In this paper, we intend to continue the study on nonlinear Schrödinger systems for saturated optical materials that was recently initiated by Maia, Montefusco and Pellacci . In their paper, the system of elliptic partial differential equations
was suggested in order to model the interaction of two pulses within the optical material under investigation. Here, the parameters satisfy and . One way to find classical fully nontrivial solutions of (1.1) is to use variational methods. The Euler functional associated to (1.1) is given by
where and for all . The symbol denotes the standard norm on and the norms , are defined via
Since we are interested in minimal energy solutions (that is, ground states) for (1.1), the ground states of the scalar problems associated to (1.1) turn out to be of particular importance. These are positive radially symmetric and radially decreasing smooth functions satisfying
Since we will encounter these solutions many times, let us recall some facts from the literature. The existence of positive finite energy solutions of (1.3) for parameters and can be deduced from [19, Theorem 2.2] for or from [5, Theorem 1 (i)] for , respectively. In the case , the positive functions are given by , for all and
where are uniquely determined by
As in the explicit one-dimensional case, it is known also in the higher-dimensional case that are radially symmetric, see [6, Theorem 2]. Finally, the uniqueness of follows from [17, Theorem 1] in the case and from [12, Theorem 1] in the case . The uniqueness result for is a direct consequence of the existence proof we gave above.
In this paper, we strengthen the results obtained by Maia, Montefusco and Pellacci  concerning ground state solutions and (component-wise) positive solutions of (1.1), so let us shortly comment on their achievements. In Theorem 3.7 of their paper, they proved the existence of nonnegative radially symmetric and nonincreasing ground state solutions of (1.1) for all and for parameter values , where the upper bound for s is in fact optimal by Lemma 3.2 in the same paper. It was conjectured that each of these ground states is semitrivial except for the special case , , where the totality of ground state solutions is known in a somehow explicit way, see [10, Theorem 2.1] or Theorem 1.1 (i) below. In , this conjecture was proved for parameters , see Theorem 3.15 and Theorem 3.17 therein. Our first result shows that the full conjecture is true even in the case , which was not considered in .
Let , and . Then, the following holds.
The proof of this result will we presented in Section 2. Our approach is based on a suitable min-max characterization of the mountain pass level associated to (1.1) involving a fibering map technique as in . This method even allows to give an alternative proof for the existence of a ground state solution of (1.1) which is significantly shorter than the one presented in  and which, moreover, incorporates the case , see Proposition 2.1. More importantly, this approach yields the optimal result.
In view of Theorem 1.1, it is natural to ask how the existence of fully nontrivial solutions of (1.1) can be proved. In , Maia, Montefusco and Pellacci found necessary conditions and sufficient conditions for the existence of positive solutions of (1.1) which, however, partly contradict each other. For instance, [10, Theorem 3.21] claims that positive solutions exist for parameters , and sufficiently small contradicting the nonexistence result from [10, Theorem 3.10]. The error leading to this contradiction is located on [10, p. 338, l. 13], where the number must be replaced by , which makes the results from Theorem 3.19 and Theorem 3.21 in that paper break down. Our approach to finding positive solutions and, more generally, seminodal solutions of (1.1) is to apply bifurcation theory to the semitrivial solution branches
which was motivated by the papers of Ostrovskaya and Kivshar  and Champneys and Yang . In the case and , , , they numerically detected a large number of solution branches emanating from containing seminodal solutions. Moreover, they conjectured that the bifurcation points on accumulate near , see [3, p. 2184 ff.]. Our results confirm these observations. For simplicity, we will only discuss the bifurcations from since the corresponding analysis for is the same up to interchanging the roles of and . Investigating the linearized problems associated to (1.1) near for parameters close to the boundary of the parameter interval , we prove the existence of infinitely many bifurcating branches containing fully nontrivial solutions of a certain nodal pattern. Despite the fact that the question whether fully nontrivial solutions bifurcate from makes perfect sense for all space dimensions , our bifurcation result is restricted to . Later, we will comment on this issue in more detail, see Remark 3.6. In order to formulate our bifurcation result, let us define the positive numbers to be the k-th eigenvalues of the linear compact self-adjoint operators mapping to itself, where denotes the positive ground state solution of the first equation in (1.3) for . By Sturm–Liouville theory, we know that these eigenvalues are simple and that they satisfy
Deferring some more or less standard notational conventions to a later stage, we come to the statement of our result.
