Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

Stabilit y of the Blo c h w all via the Bogomoln yi decomp osition in elliptic co ordinates S.R. W oo dford ∗ and I.V. Barashenk ov † Dep artment of Mathematics and Applie d Mathematics, University of Cap e T own, R onde b osch 7701, South Afric a (Dated: No vem b er 19, 2018) Abstract W e consider the one-dimensional anisotropic X Y mo del in the co n tin uum limit. Sta bilit y analysis of its Bloch wall solution is hindered by the n on d iagonalit y of the asso ciated linearised op erator and the hessian of energy . W e circumv en t this d ifficult y b y sho wing that the energy admits a Bogomol nyi b ound in elliptic co ordinates and that the Bloch wall satur ates it — that is, the Blo c h w all renders the ener gy minim um. Our analysis provi des a simp le but nontrivia l application of the BPS (Bogomoln yi - Pr asad - Sommerfield) construction in one dimension, wh ere its use is often b eliev ed to b e limited to repr o ducing results obtainable by other means. ∗ On leave of absence from F or sch ungs zentrum J¨ ulich. Postal address : Theorie I, Institut f¨ ur F estk¨ orp erfors ch ung, F orsch ungszentrum J ¨ ulic h, D-5242 8 J¨ ulic h, German y; Electronic address: s.woo dford@fz- juelic h.de † Electronic addres s: Igor.Bar ashenko v@uct.ac.za; igor@o dette.mth.uct.ac.za 1 I. INTR ODUCTION Although the Bogomo lnyi decomp osition is an indisp ensable to o l for the study of higher- dimensional field theories [1], it has seldom b een used in one dimension. It do es allow one to reduce the order and find top o logical solitons of one- comp onen t mo dels suc h as the sine-Gordon and φ 4 -theory , but these can b e obtained with less effort simply b y using an in tegrating factor. It is of more v alue for m ulticomp onen t [2] and lat t ice [3] systems, but all mo dels that b enefited fro m its use had to be sp ecifically designed to admit s uc h a decomp osition. In this pap er, w e apply the Bogomolnyi construction to a system t hat has b een studied for mo r e than fo r t y y ears: the a nisotropic X Y m o del. It has long b een believ ed that the Blo c h-w all solutions of this mo del a re stable; this fact ha s b een demonstrated numerically but nev er prov en analytically . Here, b y transforming to elliptic co ordinates, we find the Bogomoln yi b ound for the energy and sho w that the Blo ch w all minimizes the energy in the corresponding top o logical sec tor; this prov es its s tabilit y . (Note that the Bogo moln yi construction cannot b e carried o ut in t he original co ordinates.) An outline of the pap er is as fo llo ws. In the next section w e in tro duce the (station- ary) anisotropic X Y mo del and its domain-wall solutions. W e also describe three differen t time-dep enden t extensions of the mo del whic h corresp ond to three different types of b e- ha viour, viz . , dissipativ e, conse rv ativ e relativistic and conserv ative nonrelativistic dynamics. In section I I I w e s ho w that because of the nondiagonalit y of the linearised opera t o r and the second v ariation of energy , the standar d metho ds are inadequate for proving the s tabilit y of the Blo c h wall — in eac h of the three cases. Section IV con ta ins the main result of this pa- p er: the pro o f of the energy minimization b y the Blo ch w all. F ina lly , section V summarises the results o f our w ork. I I. THE MO DEL A. Anisotropic easy-axis f e rromagnet near t he Curie temp erature The a nisotropic X Y model w as originally in tro duced to describe domain walls in an easy- axis ferromagnet near the Curie p oin t [4, 5 ]. In the study of domain walls, the magnetization is assumed to v ary only in x direction and the ma g netization v ector M is tak en to lie in the 2 y z plane. (With this notation, the system should ha v e b een c alled the Y Z mo del but w e k eep the traditional na me to a v oid