Symmetry, Integrability and Geometry: Methods and Applications Solitary Waves in Massive

The solitary waves of massive (1+1)-dimensional nonlinear S N -sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspon- dence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological kinks is established applying the Morse index theorem.


Introduction
The existence and the study of solitary waves in (1 + 1)-dimensional relativistic field theories is of the greatest interest in several scientific domains. We specifically mention high energy physics, applied mathematics, condensed matter physics, cosmology, fluid dynamics, semiconductor physics, etcetera. Solitary waves have been discovered recently in massive nonlinear sigma models with quite striking properties, see e.g. [1,2,3] and references quoted therein. The nonlinear S N -sigma models are important in describing the low energy pion dynamics in nuclear physics [4] and/or featuring the long wavelength behavior of O(N + 1)-spin chains in ferromagnetic materials [5].
In this work, we present a mathematically detailed analysis of the solitary wave manifold arising in a massive S N -sigma model that extends and generalize previous work performed in [1] and [2] on the N = 2 case. In these papers the masses of the N = 2 meson branches are chosen to be different by breaking the O(N ) degeneracy of the ground state to a Z 2 subgroup. Remarkably, we recognized in [1] and [2] that the static field equations of this nonisotropic massive nonlinear S 2 -sigma model are the massive version of the static Landau-Lifshitz equations governing the high spin and long wavelength limit of 1D ferromagnetic materials. This perspective allows to interpret the topological kinks of the model as Bloch and Ising walls that form interfaces between ferromagnetic domains, see [6,7,8].
where y 0 , y 1 stand for the standard coordinates in the Minkowski space R 1,1 , and the fields are constrained to be in the S N -sphere, i.e. Φ ∈ Maps R 1,1 , S N , Φ · Φ = φ 2 1 + · · · + φ 2 N +1 = R 2 . (2.1) The dynamics of the model is governed by the action functional: where V ( Φ) is the quadratic polynomial function in the fields: We follow standard Minkowski space conventions: y µ y µ = g µν y µ y ν , g µν = diag(1, −1), The field equations of the system are: λ being the Lagrange multiplier associated to the constraint (2.1). λ can be expressed in terms of the fields and their first derivatives: We shall address the maximally anisotropic case: α a = α b , ∀ a = b, and, without loss of generality, we label the parameters in decreasing order: α 2 1 > α 2 2 > · · · > α 2 N +1 ≥ 0, in such a way that V ( Φ), on the S N -sphere (2.1), is semi-definite positive.
According to Rajaraman [9] solitary waves (kinks) are non-singular solutions of the field equations (2.3) of finite energy such that their energy density has a space-time dependence of the form: ε(y 0 , y 1 ) = ε(y 1 − vy 0 ), where v is some velocity vector. The energy functional is: and the integrand ε y 0 , y 1 is the energy density. Lorentz invariance of the model implies that it suffices to know the y 0 -independent solutions Φ y 1 in order to obtain the traveling waves of the model: Φ y 0 , y 1 = Φ y 1 − vy 0 . For static configurations the energy functional reduces to: (2.5) and the PDE system (2.3) becomes the following system of N +1 ordinary differential equations: The finite energy requirement selects between the static and homogeneous solutions (2.4) only two, the set M of zeroes (and absolute minima) of V ( Φ) on S N : i.e. the "North" and the "South" poles of the S N -sphere. Moreover, this requirement forces in (2.5) the asymptotic conditions: Consequently, the configuration space, the space of finite energy configurations: is the union of four disconnected sectors: where the different sectors are labeled by the element of M reached by each configuration at y 1 → −∞ and y 1 → ∞. C NS and C SN solitary waves will be termed as topological kinks, whereas non-topological kinks belong to C NN or C SS .

