Stability of periodic waves

Stability of periodic traveling waves
for nonlinear dispersive equations

Vera Mikyoung Hur Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA verahur@math.uiuc.edu  and  Mathew A. Johnson Department of Mathematics, University of Kansas, Lawrence, KS 66045 USA matjohn@ku.edu
July 14, 2019
Abstract.

We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure. We then discuss when the linearized equation admits solutions exponentially growing in time.

Key words and phrases:
stability; periodic traveling waves; nonlinear dispersive; nonlocal
2010 Mathematics Subject Classification:
35B35, 35Q53, 35B10

1. Introduction

We study the stability and instability of periodic traveling waves for a class of nonlinear dispersive equations, in particular, equations of Korteweg-de Vries (KdV) type

(1.1)

Here denotes the temporal variable and is the spatial variable in the predominant direction of wave propagation; is real valued, representing the wave profile or a velocity. Throughout we express partial differentiation either by a subscript or using the symbol . Moreover is a Fourier multiplier, defined as and characterizing dispersion in the linear limit, while is the nonlinearity. In many examples of interest, obeys a power law.

Perhaps the best known among equations of the form (1.1) is the KdV equation

itself, which was put forward in [Bou77] and [KdV95] to model the unidirectional propagation of surface water waves with small amplitudes and long wavelengths in a channel; it has since found relevances in other situations such as Fermi-Pasta-Ulam lattices. Observe, however, that (1.1) is nonlocal unless the dispersion symbol is a polynomial of ; examples include the Benjamin-Ono equation (see [Ben67, Ono75], for instance) and the intermediate long wave equation (see [Jos77], for instance), for which and , respectively, while . Another example, proposed by Whitham [Whi74] to argue for breaking of water waves, corresponds to and . Incidentally the quadratic nonlinearity is characteristic of many wave phenomena.

A traveling wave solution of (1.1) takes the form , where and satisfies by quadrature that

for some . In other words, it steadily propagates at a constant speed without changing the configuration. Periodic traveling waves of the KdV equation are known in closed form, namely cnoidal waves; see [KdV95], for instance. Moreover Benjamin [Ben67] calculated periodic traveling waves of the Benjamin-Ono equation. For a broad range of dispersion symbols and nonlinearities, a plethora of periodic traveling waves of (1.1) may be attained from variational arguments. To illustrate, we shall discuss in Section 2 a minimization problem for a family of KdV equations with fractional dispersion.

Benjamin in his seminal work [Ben72] (see also [Bon75]) explained that solitary waves of the KdV equation are nonlinearly stable. By a solitary wave, incidentally, we mean a traveling wave solution which vanishes asymptotically. Benjamin’s proof hinges upon that the KdV “soliton” arises as a constrained minimizer for a suitable variational problem and spectral information of the associated linearized operator. Later it developed into a powerful stability theory in [GSS87], for instance, for a general class of Hamiltonian systems and led to numerous applications. In the case of , , and , , in (1.1), in particular, solitary waves were shown in [BSS87] (see also [SS90, Wei87]) to arise as energy minimizers subject to the conservation of the momentum and to be nonlinearly stable if whereas they are constrained energy saddles and nonlinearly unstable if .

We shall take matters further in Section 4 and establish that a periodic traveling wave of a KdV equation with fractional dispersion is nonlinearly stable with respect to period preserving perturbations, provided that it locally minimizes the energy subject to conservations of the momentum and the mass and that the associated linearized operator enjoys a Jordan block structure. Moreover we relate the latter condition with the momentum and the mass as functions of Lagrange multipliers arising in the traveling wave equation, generalizing that in [BSS87], for instance, in the solitary wave setting. In the case of generalized KdV equations, i.e., in (1.1), the nonlinear stability of a periodic traveling wave to same period perturbations was determined in [Joh09], for instance, through spectral conditions, which were expressed in terms of eigenvalues of the associated monodromy map (or the periodic Evans function); see also [AP07, APBS06, BJK11, DK10, DN11]. Confronted with nonlocal operators, however, spectral problems may be out of reach by Evans function techniques. Instead we make an effort to replace ODE based arguments by functional analytic ones. The program was recently set out in [BH14].

