( x )( u † ) + γ u † φ † x + γ ( u † ) , and φ x = ( u † ) + 2 sech( x ) u , v = 0 , ψ = 0 . The result thus follows from Proposition 3.1(i) and the fact that f and ( u , u † ) (cid:55)→ ( u † ) + 2 sech( x ) u define smooth mappings ˜ D (cid:63) → L ( R ) and H ( R ) × H ( R ) → H ( R ). (cid:50) . Transverse linear instability Finally, we demonstrate the transverse linear instability of the line soliton using the following generalresult due to Godey [5].
Theorem 5.1 (Godey)
Consider the differential equation v τ = T v t + Lv, (25) in which v ( τ, t ) belongs to a real Banach space X . Suppose that Y , Z are further real Banach spaces withthe properties that(i) L : Z ⊆ X → X and T : Y ⊆ X → X are closed linear operators with
Z ⊆ Y ,(ii) the equation is reversible: there exists an involution S ∈ L ( X ) ∩ L ( Y ) ∩ L ( Z ) with LSv = − SLv and
T Sv = − ST v for all v ∈ Z ,(iii) L has a pair ± i ω of isolated purely imaginary eigenvalues with odd algebraic multiplicity.Under these hypotheses equation (25) has a solution of the form e λt v λ ( τ ) , where v λ ∈ C ( R , X ) ∩ C ( R , Z ) is periodic, for each sufficiently small positive value of λ ; its period tends to π/ω as λ → . Theorem 1.2 is proved by applying Godey’s theorem to (19), taking X = X , Y = X and Z = D (and ofcourse τ = y and S ( u , v , u , v , φ, ψ ) = ( u , − v , u , − v , φ, − ψ )). The spectral hypothesis (iii) followsfrom Corollary 3.3. Acknowledgements
E. W. was supported by the Swedish Research Council (grant no. 621-2012-3753).
References [1] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London MathematicalSociety Lecture Note Series 149, Cambridge University Press, 1991.[2] M. J. Ablowitz, H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979) 691–715.[3] T. Arai, K. Takeuchi, M. Tajiri, Note on periodic soliton resonance: Solutions to the Davey-Stewartson II equation, J.Phys. Soc. Japan 70 (2001) 55–59.[4] P. G. Drazin, Solitons, London Mathematical Society Lecture Note Series 85, Cambridge University Press, 1983.[5] C. Godey, A simple criterion for transverse linear instability of nonlinear waves, C. R. Acad. Sci. Paris, Ser. I 354 (2016)175–179.[6] M. D. Groves, S. M. Sun, E. Wahl´en, A dimension-breaking phenomenon for water waves with weak surface tension,Arch. Rational Mech. Anal. 220 (2016) 747–807.[7] G. Iooss, Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinitedepth, J. Math. Fluid Mech. 1 (1999) 24–63.[8] T. Kato, Perturbation Theory for Linear Operators, 2nd edn, Springer-Verlag, New York, 1976.[9] Y. Watanabe, M. Tajiri, Periodic soliton resonance: Solutions to the Davey-Stewartson I equation, J. Phys. Soc. Japan67 (1998) 705–708.[1] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London MathematicalSociety Lecture Note Series 149, Cambridge University Press, 1991.[2] M. J. Ablowitz, H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979) 691–715.[3] T. Arai, K. Takeuchi, M. Tajiri, Note on periodic soliton resonance: Solutions to the Davey-Stewartson II equation, J.Phys. Soc. Japan 70 (2001) 55–59.[4] P. G. Drazin, Solitons, London Mathematical Society Lecture Note Series 85, Cambridge University Press, 1983.[5] C. Godey, A simple criterion for transverse linear instability of nonlinear waves, C. R. Acad. Sci. Paris, Ser. I 354 (2016)175–179.[6] M. D. Groves, S. M. Sun, E. Wahl´en, A dimension-breaking phenomenon for water waves with weak surface tension,Arch. Rational Mech. Anal. 220 (2016) 747–807.[7] G. Iooss, Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinitedepth, J. Math. Fluid Mech. 1 (1999) 24–63.[8] T. Kato, Perturbation Theory for Linear Operators, 2nd edn, Springer-Verlag, New York, 1976.[9] Y. Watanabe, M. Tajiri, Periodic soliton resonance: Solutions to the Davey-Stewartson I equation, J. Phys. Soc. Japan67 (1998) 705–708.