Embedded sine-Gordon models
Before of embarking ourselves in the general solution of the ODE system (2.6), we explore Rajaraman's trial-orbit method [9]. We search for solitary wave solutions living on special meridians of the S N -sphere such that the associated trajectories connect the North and South poles, to comply with the asymptotic conditions of finite energy (2.7). In [1] and [2] this strategy was tested in the N = 2 case. It was easily proved that the nonlinear sigma model reduces to a sine-Gordon model only over the meridians in the intersection of the coordinate planes in R 3 with the S 2 -sphere, that contain the North and South poles. Consequently, there are sine-Gordon kinks (and multi-solitons) embedded in the nonlinear sigma model living in these meridians. The generalization to any N is immediate: the N meridians m a , a = 1, . . . , N , determined by the intersection of the φ a − φ N +1 plane with the S N -sphere, all of them containing the points N and S, . . , N, are good trial orbits on which the field equations (2.3) reduce to the system of two PDE's: Use of "polar" coordinates in the φ a − φ N +1 plane: reveals that the two equations (3.1), (3.2) are tantamount to the sine-Gordon equation in the half-meridians sign(φ a ) = ±1: For the sake of simplicity it is convenient to define non dimensional parameters and coordinates in the Minkowski space R 1,1 : Thus, the N sine-Gordon equations (3.3) become: that look at rest in their center of mass system v = 0: The kinks θ + Ka belong to the topological sector C NS of the configuration space, whereas the antikink θ − Ka lives in C SN . Henceforth, we shall term as topological kinks to these solitary waves. In the original coordinates in field space, φ a and φ N +1 , θ ± Ka correspond to two kinks and their two anti-kinks, depending of the choice of half-meridian. We will keep the K a notation for the φ a > 0 half-meridian and introduce K * a for the φ a < 0 choice: The energy functional, for static configurations, is written, in non-dimensional coordinates as: where ν = α 2 1 − α 2 N +1 . The embedded sine-Gordon kink energies are:

The moduli space of generic solitary waves
The system of ordinary differential equations (2.6) is the system of motion equations of the repulsive Neumann problem with N degrees of freedom if one interprets the coordinate y 1 as the "mechanical time" and the field components φ a as the coordinates setting the "particle position", the so-called mechanical analogy [9]. It is a well known fact that the Neumann system is Hamilton-Jacobi separable using sphero-conical coordinates [13,14,15]. Let us introduce sphero-conical coordinates (u 1 , u 2 . . . , u N ) in S N with respect to the set of constantsσ 2 a = 1 − σ 2 a : The Cartesian coordinates in R N +1 are given in terms of the sphero-conical coordinates by: , The algebraic equation characterizing the S N -sphere in R N +1 is simply u 0 = R 2 in sphero-conical coordinates.
The change of coordinates (4.2) provides a 2 N +1 to 1 map from the S N sphere to the interior of the hyper-parallelepiped (in the (u 1 , . . . , u N ) space) P N characterized by the inequalities (4.1). The map from each "2 N +1 -tant" of the S N -sphere to the interior of P N is one-to-one. One "2 N +1 -tant" is the piece of S N that lives in each of the 2 N +1 parts in which R N +1 is divided by the coordinate N -hyperplanes. The intersections of the S N -sphere with the coordinate Nhyperplanes, N + 1 S N −1 -spheres, are 2 N to 1 mapped into the boundary of P N , a set of (N − 1)-hyperplanes in the (u 1 , . . . , u N ) space. Simili modo S N −k -spheres, intersection of S N with coordinate (N − k + 1)-hyperplanes of R N +1 , are 2 N −k+1 to 1 mapped into (N − k)hyperplanes of the boundary of P N . Finally, the North and South poles of S N are mapped into a single point in the boundary of P N : v ± ≡ (u 1 = 0, u 2 =σ 2 2 , . . . , u N =σ 2 N ). The Euclidean metric in R N +1 ds 2 = dφ 2 1 + · · · + dφ 2 N +1 becomes in P N : Calculations using sphero-conical coordinates remarkably simplify recalling the Jacobi lemma, see e.g. [14]. Let be a collection of p distinct constants (a 1 , a 2 , . . . a p ), a i = a j , ∀ i = j. Let us The following identities hold: Now, use of (4.4) allows us easily determine the V ( Φ) function (2.2) in sphero-conical coordinates: The energy functional for static configurations (3.5) in this system of coordinates can be written as follows: The p−2 identities on the left in (4.4) allows us to write the potential energy density in different ways. We choose an expression (4.5) in the form of a sum of products in the factors (u a −σ 2 b ) that guarantees a direct verification of the asymptotic conditions (2.7).