confusion.) In the con tin uum limit, the mo del is defined b y its free e nergy expansion [4, 5 ]: E = Z  1 2 ( ∂ x M ) 2 − (1 + h ) M 2 + 1 2 M 4 + 2 hM 2 y + E 0  dx. (1) The first term in (1) is the exc ha ng e energy , whic h is minimized when M = const . The com binat io n of the second and third terms mak es the | M | 6 = 0 ground state energetically preferable to the state with | M | = 0. Anisotrop y is caused b y the term 2 hM 2 y ; the parameter h is assumed p ositiv e, making the z direction the easy-axis. Finally , the co efficien t (1 + h ) is in tro duced to simplify the subse quen t f o rm ulas and a constant E 0 has b een added to the in t egr a nd to ensure that the free energy is finite. Defining ψ = M y + iM z , w e can rec ast the free energy in the f orm E = Z  1 2 | ψ x | 2 + 1 2 | ψ | 4 − | ψ | 2 + h 2 ( ψ 2 + ψ ∗ 2 ) + E 0  dx. (2) This is the form that w e shall b e w orking with in this pap er. B. Anisotropic easy-plane f e rromagnet in externa l field The same f r ee energy ex pansion (2) describ es one more, unrelated, magnetic system : a w eakly anisotropic e asy-plane ferromagnet in an external field p erp endicular to the easy plane [6 ]. In terms of the magnetization v ector M , the free energy of this system is giv en b y E = Z  1 2 ( ∂ x M ) 2 + β 2 M 2 z + ǫβ 2 M 2 x − H M z + E 0  dx, (3) where H > 0 is an applied field, the a nisotrop y parameters β and ǫβ are p o sitiv e, and ǫ ≪ 1. In this mo del, the xy plane is the easy plane, with a w eak anisotrop y fa vouring the y direction. F ar from the Curie p oint, the magnetization is ha r d, i.e. the magnitude of the magnetisation v ector is constan t: M 2 = M 2 0 = const. [F or this reason equation ( 3) has no terms in M 2 and M 4 .] W e assume that the the magnitude of the external magnetic field is close to β M 0 : H = β M 0 − ǫq , where q is a quan tit y of o r der 1 . The (degenerate) g r o und states will then ha v e the magnetization v ector almost parallel to the field: M ( ± ) ≈ (0 , ± p 2 ǫq M 0 /β , M 0 − ǫq /β ) . 3 The deviation from the reference vec tor M = (0 , 0 , M 0 ) p ointing in the direction of the field can b e characterise d using the complex v ariable ψ = ( β / 2 ǫs ) 1 / 2 ( M x − iM y ), where s ≡ q M 0 − β M 2 0 / 2. The M z comp onen t o f the mag netisation is expressible using M 2 = M 2 0 = const: M z = M 0 s 1 − 2 ǫs β M 2 0 | ψ | 2 . T ransforming to ˜ x = ( ǫs/ 2 α M 2 0 ) 1 / 2 x and k eeping t erms up to ǫ 2 , the free energy (3) reduces to equation (2) where h = β M 2 0 / 2 s and w e hav e dropp ed the tilde ov er x . Unlik e t he anisotropic X Y mo del considered in the previous subsection, the domain w a lls in the presen t system interpolat e b et w een the nearly-parallel ground states M (+) and M ( − ) . It is fitting to note here that due to the relativ e w eakness of magnetic anisotrop y , the field H ≈ β M 0 will generally b e we ll within t he ra ng e of mo dern experiments . F or example, the martensite phase of NiMnGa is a w eakly anisotro pic easy-plane ferromagnet with β ≈ 2 × 10 − 6 Tm/A and M 0 ≈ 5 × 10 5 A/m [7]. Therefore, when sub jected to a magnetic field H ≈ 1T, it will b e described b y equation (2). Finally , we no t e that the expansion (1)-(2) arises in y et another magnetic con text, viz. a strongly anisotropic ferromagnet with spin S = 1 [8]. C. Blo c h and Ising walls The free energy (2) is ex tremised b y solutions o f the following stationa ry equation: 1 2 ψ xx − | ψ | 2 ψ + ψ − hψ ∗ = 0 . (4) There a re tw o solito n, or kink, solutions av ailable in lit era t ure; eac h describes an in terfa ce b et w een t wo ferromagnetic domains. One is commonly kno wn as the Ising, or N ´ e el, w all [4, 5, 9, 10]: ψ I ( x ) = iA tanh( Ax ) , (5) where A = √ 1 + h . The second kink s olution has the for m ψ B ( x ) = iA t anh( B x ) ± C sec h( B x ) , (6) where B = √ 4 h and C = √ 1 − 3 h [5 , 11, 12]. This solution is referred to a s the Bloch w all. The t wo signs in fr o n t of the real part in (6) distinguish Blo c h w alls of opp osite c hiralities. The Ising wall exists for all positiv e h w hereas the Blo c h w all exists o nly for h < 1 3 . 