As a key intermediate step we shall demonstrate in Section 3 that the linearized operator associated with the traveling wave equation is nondegenerate at a periodic, local constrained minimizer for a KdV equation with fractional dispersion. That is to say, its kernel is spanned merely by spatial translations. The nondegeneracy of the linearization proves a spectral condition, which plays a central role in the stability of traveling waves (see [Wei87, Lin08] among others) and the blowup (see [KMR11], for instance) for the related, time evolution equation, and therefore it is of independent interest. In the case of generalized KdV equations, the nondegeneracy at a periodic traveling wave was identified in [Joh09], for instance, with that the wave amplitude not be a critical point of the period. Furthermore it was verified in [Kwo89], among others, at solitary waves. These proofs utilize shooting arguments and the Sturm-Liouville theory for ODEs, which may not be applicable to nonlocal operatorors. Nevertheless, Frank and Lenzmann [FL13] obtained the property at solitary waves for a family of nonlinear nonlocal equations, which we follow. The idea lies in to find a suitable substitute for the Sturm-Liouville theory to count the number of sign changes in eigenfunctions for a linear operator with a fractional Laplacian.

The present development may readily be adapted to other, nonlinear dispersive equations. We shall illustrate this in Section 5 by discussing equations of regularized long wave type. We shall remark in Section 6 about Lin’s approach [Lin08] to linear instability.

2. Existence of local constrained minimizers

We shall address the stability and instability mainly for the KdV equation with fractional dispersion

(2.1)

where and is defined via the Fourier transform as .

In the case of , notably, (2.1) recovers the KdV equation, and in the case of it corresponds to the Benjamin-Ono equation. In the case Note that is non singular for . of , furthermore, (2.1) was argued in [Hur12] to have relevances to surface water waves in two dimensions in the infinite depths. Observe that (2.1) is nonlocal for . Incidentally fractional powers of the Laplacian occur in numerous applications, such as dislocation dynamics in crystals (see [CDLFM07], for instance) and financial mathematics (see [CT04], for instance).

The present treatment extends mutatis mutandis to general power-law nonlinearities; see Remark 2.4. We focus on the quadratic nonlinearity, however, to simplify the exposition.

Throughout we’ll work in the -based Sobolev spaces over the periodic interval , where is fixed although at times it is treated as a free parameter. For let

We employ the standard notation for the -inner product.

Notice that (2.1) may be written in the Hamiltonian form

(2.2)

where is the symplectic form,

(2.3)

is the Hamiltonian and denotes variational differentiation; and correspond to the kinetic and potential energies, respectively. Notice that (2.1) possesses, in addition to , two conserved quantities

(2.4)
and
(2.5)

which correspond to the momentum and the mass, respectively. Conservation of implies that (2.1) is invariant under spatial translations thanks to Noether’s theorem while is a Casimir invariant of the flow induced by (2.1) and is associated with that the kernel of the symplectic form is spanned by a constant. Notice that

(2.6)

Moreover (2.1) remains invariant under

(2.7)

for any for any .

Remark 2.1 (Well-posedness).

In the range , one may work out the local in time well-posedness for (2.1) in , combining an a priori bound and a compactness argument. Without recourse to dispersive effects, the proof is identical to that for the inviscid Burgers equation, i.e., . We omit the detail.

With the help of techniques in nonlinear dispersive equations and specific properties of the equation, the global in time well-posedness for (2.1) may be established in in the case of , namely the KdV equation (see [CKS03], for instance), and in in the case of , the Benjamin-Ono equation (see [Mol08], for instance). For non-integer values of , however, the existence matter for (2.1) seems not adequately understood in spaces of low regularities. The global well-posedness in was recently settled in [KMR11] for (2.1), in the case of and in place of , , but the proof seems to break down in the periodic functions setting.

In what follows we shall work in a suitable subspace, say, of , where the initial value problem associated with (2.1) is well-posed for some interval of time and are smooth.

A periodic traveling wave of (2.1) takes the form , where represents the wave speed, is the spatial translate and is -periodic, satisfying by quadrature that

(2.8)

for some (in the sense of distributions). Equivalently, it arises as a critical point of

(2.9)

Indeed

(2.10)

agrees with (2.8).