Alternatively, the energy can be written in the Bogomolnyi form [17]: where g aa denotes the components of the inverse tensor metric, and W (u 1 , . . . , u N ) is a solution of the partial differential equation: (4.7) Note that (4.7) is the "time"-independent Hamilton-Jacobi equation for zero mechanical energy (i 1 ≡ 0) of the "repulsive" Neumann problem. Therefore, W is the zero-energy time-independent characteristic Hamilton function.
The ansatz of separation of variables W (u 1 , . . . , u N ) = W 1 (u 1 ) + · · · + W N (u N ) reduces the PDE (4.7) to a set of N uncoupled ordinary differential equations depending on N separation constants i a . The separation process is not trivial (see [14]) and requires the use of equations (4.4). Let write explicitly (4.7) as: The identities of the Jacobi lemma (4.4), for k ≤ N − 2, allow to express (4.8) as a system of N equations: depending on N − 1 arbitrary separation constants i 2 , . . . , i N . Asymptotic conditions (2.7) imply, in sphero-conical coordinates, the limits: lim x→±∞ u a =σ 2 a . This requirement fixes, in (4.9), all the separation constants i a to be zero, that correspond to the separatrix trajectories between bounded and unbounded motion in the associated mechanical system. Thus, integration of (4.9) gives the W function, i.e. the time-independent, zero mechanical energy, characteristic Hamilton function of the repulsive Neumann problem: It is interesting to remark that general integration of the Hamilton-Jacobi equation for the Neumann problem involves hyper-elliptic theta-functions [15]. The asymptotic conditions (2.7) guaranteeing finite field theoretical energy to the solitary waves -and finite mechanical action to the associated trajectories -force theta functions in the boundary of the Riemann surface where degenerate to very simple irrational functions. The ordinary differential equations: . K 1 (red) and K 2 (blue) kink orbits over S 2 (left). Energy densities for K 1 /K * 1 (red) and K 2 /K * 2 (blue) kinks for a concrete value of 0 < σ < 1 (right).
can alternatively be seen as first-order field equations for which the field theoretical energy is of the simple form: E[u 1 , u 2 , . . . , u N ] = ν dW , see (4.6), or, as the motion equations of the mechanical system restricted to the sub-space of the phase space such that i a = 0, ∀ a = 1, 2, . . . , N . In differential form they look: Although it is possible to integrate equations (4.12) for an arbitrary value of N (see [18] for instance), we shall devote the rest of the paper to analyze the explicit solutions and the structure of the moduli space of solitary waves in the lower N = 2 and N = 3 cases. There are two sine-Gordon models embedded in the φ 1 = 0 and φ 2 = 0 meridians, see Fig. 2 (left). The non-dimensional parameters, for N = 2, are: σ 1 ≡ 1, σ 2 ≡ σ and σ 3 = 0. The K 1 /K * 1 and K 2 /K * 2 sine-Gordon kinks/antikinks profiles read: The K 1 , K * 1 , K 2 and K * 2 kinks belong to C NS , see Fig. 1; the φ 3 -component is kink shaped for the four kinks, whereas the φ 1 -and φ 2 -components are bell shaped respectively for K 1 and K 2 but anti-bell shaped for K * 1 and K * 2 . The antikinks live in C SN and have similar profiles. These solitary waves are topological kinks, and they correspond to heteroclinic trajectories in the analogous mechanical system. Their energies are: The energy of the K 1 /K * 1 kinks is greater than the energy of the K 2 /K * 2 kinks, see Fig. 2 (right).