4 The Blo c h and Ising wall ha v e the same asymptotic b eha viour: ψ B ,I ( x ) → ± iA as x → ±∞ . This determines the constan t E 0 that is added to the in tegrand in ( 2 ) to ensure the finiteness of the free energy: E 0 = A 4 / 2. The Blo c h and Ising w alls ha v e energies E B = 2 B ( A 2 − B 2 / 3) (7) and E I = 4 3 A 3 , resp ectiv ely . It is easily v erified that the energy of the Ising wall is greater than the e nergy of the Blo c h w all for all h < 1 3 . In addition to t he Blo c h and Ising w alls, equation (4) has a f amily of nontop ological solitons, av ailable in explicit form and describing Blo ch -Ising b o und states [13]. These will not be considered here. D. Three types of dynamical b e haviour The dynamics of the domain w alls is gov erned by one of three p ossible time-dep enden t extensions of equation (4) . In the case when the free ene rgy (2) is used to mo del the ferromagnet near the Curie p oint, the evolution of the field ψ is dissipativ e and gov erned b y the Ginsburg-Landau equation [1 4 ]: ψ t = ψ − | ψ | 2 ψ + 1 2 ψ xx − hψ ∗ . (8) On the o ther hand, in the case of the ferromagnet with spin S = 1, the field ψ satisfies a relativistically-in v ariant equation [8] ψ tt − ψ xx − 2 ψ + 2 | ψ | 2 ψ + 2 hψ ∗ = 0 . (9) Originally , this equation w as in tro duced b y Montonen [11 ] in a differen t con text — as an exactly s olv able special case of Ra jaraman and W ein b erg’s bag mo del [9]. Indep endently , Sark er, T rullinger and Bishop [12] prop osed it a s an in teresting in terp olate b et wee n the sine-Gordon and the φ 4 theories. Accordingly , t he Klein-Gordon equation (9) is commonly kno wn as the Mon tonen-Sarker-T rullinger-Bishop (MSTB) model. Finally , the magnetisation v ector of an anisotropic easy-plane f erromagnet in an external field satisfies the L a ndau-Lifshitz equation. The p erturbation pro cedure described in section I I B reduces it [6] to the pa r a metrically driv en nonlinear Sc hr¨ odinger equation, iψ t + 1 2 ψ xx − ψ | ψ | 2 + ψ = hψ ∗ . (10) 5 The s tabilit y of the Blo c h or Ising w all dep ends on whic h of the t hr ee equations (8), (9) or (10) go v erns the ev olution of ψ . Consequen tly , the three cases — the Ginsburg-Landau, the Klein-Gordon and the nonlinear Sc hr¨ odinger — need to b e considered separately . I I I. APPR O A C HES TO ST ABILITY In this section w e des crib e the standard methods of stability ana lysis and explain wh y they all fa il in the case of the Blo ch w all — f or eac h of the three t yp es of ev olution. A. The Ginsburg-Landau and relativistic dynamics W e start with the Ginsburg-Landau equation, equation (8), and linearise it ab out the stationary solution ψ 0 ( x ), whic h can be either the Blo c h or Ising w all. Decomp o sing the small p erturbation δ ψ ( x, t ) into its real and imaginary parts, δ ψ ( x, t ) = δ R ( x, t ) + iδ I ( x, t ), and letting δ R ( x, t ) = ˜ δ R ( x ) e λt , δ I ( x, t ) = ˜ δ I ( x ) e λt , yields an eigenv alue problem − H   δ R δ I   = λ   δ R δ I   , (11) where H is a self-adjoin t op erator H =   − 1 2 ∂ 2 x − 1 + h + 3 R 2 0 + I 2 0 2 R 0 I 0 2 R 0 I 0 − 1 2 ∂ 2 x − 1 − h + R 2 0 + 3 I 2 0   . (12) In (12), R 0 and I 0 stand for t he real and imaginary part of the stationary solution ψ 0 , resp ectiv ely . The solution ψ 0 will b e unstable if the op erator −H has at least one p ositiv e eigen v alue λ , and stable otherwis e. In the case of the MSTB m o del [equation (9)], the linearisation a b out ψ 0 pro duces the same eigen v alue problem (11), with the same op erator (12), where one just needs to replace λ with λ 2 / 2. Here w e ha v e the same stabilit y criterion a