Let and let and satisfy
Then, there is an increasing sequence of positive numbers converging to such that continua containing -nodal solutions of (1.1) emanate from at for all . In the case , we necessarily have and there is a such that all positive solutions with satisfy
In the case , we can estimate from above in order to obtain a sufficient condition for the conclusions of Theorem 1.2 to hold for . This estimate, which leads to Corollary 1.3, is based on the Courant–Fischer min-max principle and Hölder’s inequality. In the one-dimensional case, the values of all eigenvalues are explicitly known, which results in Corollary 1.4.
Let . Then, the conclusions of Theorem 1.2 are true for if
Let . Then, the conclusions of Theorem 1.2 are true in the case
As we mentioned above, one can find sufficient criteria for the existence of -nodal solutions bifurcating from by reversing the roles of and in the statement of Theorem 1.2 as well as in its corollaries.
Theorem 1.2 gives rise to many questions which would be interesting to solve in the future. A list of open problems is provided in Section 5. Before going on with the proof of our results, let us clarify the notation which we used in Theorem 1.2. The set is the closure of all solutions of (1.1) which do not belong to and a subset of is called a continuum if it is a maximal connected set within . Finally, a fully nontrivial solution of (1.1) is called -nodal if both component functions are radially symmetric and u has precisely nodal annuli and v has precisely nodal annuli. In other words, since double zeros cannot occur, is -nodal if the radial profiles of u, respectively v, have precisely k, respectively l, zeros.
2 Proof of Theorem 1.1
According to the assumptions of Theorem 1.1, we will assume throughout this section that the numbers are positive, that s lies between and , and that the space dimension is an arbitrary natural number. Furthermore, we define the energy levels
The first step towards the proof of Theorem 1.1 is a more suitable min-max characterization of the least energy level of (1.1) which, as in , gives rise to a simple proof for the existence of a ground state. To this end, we introduce the Nehari manifold
is attained at a radially symmetric and radially nonincreasing ground state of (1.1).
so that holds for if and only if . Since β is smooth and strictly concave with , a critical point of β is uniquely determined and it is a maximizer (whenever it exists). Since the supremum of β is , when there is no maximizer of β we obtain
which proves (2.1).
Due to , we can find a semitrivial function satisfying
which implies that according to (2.1). So, let be a minimizing sequence in satisfying as . Using the classical Polya–Szegő inequality and the extended Hardy–Littlewood inequality
for the spherical rearrangement taken from [1, Theorem 2.2], we may assume to be radially symmetric and radially decreasing. Since the function strictly increases on from to , we may moreover assume that are rescaled in such a way that the equality
holds for . The inequality
implies that the sequence is bounded in . Using the uniform decay rate and the resulting compactness properties of radially decreasing functions bounded in (apply, for instance, [18, Compactness Lemma 2]), we may take a subsequence, again denoted by , such that in pointwise always everywhere and
From this we infer that
and, thus, . Hence, for all , we obtain
so that is a nontrivial radially symmetric and radially decreasing minimizer. Taking for r the maximizer of the map , we obtain the ground state solution having the properties we claimed to hold. Indeed, the Nehari manifold may be rewritten as
so that the Lagrange multiplier rule applies due to
for all . ∎
Proof of Theorem 1.1.
Part (i) was proved in [10, Lemma 3.2], so let us prove (ii). First, we show that the ground state energy level equals . Since we have by definition, we have to show that
To this end, let be arbitrary but fixed. From (2.3) we infer that the numbers
satisfy and as well as
The concavity of the logarithm yields
Combining this inequality with (2.4) gives
Taking the supremum with respect to , gives (2.2) and, therefore, , which is what we had to show.