Generic solitary waves on S 2
We shall denote (λ 1 , λ 2 ) the sphero-conical coordinates in the S 2 -sphere, λ 0 = R 2 , to avoid confusion with the N = 3 case. In terms of these coordinates the fields read: We recall the definition:σ 2 = 1 − σ 2 , and the (λ 1 , λ 2 ) ranges: The change of coordinates (5.3) maps the S 2 -sphere into the interior of the rectangle P 2 defined by the inequalities (5.4) in the (λ 1 , λ 2 )-plane. In fact, the map is eight to one, due to the squares that appears in (5.3). Each octant of the S 2 -sphere is mapped one to one into the interior of P 2 , whereas the meridians of S 2 contained in the coordinate planes are mapped into the boundary of P 2 . The set of zeroes M of V (φ 1 , φ 2 , φ 3 ) in S 2 , the North and South poles, becomes a unique point in becomes linear in sphero-conical coordinates: and the energy functional for static configurations in terms of (λ 1 , λ 2 ) is: Here, the components of the metric tensor (4.3) read: The knowledge of the solution (4.10) for N = 2: suggests to write the energy functional (5.5) in the Bogomolnyi form (4.6) leading to the system of first-order ODE's (4.11): .
These equations can be separated in a easy way: The Hamilton-Jacobi procedure precisely prescribes the equation (5.6) as the rule satisfied by the orbits of zero mechanical energy whereas (5.7) sets the mechanical time schedule of these separatrix trajectories between bounded and unbounded motion.
Instead of attempting a direct solution of the system (5.6), (5.7) we introduce new variables In terms of s 1 and s 2 the ODE's system (5.6), (5.7), after decomposition in simple fractions, becomes: Integrating equations (5.8) in terms of the inverse of hyperbolic tangents/cotangents, and using the addition formulas for these functions we find the following general solution depending on two real integration constants γ 1 and γ 2 : To solve the system of equations (5.9) separately in s 1 and s 2 we introduce the new variables A = s 1 + s 2 and B = s 1 s 2 . The subsequent linear system in A and B and its solution are: Therefore, s 1 and s 2 are the roots of the quadratic equation Because λ 1 = 1 − s 2 1 , λ 2 = 1 − s 2 2 , the come back to Cartesian coordinates (5.3) provides the explicit analytical formulas for the two-parametric family of solitary waves:

The structure of the moduli space of kinks
To describe the structure of the two-dimensional moduli space of solitary waves solutions (5.10) it is convenient to rely on a re-shuffling of the coordinates of the moduli space: γ = −γ 1 , γ = γ 2 − γ 1 . γ determines the "center" of the energy density of a given kink andγ distinguishes between different kinks in the moduli space by selecting the kink orbit in S 2 , see Figs. 5 and 6. In the Figs. 3 and 4, however, we have plotted the components of the kink profiles for two different choices ofγ and γ = 0. We remark that none of the three components are kink-shaped, the φ 1 and φ 3 components are bell-shaped and the φ 2 -components have a maximum and a minimum. The difference is that the x → −x reflection symmetry is lost forγ = 5. The profiles of these two solitary waves tend to the South pole in both x → ±∞. Therefore, these kinks are non-topological kinks living in C SS . (γ,γ) are good coordinates in the solitary wave moduli space characterized by the orbits in P 2 . The inverse mapping to S 2 is classified by the different choices of 1 , 2 and 3 : • The value of 3 selects the topological sector of the solution. Clearly: and the equations (5.10) determine two families of non-topological (N T K) kinks, one family living in C SS , 3 = 0, and the other one belonging to C NN , 3 = 1. They are homoclinic trajectories of the analogous mechanical system.  • 1 = 0 provides the N T K kink orbits running in the φ 1 > 0 hemisphere and 1 = 1 corresponds to the N T K orbits passing through the other (φ 1 < 0) hemisphere of S 2 .
• 2 , however, does not affect to the kink orbit and merely specifies the kink/antikink character of the solitary wave, i.e. the sense in which the orbit is traveled.
In Fig. 5 (right) several N T K orbits are shown in S 2 . The sector C SS and the φ 1 > 0 hemisphere have been selected. All of them cross the φ 2 = 0 meridian at the same point: F ≡ (Rσ, 0, Rσ) -in sphero-conical coordinates this point is the low right corner λ 1 = λ 2 =σ 2 of P 2 ). Therefore the point F is a conjugate point 1 of the South pole S of S 2 .