Henceforth we shall write a periodic traveling wave of (2.1) as . In a more comprehensive description, it is specified by four parameters , and , . Note, however, that is arbitrary and fixed. Corresponding to translational invariance (see (2.7)), moreover, is inconsequential in the present development. Hence we may mod it out.

In the present notation, a solitary wave whose profile vanishes asymptotically corresponds, formally, to and .

In the case of , periodic traveling waves of (2.1), namely the KdV equation, are well known in closed form, involving Jacobi elliptic functions; see [KdV95], for instance. In the case of , moreover, Benjamin [Ben67] exploited the Poisson summation formula and explicitly calculated periodic traveling waves of (2.1). In general, the existence of periodic traveling waves of (2.1) follows from variational arguments, although one may lose an explicit form of the solution. In the energy subcritical case, in particular, a family of periodic traveling waves of (2.1) locally minimizes the Hamiltonian subject to conservations of the momentum and the mass, generalizing “ground states” in the solitary wave setting.

Proposition 2.2 (Existence, symmetry and regularity).

Let . A local minimizer for subject to that and are conserved exists in for each and it satisfies (2.8) for some and . It depends upon and in the manner.

Moreover may be chosen to be even and strictly decreasing over the interval , and .

Below we develop integral identities which a periodic solution of (2.8), or equivalently (2.10), a priori satisfies and which will be useful in various proofs.

Lemma 2.3 (Integral identities).

If satisfies (2.8), or equivalently (2.10), then

(2.11)
(2.12)
Proof.

Integrating (2.8), or equivalently (2.10), over the periodic interval leads to (2.11). Multiplying it by and integrating over lead to (2.12). ∎

Proof of Proposition 2.2.

We claim that it suffices to take and . Suppose on the contrary that . We then assume without loss of generality that and are of opposite sign and . For, in case and are of the same sign, since (2.1) is time reversible, we make the change of variables in (2.1) to reverse the sign of in (2.8) while leaving other components of the equation invariant. Once we accomplish that and are of opposite sign, must follow since and by definition. We shall then devise the change of variables and rewrite (2.8) as

(2.13)

Therefore it suffices to take in (2.8). This is reminiscent of that (2.1) enjoys Galilean invariance under for any . By virtue of scaling invariance (see (2.7)), we shall further devise the change of variables and rewrite (2.13) as

(2.14)

To recapitulate, it suffices to take and in (2.8) and seek a local minimizer for . (But we shall not a priori assume that or in the stability proof in Section 4.)

Since in the range is compactly embedded in by a Sobolev inequality, it follows from calculus of variations that for each parameter (abusing notation) Note from (2.12) that if , , satisfies (2.14) then and unless . there exists such that

(2.15)

The proof is rudimentary. We merely pause to remark that amounts to and the constraint is compact in . Moreover, satisfies

for some in the sense of distributions. By a scaling argument, we may choose to ensure that . Consequently (abusing notation) attains the constrained minimization problem (2.15) and satisfies (2.14). Note from (2.12) that .

Furthermore we claim that

(2.16)

Since

(2.17)

and whenever , , it suffices to show that

(2.18)

Suppose that , and . We define

and observe that (2.18) follows if so that . Indeed we infer from (2.17) that

Moreover, since and since attains the constrained minimization problem (2.15), it follows that

Consequently

whence . This proves the claim. Since

for all , furthermore, solves the constrained minimization problem (2.16) if and only if minimizes among its critical points. The existence assertion therefore follows. Clearly, depends upon and in the manner.

To proceed, since the symmetric decreasing rearrangement of does not increase for (see [Par11], for instance, for a proof in the solitary wave setting) while leaving invariant, it follows from the rearrangement argument that a local minimizer for subject to conservations of and must symmetrically decrease away from a point of principal elevation. The symmetry and monotonicity assertion then follows from translational invariance in (2.7). (Note that unlike in the solitary waves setting, for which and , a periodic, local constrained minimizer needs not be positive everywhere.)

It remains to address the smoothness of a periodic solution of (2.8), or equivalently,

(2.19)

after reduction to , and after inversion. The validity of (2.19) is to be specified in the course the proof. We claim that if satisfies (2.19) then . In the case of this follows immediately from a Sobolev inequality, whereas in the case of a proof based upon resolvent bounds for is found in [FL13, Lemma A.3], for instance, albeit in the solitary wave setting. Indeed, the Fourier series lies in for for by the Hausdorff-Young inequality, whence after iterating (2.19) sufficiently many times.