s in the Ginsburg- L andau case: the solution ψ 0 will b e unstable if −H has at least one p ositiv e eigen v alue λ 2 / 2. An alternativ e appro ac h (whic h still leads to the same criterion, though) is based on considering the Ly a puno v functional (see e.g. [15]). The Ginsburg-Landau equation (8 ) can 6 b e written a s ψ t = − δ E /δ ψ ∗ , w here E is the functional (2). Hence this functional satisfi es E t = − 2 Z | ψ t | 2 dx, and so w e ha v e E t < 0 unless ψ is a static solution, ψ t = 0. Now if w e could prov e that E [ ψ ] > E [ ψ 0 ] for a ll ψ in some neigh b o urho o d of ψ 0 , the functional E w ould b e the Ly apuno v functional for this solution and hence ψ 0 w ould b e pro ve n to be stable. Here w e need to mak e a standard remark on the translatio na l inv ariance of equations (8), (9) and (10). The doma in w all cen tred at the origin has the same energy as the w all cen tred at any other p oint x 0 and therefore can nev er b e an isolated minim um of E . Ho w eve r, a mere translation of the w all f r o m the origin to the p oint x 0 do es not imply instability . Therefore, it is phy sically reasonable t o gro up a ll config ura tions obtained from a give n ψ ( x ) b y t r a nslations ψ ( x ) → ψ ( x − x 0 ) with −∞ < x 0 < ∞ , in to equiv alence classes. Unlik e the wall with an y particular x 0 , the equiv alence c lass consisting of all translated w alls c an b e an isolated minim um of the energy — defined on the corresp onding q uotien t manifold. If the p erturba t ions of the w all are assumed to b e infinitesimal, the quotien t is a linear subspace; namely , it is the quotien t of the space of all infinitesimal p erturbatio ns of the w all b y the subspace spanned b y its translation mo de ( ∂ x R 0 , ∂ x I 0 ). This quotien t space can b e con venie n tly characteris ed b y the orthogonality constraint Z ( ∂ x R 0 , ∂ x I 0 )   δ R δ I   dx = 0 . (13) In what follo ws, the subspace of p erturbations defined b y the constrain t (13) will b e denoted S . T o che c k whe ther ψ 0 renders E [ ψ ] a minim um in S , the functional is expanded ab out the stationary p oin t ψ 0 : E [ ψ ] = E 0 + 1 2 δ 2 E + .... Here ψ = ψ 0 + δ R + iδ I , and the second v ariation has the form 1 2 δ 2 E = Z ( δ R , δ I ) H   δ R δ I   dx, (14) where the hessian H coincides with the linearised op erator (12). The solution ψ 0 will min- imise E [ ψ ] in S pro vided H has no negativ e or zero eigen v alues other than the one asso ciated with the translatio n mode. 7 A Ly apunov functional can also b e used in the case o f the relativistic dynamics (9) (see e.g. [16]). Here, as a candidate functional one c onsiders the total energy E total [ ψ , ψ t ] = 1 2 Z | ψ t | 2 dx + E [ ψ ] , (15) where E [ ψ ] is as in (2). The energy is conserv ed, and henc e E total will define the Ly apunov functional for the solution ψ 0 if it renders this functional a minim um in S . Since the first term in (1 5) is minimised by an y static configuration, it is sufficien t to c hec k whether ψ 0 ( x ) renders the functional E [ ψ ] a minim um. Consequen tly , the minimisation problem reduces to the eigen v alue problem (11). Th us, w e hav e the same stabilit y criterion in the case of the Ginsburg-La nda u and rela- tivistic dynamics — one ha s to pro v e that the solution minimises the functional ( 2) under the constrain t (13), or, equiv alen tly , show tha t t he op erator H has no negative o r zero eigen v alues ot her than the translation mo de. In the case of the Ising w a ll (5), w e hav e R 0 ( x ) = 0 and t he op erator (12) is diag onal. Its eigen v alues can b e readily found and the ab ov e criterion easily implemen ted. Th us, it w a s sho wn in [8, 9, 17, 18] that the Ising w all is stable for h > 1 3 and unstable for h < 1 3 . (See [19] for the generalisation to a nonv ariat io nal case.) In the c ase of the Blo c h w all, on the other hand, the op