It remains to prove that every ground state is semitrivial unless , . To this end, assume that is a fully nontrivial ground state solution of (1.1), so that in particular holds. Then, implies that the inequalities above are equalities for some . In particular, since the logarithm is strictly concave and , we get
for some . This implies that , so that have to be positive multiples of each other. From the Euler–Lagrange equation (1.1) we deduce that and , which finishes the proof. ∎
3 Proof of Theorem 1.2
In this section, we assume as before but the space dimension n is supposed to be or 3. In Remark 3.6, we will comment on the reason for this restriction. Let us first provide the functional analytic framework we will be working in. In the case , we set to be the product of the radially symmetric functions in and define by
Hence, finding solutions of (1.1) is equivalent to finding zeros of F. Using the compactness of the embeddings for and , one can check that the function is a smooth compact perturbation of the identity in X for all s, so that the Krasnosel’skii–Rabinowitz global bifurcation theorem [9, 15] is applicable. In the case , however, this structural property is not satisfied, which motivates a different choice for X. In Appendix A, we show that one can define a suitable Hilbert space X of exponentially decreasing functions such that is again a smooth compact perturbation of the identity in X. Except for this technical inconvenience, the case can be treated in a similar way to the case , so we carry out the proofs for the latter case only. Furthermore, we always assume that according to the assumption of Theorem 1.2.
The first step in our bifurcation analysis is to investigate the linearized problems associated to the equation around the elements of the semitrivial solution branch . While doing this, we make use of a nondegeneracy result for ground states of semilinear problems which is due to Bates and Shi . Amongst other things, it tells us that is a nondegenerate solution of the first equation in (1.3), that is, we have the following result.
The linear problem
only admits the trivial solution .
In order to apply [2, Theorem 5.4 (6)], we set
so that is the ground state solution of in which is centered at the origin. In the notation of , one can check that g is of class (A). Indeed, the properties (g1), (g2), (g3A), (g4A), (g5A) from [2, p. 258] are satisfied for
and the unique positive number satisfying
Notice that (g4A), (g5A) follow from the fact that decreases from 1 to on the interval and that it decreases from to on . Having checked the assumptions of [2, Theorem 5.4 (6)], we obtain that the space of solutions of in is spanned by , implying that the linear problem only has the trivial solution in . Due to
this proves the claim. ∎
Using this preliminary result, we can characterize all possible bifurcation points on which are, due to the implicit function theorem, the points where the kernel of the linearized operator is nontrivial. For notational purposes, we introduce the linear compact self-adjoint operator for parameters by setting
for . Denoting by the decreasing null sequence of eigenvalues of , we will observe that finding bifurcation points on amounts to solving for and . In fact, we have the following result.
For , we have
Plugging in , and gives
From these formulas and Proposition 3.1, we deduce the claim. ∎
Given this result, our aim is to find sufficient conditions for the equation to be solvable. Since there is only few information available for any given , our approach consists of proving the continuity of and calculating the limits of as s approaches the boundary of . It will turn out that the limits at both sides of the interval exist and that they lie on opposite sides of the value 1 provided our sufficient conditions from Theorem 1.2 are satisfied. As a consequence, these conditions and the intermediate value theorem imply the solvability of and it remains to add some technical arguments in order to apply the Krasnosel’skii–Rabinowitz global bifurcation theorem to prove Theorem 1.2. Calculating the limits of at the ends of requires Proposition 3.3 and Proposition 3.4.
where the convergence is uniform in .
As in Lemma A.1 in Appendix A, one shows that on every interval with , there is an exponentially decreasing function which bounds each of the functions with from above. In particular, the Arzelà–Ascoli theorem shows that and as locally uniformly in , so that the uniform exponential decay gives and uniformly in . ∎
where the convergence is uniform on bounded sets in .
First, we show that
Otherwise, we would observe that for some subsequence, where . In the case , a combination of elliptic regularity theory for (1.3) and the Arzelà–Ascoli theorem would imply that converges locally uniformly to a nontrivial radially symmetric function satisfying
in the weak sense and . As in Lemma A.1, we conclude that the functions are uniformly exponentially decaying, so that u even lies in . Hence, we may test the differential equation with u and obtain
which is impossible. It therefore remains to exclude the case . In this case, the functions would converge uniformly in to the trivial solution, implying that would converge to a nonnegative bounded function satisfying in and . Hence, ϕ is smooth, so that Liouville’s theorem applied to the function defined on implies that ϕ is constant and, thus, , contradicting . This proves (3.3).
Now, set . Using
and the fact that remains bounded as , we get that the functions converge locally uniformly as to some nonnegative radially nonincreasing function satisfying . In order to prove our claim, it is sufficient to show that , since this implies locally uniformly and, in particular, locally uniformly.
First, we show that . If this were not true, then there would exist a number such that and for all . In , we have and implies in and , in contradiction to the maximum principle. Hence, we must have in . Repeating the above argument, we find in and , so that Liouville’s theorem implies . ∎
The previous propositions enable us to calculate the limits of the eigenvalue functions as s approaches the boundary of .