We then promote to since the Plancherel theorem leads to that

Furthermore the fractional product rule (see [CW91], for instance) leads to that

for a constant independent of . After iterations, therefore, follows. ∎

Remark 2.4 (Power-law nonlinearities).

One may rerun the arguments in the proof of Proposition 2.2 in the case of the general power-law nonlinearity

(2.20)

and obtain a periodic traveling wave, where and is an integer such that

(2.21)

It locally minimizes in the Hamiltonian

subject to conservations of and , defined in (2.4) and (2.5), respectively. Note that , which is vacuous if , ensures that (2.20) is -subcritical and compactly. In the case of , it is equivalent to that .

Remark 2.5 (Periodic vs. solitary waves).

In the non-periodic functions setting, Weinstein [Wei87] (see also [FL13]) proved that (2.8) in the range admits a solitary wave, for which and . In the case of so that (2.8) is -subcritical, the solitary wave further arises as an energy minimizer subject to the conservation of the momentum. Periodic, local constrained minimizers for (2.8), constructed in Proposition 2.2, are then expected to tend to the solitary wave as their period increases to infinity. This in some sense generalizes the homoclinic limit in the case of .

In the case of , on the other hand, local constrained minimizers for (2.8) exist in the periodic wave setting, but they are unlikely to achieve a limiting state with bounded energy (the -norm) as the period increases to infinity.

In the -critical case, i.e. , periodic traveling waves with small energy tend to the solitary wave as their period increases to infinity. Their stability is, however, delicate and outside the scope of the present development. We refer the reader to [KMR11], for instance.

For a broad range of dispersion operators and nonlinearities, including in (2.1), one is able to construct periodic traveling waves of (1.1) at least with small amplitudes from perturbation arguments such as the Lyapunov-Schmidt reduction; see [HJ14], for instance. In the solitary wave setting, in stark contrast, Pohozaev identities techniques dictate that (2.8) () in the range does not admit any nontrivial solutions in .

3. Nondegeneracy of the linearization

Throughout the section, let be a periodic traveling wave of (2.1), whose existence follows from Proposition 2.2. We shall examine the -null spaces of the linearizations associated with (2.8) and (2.1).

Proposition 3.1 (Nondegeneracy).

Let . If for some , and for some locally minimizes subject to that and are conserved then the associated linearized operator

(3.1)

acting on is nondegenerate. That is to say,

The nondegeneracy of the linearization associated with the traveling wave equation is of paramount importance in the stability of traveling waves and the blowup for the related, time evolution equation; see [Wei87, Lin08, KMR11], among others. To prove the property is far from being trivial, however. Actually, one may cook up a polynomial nonlinearity, say, , for which the kernel of at a periodic traveling wave is two dimensional at isolated points.

In the case of generalized KdV equations, for which in (2.1) but the nonlinearity is arbitrary, the nondegeneracy of the linearization at a periodic traveling wave was shown in [Joh09], for instance, to be equivalent to that the wave amplitude not be a critical point of the period; the proof uses the Sturm-Liouville theory for ODEs. Furthermore it was verified in [Kwo89], among others, at solitary waves (in all dimensions). Amick and Toland [AT91] demonstrated the property in the case of in (2.1), namely the Benjamin-Ono equation, in the periodic and solitary wave settings, by relating via complex analysis techniques the nonlocal, traveling wave equation to a fully nonlinear ODE; unfortunately, the arguments are specific to the equation. Angulo Pava and Natali [APN08] made an alternative proof based upon the theory of totally positive operators, but it necessitates an explicit form of the solution. A satisfactory understanding of the nondegeneracy of the linearization thus seems largely missing for nonlocal equations. The main obstruction is that shooting arguments and other ODE methods, which seem crucial in the arguments for local equations, may not be applicable.

Nevertheless, Frank and Lenzmann [FL13] recently obtained the nondegeneracy of the linearization at solitary waves for a family of nonlinear nonlocal equations with fractional derivatives. Their idea is to find a suitable substitute for the Sturm-Liouville theory to estimate the number of sign changes in eigenfunctions for a fractional Laplacian with potential. Our proof of Proposition 3.1 follows along the same line as the arguments in [FL13, Section 3], but with appropriate modifications to accommodate the periodic nature of the problem.