In Fig. 6 the graphics of the kink energy densities of two members of the N T K family are plotted. We remark that the energy density appears to be composed by two basic elements. This behavior is more evident for |γ| 0 because in such a range the energy density resembles the superposition of the energy densities of the two embedded sine-Gordon kinks (compare with Fig. 2 (right)). All the N T K kinks, however, have the same energy that can be computed from the formula (4.6): There is a kink energy sum rule: the energy of any N T K kink is exactly the the sum of the two sine-Gordon, K 1 and K 2 , topological kink energies. These sum rules arises in every field theoretical model with mechanical analogue system which is Hamilton-Jacobi separable and has several unstable equilibrium points, see, e.g., [16]. Moreover, looking at Fig. 4 as a posterior picture to Fig. 3 in a movie that evolves with increasingγ, it is clear that we find almost a K 1 anti-kink in the x 0 region and a K * 2 kink in the x 0 region whenγ → ∞. This means that the solitary waves (5.10) tend to a K 1 kink/antikink plus a K 2 antikink/kink whenγ → ±∞. Thus the combinations K 1 +K 2 (kink/antikink and antikink/kink) form the closure of the N T K moduli space and belong to the boundary. Alternatively, one can tell that there exist only two basic (topological) kink solutions and the rest of the kinks, the N T K family, are (nonlinear) combinations of the two basic solitary waves. 6 The moduli space of S 3 -solitary waves 6

.1 Embedded N = 2 kinks
In the massive nonlinear S 3 -sigma model we first count the three embedded sine-Gordon kinks: • The K 1 /K * 1 kinks and their anti-kinks living in the meridian the intersection of the φ 2 = φ 3 = 0 plane with the S 3 sphere.
• Idem for the K 2 /K * 2 and their anti-kinks: • Idem for the K 3 /K * 3 kinks and their anti-kinks: The novelty is that all the non-topological kinks of the massive nonlinear S 2 -sigma model are embedded in the S 3 version in three different S 2 "meridian" sub-manifolds.
1. Consider the two-dimensional S 2 I sphere: the intersection of the φ 1 = 0 3-hyperplane in R 4 with the S 3 -sphere.
The restriction of the N = 3-model to these two-manifold collects all the solitary waves of the N = 2-model. We shall call N T K I kinks to the N = 2 N T K kinks of the N = 2 model that lives in S 2 I . The K 2 /K * 2 and K 3 /K * 3 sine-Gordon topological kinks/anti-kinks also live in in S 2 I and belong to the boundary of the N T K I moduli space. 2. Idem for the S 2 II sphere: The non-topological kinks running on this 2-sphere will be termed N T K II . The sine-Gordon kinks/antikinks are: K 1 /K * 1 and K 3 /K * 3 . 3. Idem for the S 2 III sphere: The non-topological kinks running on this 2-sphere will be termed N T K III . The sine-Gordon kinks/antikinks are: K 1 /K * 1 and K 2 /K * 2 .

Generic solitary waves
Away from the meridian 2-spheres S 2 I , S 2 II and S 2 III considered in the previous sub-section, there is a three-parametric family of generic kinks. In N = 3 dimensional sphero-conical coordinates (4.2): such that: 0 < u 1 <σ 2 2 < u 2 <σ 2 3 < u 3 < 1, the S 3 -sphere of radius R is characterized by the equation: u 0 = R 2 . The change of coordinates (6.1) corresponds to a sixteen-to-one map from the points of S 3 out of the coordinate 3-hyperplanes φ a = 0, a = 1, . . . , 4, to the interior of the parallelepiped P 3 , in the (u 1 , u 2 , u 3 )-space, defined by the inequalities above. The meridian 2-spheres are eight-to-one mapped into the boundary of P 3 , where some inequalities become identities: u 1 = 0 for S 2 I , {u 1 =σ 2 2 } {u 2 =σ 2 2 }, for S II , and {u 2 =σ 2 3 } {u 3 =σ 2 3 } in the S 2 III case. The one-dimensional coordinate meridians of these 2-spheres, in particular those that are the embedded sine-Gordon kink orbits are four-to-one mapped into the edges of P 3 , see Fig. 7. Finally, the North and South poles of S 3 are two-to-one mapped into one vertex of P 3 : v ± ≡ (u 1 , u 2 , u 3 ) = (0,σ 2 2 ,σ 2 3 ). The quadratic polynomial V (φ 1 , φ 2 , φ 3 , φ 4 ) becomes linear in the sphero-conical coordinates: Figure 7. Parallelepiped P 3 . The K 1 , K 2 and K 3 sine-Gordon kink orbits are depicted (in blue) at the corresponding edges of P 3 . but we write it in "separable" form: .