erator H is nondiagonal. This mak es it imp ossible to determine the sign of the low est eigen v alue of H usin g standard analytical metho ds. Consequen tly , previous studies had to resort to semi-in tuitiv e and n umerical arg umen ts. In particular, it was no t ed that the energy of the Blo c h w all is lo w er than that of the Ising w all and suggested that the Blo c h w all should b e stable [11, 12, 2 0]. This conjecture w as supp orted by results of direct n umerical sim ulations of equation (9) [21] and p erturbation theory for small C ( C = √ 1 − 3 h ) [8 ]. Ho w ev er no analytical pro o f, applicable for all parameter v alues, has b een giv en so far. B. The Sc hr¨ odinger dynamics Finally , we dis cuss the parametrically driven nonlinear Sc hr¨ odinger equation, equation (10). In this c ase, the linearisation a b out ψ 0 pro duces a s ymplectic eigen v alue problem H   δ R δ I   = λJ   δ R δ I   , J =   0 − 1 1 0   , (16) 8 where H is a s in (12 ). The pro duct op era t o r J − 1 H is non-self-adjoint and hence its eigen- v alues can b e complex. The solution ψ 0 will b e unstable if the op erator J − 1 H ha s at least one eigen v alue λ with p ositiv e real part. The nonline ar Sc hr¨ odinger equation (10) conserv es energy whic h is giv en b y the in tegral (2); hence the energy is a p oten tial Ly apuno v functional for equation (1 0). One can therefore try t o establish the stability of ψ 0 b y proving t ha t it renders the energy minimum under the constrain t (13); this happ ens when the o p erator H in t he second v ariation (14) do es not ha v e negativ e or zero eigen v alues o ther than the translation mo de. Note that the criteria based on the linearisation and the energy minimality app eal to eigen v alues of di ffer ent op erators here, J − 1 H a nd H , resp ective ly . Ho w ev er, it is not diffic ult to show that the positive definiteness of H implies that J − 1 H do es not ha ve eigen v alues with p ositive real part. W e include a pro of of this simple fact in the App endix. As we hav e p ointed out in the previous subsection, the op erator H asso ciated with the Ising wall is diagonal. Making use of t his prop ert y , it w as pro v ed in [6] that J − 1 H do es not ha v e eigen v alues with p ositiv e real part a nd the Is ing wall is stable fo r all h > 0. On the other hand, in the case of the Blo c h w all, the op erato r H in (11) and (16) is no ndiagonal. This prev ents the determination of the sign of the lo w est eigen v alue of H , or testing the existence o f unstable eigenv alues of J − 1 H , using an y of the standa r d analytical approa ches . The eigen v alue problems (11) and (16) can of course b e studied n umerically; this w as done in Ref.[6] where the Blo ch w all w as found to b e stable for all examined v alues of h . How ev er, n umerical solutions tend to o verlook subtleties (e.g. expo nentially small eigen v alues) and giv e limited insigh ts in to the structure o f the configuration space. This motiv ates our searc h for an a nalytical stabilit y pro of . W e pro vide suc h a pro of in the next section. IV. X Y MODEL IN E LLIPTIC C OORDINA TES Returning to the energy (2), w e define e lliptic co ordinates on the (Re ψ , Im ψ ) -plane: ψ = B (sinh u sin v + i cosh u cos v ) . (17) Here, u ( x ) ≥ 0 and 0 ≤ v ( x ) ≤ 2 π are contin uous fields. The use of elliptic co ordinates w as pioneered b y T rullinger and DeLeonardis who utilised 9 these in their calculatio n o f the partition f unction for the MSTB mo del [22]. Elliptic co or- dinates allo w the separation of v ariables in t he effectiv e Sch r¨ odinger equation that arises in their transfer-matrix approac h (see a lso [20]). Subsequen tly , Ito [23] used elliptic co o r dina t es to se parate v ariables in the Hamilton-Jacobi form ulation of equation (4) (see also [2 4]). T ransforming to the elliptic coordinates (17), the energy functional (2) acquires the form E = 2 h Z  (sinh 2 u + sin 2 v )( u 2 x + v 2 x ) + f 2 ( u ) + g 2 ( v ) sinh 2 u + sin 2 v  dx, (18) where f ( u ) = B sinh u  cosh 2 u − A 2 B 2  , (19a) g ( v ) = B sin v  A 2 B 2 − cos 2 v  . (19b) The in tegrand in (18) admits a Bogomoln yi-t yp e decomp osition E = 2 h Z ( µ ( u, v ) "  u x + f ( u ) µ ( u, v )  2 +  v x + g ( v ) µ ( u, v )  2 # + Φ x ) dx, (20) where µ ( u, v ) = sinh 2 u + sin 2 v ≥ 0 and Φ x = − 2 f ( u ) u x − 2 g ( v ) v x . Since b o th terms in the square brac k ets in (20) are nonnegative, the energy is b ounded from b elo w: E ≥ 2 h Φ( x )     ∞ −∞ . (21) Ev aluating the r ig h t -hand side of (21) using (19), this inequality is tra nsformed in to E ≥ B 3  3 A 2 cosh u ( x ) − B 2 cosh 3 u ( x )  +  3 A 2 cos v ( x ) − B 2 cos 3 v ( x )      ∞ −∞ . (22) In terms of the elliptic co ordinat es, the domain walls’ b oundary conditions ψ ( ±∞ ) = ± iA acquire the form u ( −∞ ) = arccosh  A B  , v ( −∞ ) = π ; (23a) u (+ ∞ ) = a r ccosh  A B  , v (+ ∞ ) = 0 , (23b) 10 or u (+ ∞ ) = a r ccosh  A B  , v (+ ∞ ) = 2 π . (23c) Using these , the inequalit y (22) becomes simply E [ ψ ] ≥ E B , (24) where E B is the energy of the Bloch w all giv en by (7). The bound (24) is obviously saturated by the Blo ch w alls (6). W e no w sho w that, giv en the b oundary conditions (23), the tw o Blo c h w a lls are the only s olutions with the minim um energy . The pro of app eals to the Bogomoln yi equations u x = − B µ ( u, v ) sinh u  cosh 2 u − A 2 B 2  , (25a) v x = − B µ ( u, v ) sin v  A 2 B 2 − cos 2 v  , (25b) whic h hav e to b e satisfied b y any configura tion with E [ ψ ] = E B . In order to pro v e the uniqueness , it is sufficien t to demonstrate that the dynamical system (25) has a unique hetero clinic tra jectory connecting the p oin t (23 a) to the p oin t (23b) and another unique tra jectory connecting (23 a) to (23c). T o this end, we note that the line u = arccosh( A/B ) is an in v arian t manifold and that this manifold is attractive : tra jectories flow to wards this line but no tra jectories can lea v e it. Therefore, the only tra jectories connecting the p oints (23) hav e to b e segmen ts of this straight line. Letting u = arccosh( A/B ) , equation (25 b) simplifies to v x = − B sin v . Sub ject to the b oundary conditions v ( −∞ ) = π , v ( ∞ ) = 0 and v ( −∞ ) = π , v ( ∞ ) = 2 π , this equ ation has a uniq ue pair of solutions sin v = ± sec h ( B x ) , cos v = tanh( B x ) . (26) (More precisely , these solutions are unique up to translations x → x − x 0 .) Inserting equa- tions (26) in to (17) yields the righ t- and left-handed Blo ch w alls, equation (6). Consequen tly , the Blo c h w alls are indeed the unique minimal energy solutions (mo dulo translations). This pro v es their stabilit y — within eac h of the three ev olution equations (8), (9 ) and (10). 11 V. CONC LUDING REMARKS W e ha v e utilised the Bog o moln yi construction to prov e that the Blo ch walls of equation (2) are energy-minimizing kinks. W e ha ve also sho wn that t hey are unique energy minimizers. Th us for all three ev olution equations (8) , (9), and (10), the Blo c h w alls are prov en t o b e stable. The energy (2) w as in t r o duced to describ e the anisotropic X Y mo del. How ev er, the related ev olution equations (8), (9) and (10) emerge in sev eral other areas where our results will also b e a pplicable. In particular, the Ginsburg-Landau equation (8) app ears as a generic amplitude equation in resonan tly forced oscillatory media near t he Hopf bifurcation [25]. The Blo c h and Ising w alls are often regarded as the basic building blo ck s for the one- a nd t wo- dimensional patterns arising in suc h media [26]. Our stabilit y r esult puts this interpretation on a firmer ground. Next, the MSTB mo del (9) w as studied in the con text of quan tum field theory [27]. Here, t he fact that the mo del admits a