Lemma 3.2 (Oscillation of eigenfunctions).

Under the hypothesis of Proposition 3.1, an eigenfunction in corresponding to the -th eigenvalue of , , changes its sign at most times over the periodic interval .

We shall present the proof in Appendix A.

Remark 3.3 (Oscillation of higher eigenfunctions).

Lemma 3.2 holds for all . See [HJM15], where the proof and applications are studied.

Below we gather some facts about .

Lemma 3.4 (Properties of ).

Under the hypothesis of Proposition 3.1, the followings hold:

  • and it corresponds to the lowest eigenvalue of restricted to the sector of odd functions in ;

  • , where means the number of negative eigenvalues of acting on ;

  • .

Proof.

Differentiating (2.8) with respect to implies that . Moreover Proposition 2.2 implies that may be chosen to satisfy for . The lowest eigenvalue of acting on the sector of odd functions in , denoted , on the other hand, must be simple and the corresponding eigenfunction is strictly positive (or negative) over the half interval ; a proof based upon the Perron-Frobenious argument is rudimentary and hence we omit the detail. Therefore zero is the lowest eigenvalue of restricted to and is a corresponding eigenfunction.

To proceed, recall that belongs to the kernel of and attains zero twice over the periodic interval . Since an eigenfunction associated with the lowest eigenvalue of is strictly positive (or negative), acting on must have at least one negative eigenvalue.

Moreover, since locally minimizes , and hence , subject to conservations of and , necessarily,

(3.2)

This implies by Courant’s mini-max principle that has at most two negative eigenvalues, asserting (L2).

Lastly, differentiating (2.10) with respect to and , respectively, we use (2.6) to obtain that

(3.3)

Therefore . Incidentally

(3.4)

Since

by (2.8), moreover, . ∎

Proof of Proposition 3.1.

Consider the orthogonal decomposition

Since may be chosen to be even by Proposition 2.2, it follows that and are invariant subspaces of . Since (L1) of Lemma 3.4 implies that

moreover, it remains to show that .

Suppose on the contrary that there were a nontrivial function such that . Since has at most two negative eigenvalues by (L2) of Lemma 3.4, it follows from Lemma 3.2 that changes its sign at most twice over the half interval . Consequently, unless is positive (or negative) throughout the periodic interval , either there exists such that is positive (or negative) for and negative (or positive, respectively) for , or there exist in such that is positive for and (with the understanding that the first interval is empty in case ) and is negative for .

Since lies in the kernel of , on the other hand, it must be orthogonal to and, in turn, to the subspace by (L3) of Lemma 3.4. In particular, , whence cannot be positive (or negative) throughout . In case positive for and negative for , for instance, since is symmetrically decreasing away from the origin over the interval , we find that

Consequently , and cannot be orthogonal to . In case changes signs at and , where , correspondingly, we find that is positive in and negative in , deducing that cannot be orthogonal to . A contradiction therefore proves that . ∎

Remark 3.5 (Power-law nonlinearities).

One may rerun the arguments in the proof of Proposition 3.1 for (2.20) in the range and , where is in (2.21), and establish the nondegenracy of the linearization associated with the traveling wave equation at a periodic, local constrained minimizer, provided that

for changes its sign at and but nowhere else over the interval . Indeed in place of (L3) of Lemma 3.4, but otherwise the proof is identical to that in the case of the quadratic nonlinearity.

Unlike in the solitary wave setting, where at a ground state, may have up to two negative eigenvalues at a periodic, local constrained minimizer, which is characterized by

(3.5)

provided that

(3.6)

A proof based upon “an index formula” may be found in [BH14, Lemma 19].

Notice that (3.6) ensures that the mapping is of and locally invertible. Below we show that it further ensures that the generalized -null space of the linearized operator associated with (2.1) supports a Jordan block structure, which will play a central role in the stability proof in the subsequent section.

Proposition 3.6 (Jordan block structure).

Let . If for some , and for some locally minimizes subject to that and are conserved and if it satisfies (3.6) then zero is an -generalized eigenvalue of the linearized operator associated with (2.1),

(3.7)

with algebraic multiplicity three and geometric multiplicity two.