For all , the functions are positive and continuous on . Moreover, we have
As in Proposition 3.3, the uniform exponential decay of the functions for for implies that , uniformly in whenever . Hence, the Courant–Fischer min-max characterization for the eigenvalues implies the continuity of as well as as .
In order to evaluate for , we apply Lemma C.1 from Appendix C. The conditions (i) and (ii) of the lemma are satisfied since we have and locally uniformly as by Proposition 3.4. From the lemma we get as , which is all we had to show. ∎
When , the statement of Proposition 3.3 is not meaningful since does not exist in this case by Pohožaev’s identity. So, it is natural to ask how and behave when s approaches zero and . Having found an answer to this question, it might be possible to modify our reasoning in order to prove sufficient conditions for the existence of bifurcation points from in the case .
The above propositions are sufficient for proving the mere existence of the continua from Theorem 1.2. So, it remains to show that positive solutions lie to the left of the threshold value and that they are equibounded in X. The latter result will be proved in Lemma A.1 whereas the first claim follows from the following nonexistence result which slightly improves [10, Theorem 3.10 and Theorem 3.11].
If positive solutions of (1.1) exist, then we either have
Hence, the function vanishes identically or it changes sign in . In the first case, we get (i), so let us assume that the function changes sign. Then, we have and , so that [10, Theorem 3.11 and Remark 3.18] imply that
Moreover, would imply that
contradicting the assumption that changes sign. Hence, we have , which concludes the proof. ∎
Proof of Theorem 1.2.
The main ingredient of our proof is the Krasnosel’skii–Rabinowitz global bifurcation theorem (cf. [9, 15] or [8, Theorem II.3.3]) which, roughly speaking, says that a change of the Leray–Schauder index along a given solution curve over some parameter interval implies the existence of a bifurcating continuum emanating from the solution curve within this parameter interval. In our application, the solution curve is and the first task is to identify parameter intervals within where the index changes. For notational purposes, we set .
The continuity of the eigenvalue functions on as well as the fact that for all , , therefore implies that for the numbers given by
By the definition of , we can find such that the following inequalities hold:
In fact, one first chooses such that (ii) is satisfied and then sufficiently close to such that (i) and (iii) hold.
Now, let us show that the Leray–Schauder index changes sign on each of the mutually disjoint intervals . The index of near is computed using the Leray–Schauder formula which involves the algebraic multiplicities of the eigenvalues of the compact linear operator , see [8, (II.2.11)]. From the formulas appearing in Proposition 3.2 we find that is such an eigenvalue if and only if one of the equations
is solvable. If , then the second equation is solvable with if and only if μ is an eigenvalue of larger than 1. By (3.4) (i), this is equivalent to . Due to Sturm–Liouville theory, each of these eigenvalues is simple. The first equation is solvable with if and only if has a negative eigenvalue in , where g is defined as in (3.2). From [2, Theorem 5.4 (4)–(6)] we infer that there is precisely one such eigenvalue and μ has algebraic multiplicity one. Denoting the spectrum with σ, we arrive at the formula
The Krasnosel’skii–Rabinowitz theorem implies that the interval contains at least one bifurcation point , so that the maximal component in satisfying is nonempty. By Proposition 3.2, this implies for some and (3.4) implies , that is, . Indeed, property (ii) gives and (i) gives .
Step 2. as . If the claim did not hold, then we would have from below for some . From , the inequality and the definition of , we deduce that whenever , , and, thus,
for some sufficiently large . This contradicts as for all and the claim is proved.
Step 3. Existence of Seminodal Solutions within . We briefly show that fully nontrivial solutions of (1.1) belonging to a sufficiently small neighbourhood of are -nodal. Indeed, if solutions of (1.1) converge to , then converges to the eigenfunction ϕ of with which is associated to the eigenvalue 1. Due to the fact that and Sturm–Liouville theory, ϕ has precisely nodal annuli, so that the same is true for and sufficiently large . On the other hand, the convergence implies that must be positive for large m, which proves the claim.