BPS b ound is o f fundamen tal imp ortance as it means that it admits a natural sup ersymmetric extension. Finally , the parametrically driv en NLS equation (10) describ es F arada y resonance in a wide s hallo w w ater c hannel in the lo w-viscosit y limit [10, 28]. Since the Ising w all has already b een sho wn to b e stable within the NLS equation [6], a similar conclusion obtained now fo r the Blo c h w all rev eals an interesting bistability of the tw o solitons. This bistabilit y should allow exp erimen tal realization. Ac knowledgmen ts It is a pleasure to thank Alexander Iano vsky and Dmitry P elino vsky for useful remarks. S.W. w as supported b y a grant from the Science F a culty of the Univ ersit y of Cap e T ow n. I.B. w as suppor t ed by the NRF of South Africa under gran t 2053723 . APPENDIX A: T HE ST ABILITY-MINIMALITY C O RRESPONDENCE FOR THE NONLI NEAR SCHR ¨ ODINGER D YNAMICS While the linearised op erator coincides with the hessian of energy in the case of the Ginsburg-Landau and relativistic dynamics, the t wo op erators are differen t in the nonlinear Sc hr¨ odinger case. The hessian H is g iv en by equation (12), whereas the linearised op erator 12 is J − 1 H , with J the sk ew-symmetric matr ix (16). The m ultiplication by a sk ew-symmetric matrix c hanges t he sp ectral prop erties of an op erator; for example, the con tin uo us sp ectrum of H lies on the p ositiv e real axis whereas the contin uous sp ectrum of J − 1 H consists of pure imaginary λ . Therefore it is not ob vious ho w the energy minimalit y translates into the absence of unstable eigen v alues. In this app endix w e pro vide a simple pro of that the absence of negativ e and zero eigen- v alues of H   S is sufficien t for J − 1 H not to ha v e eigenv alues w ith p ositiv e real part. This fact is usually familiar to w ork ers in t his field; for the comprehens iv e treatmen t , including the nece ssary conditions for stabilit y and the eigenv alue coun t, see [29]. First of all, w e note that if λ is an eigenv alue of the o p erator J − 1 H in (16 ), asso ciated with an eigenv ector z ( x ) =   a ( x ) b ( x )   , then − λ is also an eigenv alue, asso ciated with the eigen v ector ˜ z ( x ) =   a ( − x ) − b ( − x )   . (A1) [This follows from the fact tha t b o th for the Blo c h and Ising w all, the off-diagonal elemen t s of the matrix H in (12) are odd functions of x : H 12 ( x ) = H 21 ( x ) = −H 21 ( − x ).] Hence real eigen v alues a lw ays come in ( λ, − λ ) pairs. On the other hand, if λ is a complex eigenv alue with an eigen ve ctor z ( x ), then λ ∗ is an eigenv alue with an eigen v ector z ∗ ( x ). Therefore, complex eigen v alues app ear in ( λ, − λ, λ ∗ , − λ ∗ ) quadruplets. Next, if H z = λJ z , Re λ 6 = 0 , the eigen v ector z satisfie s an iden tity ( z ∗ , J z ) = 0 , (A2) whic h follows from the self-adjoin tness of the op erator H . In (A2), w e used t he notation ( z 1 , z 2 ) = Z [ a 1 ( x ) a 2 ( x ) + b 1 ( x ) b 2 ( x )] dx, where a i and b i are the (complex) comp onen ts of the v ector z i , i.e. z i ( x ) =   a i ( x ) b i ( x )   , i = 1 , 2 . 13 Assume no w that the op erator J − 1 H has an eigen v alue λ with Re λ > 0, with the eigen- v ector z . It is not difficult to sho w that the quadratic fo r m (14) calculated on the function y = C z + C ∗ z ∗ + ˜ C ˜ z + ˜ C ∗ ˜ z ∗ , (A3) where C and ˜ C are complex co efficien ts, is either sign-indefinite, or equals zero. Indeed, substituting (A3) in t o (14) and making use of (A2), we get 1 2 δ 2 E [ y ] = 2 C ˜ C λ ( ˜ z , J z ) + c.c. + C ∗ ˜ C ( λ + λ ∗ )( ˜ z , J z ∗ ) + c.c., (A4) where ˜ z is as in (A1) and c.c. stands for the complex conjugate of the immediately preceding term. The expression (A4) is either zero or c hang es its sign under C → − C . (This conclusion ob viously remains v alid if the e igen v alue λ is real.) 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