Proof.

The proof follows from the Fredholm alternative and may be found in [BH14, Lemma 6]; see [BJK11] in the case of generalized KdV equations. Here we merely hit the main points.

Differentiating (2.2) with respect to , , and , we use (2.6) to find that

Note from Proposition 2.2 that , are even and is odd. Since is at most two dimensional by Proposition 3.1, therefore, . By a duality argument and the Fredholm alternative, we then find that , where the dagger means adjoint. Thus, if then by the Fredholm alternative and (3.6), which, in turn, has no solution other than by the Fredholm alternative and (3.6). ∎

In the solitary wave setting, zero is an -eigenvalue of with algebraic multiplicity two and geometric multiplicity one, provided that at the underlying wave. Incidentally a solitary wave corresponds to and in (2.8), and hence it depends, up to spatial translations, merely upon the wave speed. Proposition 3.6 therefore indicates that (3.6) is a natural analogue in the periodic wave setting of the familiar condition in the solitary wave setting.

In the case of , namely the Benjamin-Ono equation, (3.6) holds for all periodic traveling waves; see [BH14, Section 3.4] for the detail. Concluding the section we shall verify (3.6) in the solitary wave limit.

Lemma 3.7 (Solitary wave limit).

Let . If locally minimizes in subject to that and are conserved for some , and then

for sufficiently small and sufficiently large.

Proof.

The proof may be found in [BH14, Lemma 20]. Here we include the detail for completeness.

We recall from Remark 2.5 that in the range periodic traveling waves of (2.1), constructed in Proposition 2.2 as local constrained minimizers, tend to the solitary wave as and satisfying , namely in the solitary wave limit, which minimizes the Hamiltonian subject to the conservation of the momentum. It follows from (2.7) that (2.8) remains invariant under

Accordingly we may take without loss of generality and we find that

for sufficiently small and sufficiently large; see [BH14, Lemma 3.10] for the detail. Differentiating (2.11) with respect to and evaluating near the solitary wave limit, we use (3.4) to obtain that

for sufficiently small and sufficiently large. Since an explicit calculation dictates that , furthermore,

for sufficiently small and sufficiently large. ∎

4. Stability of constrained energy minimizers

We turn the attention to the stability of a periodic, local constrained minimizer for (2.8) with respect to period preserving perturbations.

Recall from Section 2 that the initial value problem associated with (2.1) is well-posed in for some interval of time, where are smooth. It suffices to take , .

Throughout the section let , fixed, and let locally minimize subject to that and are conserved for some , and for some . In light of Proposition 2.2, and it makes a -periodic, traveling wave of (2.1).

Notice that the evolution of (2.1) remains invariant under a one-parameter group of isometries corresponding to spatial translations. This motivates us to define the group orbit of as

Roughly speaking, is said orbitally stable if a solution of (2.1) remains close to under the norm of for all future times whenever the initial datum is sufficiently close to the group orbit of under the norm of . We shall elaborate this below in Theorem 4.1.

The present account of orbital stability is inspired by the Lyapunov method. Let

(4.1)

Proposition 2.2 implies that , i.e., is a critical point of . Moreover, Proposition 3.1 implies that the kernel of is spanned by . Intuitively, is expected to be orbitally stable if is “convex” at . As a matter of fact, one may easily verify that if the spectrum of , except the simple eigenvalue at the origin generated by translation invariance, were positive and bounded away from zero then would indeed be orbitally stable.

However, (L2) of Lemma 3.4 indicates that admits one or two negative eigenvalues and one zero eigenvalue. In other words, is a degenerate saddle point of on . The Lyapunov method therefore may not be directly applicable. In order to control potentially unstable directions and achieve stability, nevertheless, observe that the evolution under (2.1) does not take place in the entire space , but rather on a smooth submanifold of co-dimension two, along which the momentum and the mass are conserved. Specifically let

where

(4.2)

Note that and a solution of (2.1) with initial datum in remains in at all future times. We shall then demonstrate the “convexity” of on , provided that the associated linearization admits a Jordan block structure.

Theorem 4.1 (Orbital stability).

Let . If for some , and for some locally minimizes subject to that and are conserved and if the matrix

(4.3)