The reason is that in this form it is obvious that the PDE (4.7) reduces to the system of three uncoupled ODE that can be immediately integrated to find (4.10): The Bogomolnyi splitting of the energy (4.6) prompts the first-order equations (4.11): It is convenient to pass to the variables: because the quadratures in (6.2) become: where only enter rational functions. The integration of (6.3) is straightforward: The addition formulas for the inverse hyperbolic functions, and the definition of "Vieta variables": reduce the system of transcendent equations (6.4) to the linear system: where: t 1 = tanh(−(x + γ 1 )), t 2 = tanh(−σ 2 (x + γ 2 )), t 3 = tanh(−σ 3 (x + γ 3 )). From the solutions A, B and C of this linear system we obtain the s 1 , s 2 and s 3 variables as the roots of the cubic equation: The standard Cardano parameters provide the three roots s 1 , s 2 and s 3 by Cardano's formulas, and recalling that: u a = 1 − s 2 a , a = 1, 2, 3 we finally obtain: which in turn can be mapped back to cartesian coordinates. The explicit analytical expressions for the generic kinks are: where a = 0, 1, ∀ a = 1, . . . , 4. Equations (6.6) define a three-parameter family of kinks in the S 3 -sphere.
show that 4 = 0 implies that the T K kink family lies in C NS , whereas 4 = 1 sends the T K family to live in C SN . The other sign options, (−1) 1 , (−1) 2 and (−1) 3 , determine the three-dimensional hemispheres accommodating the different T K kink orbits, as well as the kink/antikink character of the solutions, in a smooth generalization of the N = 2 N T K kinks. It is possible to draw the kink orbits in the P 3 parallelepiped using the solutions (6.5) in sphero-conical coordinates. In the Fig. 10 (right) several T K orbits are drawn together. The  Figure 10. T K orbit corresponding to: γ 1 = γ 2 = γ 3 = 0 (left). The case: γ 1 = 0.8, γ 2 = 2 and γ 3 = 2 (middle). Three orbits corresponding to different values of the parameters (right). energy of all the T K kinks is: There is a new kink energy sum rule specifically arising in the N = 3 model. In the Fig. 11 it can be viewed how the energy density reflects this energy sum rule: the energy density of a given member of the T K family is a "composition" of the energy densities of the sine-Gordon kinks K 1 , K 2 and K 3 , see Fig. 11 (middle) and (right). This structure is completely similar to the kink manifold of the field theoretical model with target space flat R 3 and the Garnier system as mechanical analogue problem, see [12] and [16]. In fact, the equations are identical, even thought in that case the problem is separable using elliptical coordinates in R 3 , [12,16]. The calculation of the limits of the T K family when the parameters γ 1 , γ 2 and γ 3 tend to ±∞ is not a simple task, but it is possible to show that all the possibilities arising from the energy sum rule are obtained for the different limits (see [16] for details; the model is different but the equations are the same and thus the analytical calculation of the limiting cases coincides). In the Fig. 10 (right), for instance, a T K kink orbit (in red color) close to the limiting combination K 3 plus a member of the N T K III family, is plotted. Finally, the structure of the kink manifolds for the N = 2 and N = 3 models allows us to foresee that in higher N , the generic kinks will be topological if N is an odd number, and non-topological for even N . In fact, the kink energy sum rule will establish that the energy of a generic solitary wave is the sum of the energies of the N sine-Gordon kinks embedded in the model.

Kink stability
The analysis of the kink stability requires the study of the small fluctuations around kinks. For simplicity, we will only present the explicit results for the topological kinks of the S 2 model, and also a geometrical analysis in terms of Jacobi fields that allows us to demonstrate the instability of the non-topological kinks. The generalization to the arbitrary N case is straightforward.
7.1 Small f luctuations on topological kinks, K 1 and K 2 Taking into account the explicit expressions of the K 1 and K 2 sine-Gordon kinks of the S 2 model, we will use spherical coordinates in S 2 in order to alleviate the notation. Thus we introduce the coordinates: with θ 1 ∈ [0, π], θ 2 ∈ [0, 2π). The metric tensor over S 2 is written now as: ds 2 = R 2 dθ 1 dθ 1 + R 2 sin 2 θ 1 dθ 2 dθ 2 . The associated Christoffel symbols and Riemann curvature tensor components will be: The analysis of small fluctuations around a kink θ K (x) ≡ (θ 1 K (x), θ 2 K (x)) is determined by the second-order Hessian operator: i.e. the geodesic deviation operator plus the Hessian of the potential energy density, valuated at the kink. η denotes the perturbation around the kink. Let θ(x) denote the deformed trajectory, , then we introduce the following contra-variant vector fields along the kink trajectory, η, θ K ∈ Γ(T S 2 | K ): We will use standard notation for covariant derivatives and Riemann tensor: The geodesic deviation operator and the Hessian of the potential read: ∂θ k g jl ∂ ∂θ l evaluated at θ K (x). In sum, second-order kink fluctuations are determined by the operator: The spectrum of small f luctuations around K 1 /K *  Figure 12. Components of the Jacobi fields (7.6) for σ = 0.7, 1 = 2 = 3 = 0, and γ =γ = 0 (left), γ = 1,γ = 2 (right). expressions Φ N T K (x; γ,γ) for the N T K family of solutions (5.10), we know from classical Morse-Jacobi theory about conjugate points, that the vector-fields ∂ Φ N T K ∂γ and ∂ Φ N T K ∂γ are Jacobi fields along the N T K trajectories, i.e. zero-modes of the spectrum of small fluctuations around the solutions. In fact, taking into account that the γ parameter simply determines the "center" of the kink, it is easy to conclude that ∂ Φ ∂γ is tangent to the kink trajectories, and thus only ∂ Φ N T K ∂γ is a genuine (i.e. orthogonal to the kink-orbits) Jacobi field for this family of solutions [16]. By derivation on (5.10) with respect toγ parameter, it is obtained the Jacobi field: ∂ ∂φ 1 + (−1) 2 (σ tanh(x − γ) + tanh(σ(x − γ +γ))) ∂ ∂φ 2 − (−1) 3σ sech(σ(x − γ +γ)) ∂ ∂φ 3 . (7.6) By a direct calculation, it can be checked that lim x→±∞ J N T K (x; γ,γ) = 0, J N T K (γ; γ,γ) = 0 and thus the point x = γ in the kink trajectory, i.e. Φ N T K (γ; γ,γ) = ((−1) 1 Rσ, 0, (−1) 3 Rσ) is a conjugate point of the corresponding minima S (or N depending on the choice of 3 ). In Fig. 12 there are depicted the components of (7.6) for several different cases. The existence of a conjugate point establishes, according with Morse theory, that these kink solutions are not stable under small perturbations. In fact, the Morse index theorem states that the number of negative eigenvalues of the second order fluctuation operator around a given orbit is equal to the number of conjugate points crossed by the orbit [19]. The reason is that the spectrum of the Schrödinger operator has in this case an eigenfunction with as many nodes as the Morse index, the Jacobi field, whereas the ground state has no nodes. The Jacobi fields of the N T K orbits cross one conjugate point, their Morse index is one, and the N T K kinks are unstable.
Finally, it is possible to extend these results to the N = 3 case, where two Jacobi fields appear, and the instability of the generic T K family is showed. The procedure is the same, but the extension of the expressions is considerably bigger, thus we will not include this calculation here.