Let be the unique positive function which satisfies in , so that can be rewritten as
In the case , we have and it is known (see, for instance, [4, Lemma 5.1]) that the eigenvalue problem in admits nontrivial solutions in if and only if for some . This implies that
5 Open Problems
Let us finally summarize some open problems concerning (1.1) which we were not able to solve and which we believe provide a better understanding of the equation. Especially the open questions concerning global bifurcation scenarios are supposed to be very difficult from the analytical point of view so that numerical indications would be very helpful, too. The following questions might be of interest.
As in the author’s work on weakly coupled nonlinear Schrödinger systems , one could try to prove the existence of positive solutions by minimizing the Euler functional over the “system Nehari manifold” consisting of all fully nontrivial functions which satisfy . For which parameter values are there such minimizers and do they belong to ?
What is the existence theory and the bifurcation scenario when and , ?
In the case , , the points on are connected by a smooth curve and the same is true for every semitrivial solution. Do these connections break up when the parameters of the equation are perturbed? This is related to the question whether the continuum contains .
It would be interesting to know if the eigenvalue functions are strictly monotone. The monotonicity of would imply that are the only solutions of so that the totality of bifurcation points is given by .
We expect that extend to semitrivial solution branches containing also negative parameter values s. A bifurcation analysis for such branches remains open. Let us shortly comment on why we expect an interesting outcome from such a study. In the model case and , one obtains from (1.4) the existence of for all as well as the a priori information . Using this, one successively proves that and as . This implies that as , so that one expects that as for all . In view of , this leads to the natural conjecture that there are also infinitely many bifurcating branches in the parameter range .
Our paper does not contain any existence result for fully nontrivial solutions when and or . It would be interesting to know whether there is such a nonexistence result.
A A Priori Bounds
In our proof of the a priori bounds for positive solutions of (1.1), we will use the notation and , , so that denote the radial profiles of . Notice that all nonnegative solutions are radially symmetric and radially decreasing by [10, Lemma 3.8]. We want to highlight the fact that the main ideas leading to Lemma A.1 are taken from [7, Section 2].
Let . For all , there are such that all nonnegative solutions of (1.1) for and satisfy
We will break the proof into three steps.
Step 1. Boundedness in . Assume that there is an sequence of nonnegative solutions of (1.1) for parameters and which is unbounded in . As always, we write . Passing to a subsequence, we may assume that and for some and . Let us distinguish the cases and to lead this assumption to a contradiction.
For the case , the functions
are bounded in and satisfy as well as
Using the fact that and De Giorgi–Nash–Moser estimates, we obtain from the Arzelà–Ascoli theorem that there are bounded nonnegative radially symmetric limit functions satisfying and
From and we obtain
and for some . Since the functions are nonnegative, this is only possible in the case , which contradicts . Hence, the case does not occur.
For the case , we first show that uniformly on which, due to the fact that , is equivalent to proving that . So, let κ be an arbitrary accumulation point of the sequence and without loss of generality we assume that , so that we are left to show that . To this end, set
The functions satisfy as well as
The Arzelà–Ascoli theorem implies that a subsequence converges locally uniformly to nonnegative functions satisfying and
For the case , we arrive at a contradiction as in the case , so let us assume that . Then, is nonnegative, nontrivial and the inequality implies that
where . From [14, Theorem 8.4] we infer that and, thus, . Hence, every accumulation point of the sequence is zero, so that converges to the trivial function uniformly on .
With this result at hand, one can use the classical blow-up technique by considering
These functions satisfy as well as
Then, we have uniformly in and similar arguments as the ones used above lead to a bounded nonnegative nontrivial solution of
which we may lead to a contradiction as above. This finally shows that is also impossible in the case , so that the nonnegative solutions of (1.1) are pointwise bounded by some constant depending on ε.
Step 2. Uniform Exponential Decay. Let us assume for contradiction that there is a sequence of positive solutions of (1.1) satisfying
Due to the -bounds for which we proved in the first step, we can use De Giorgi–Nash–Moser estimates and the Arzelà–Ascoli theorem to obtain a smooth bounded radially symmetric limit function of a suitable subsequence of . As a limit of positive radially decreasing functions, are also nonnegative and radially nonincreasing. In particular, we may define
Our first aim is to show that . Since decreases to some limit at infinity, we have , as , so that (1.1) implies that
The differential equation implies that
so that decreases to some limit at infinity. The monotonicity of and the fact that as imply that this limit must be . In particular, we obtain that and the pointwise convergence implies that E is a nonnegative nonincreasing function. From this we obtain that
This equation implies that and, hence, .
Now, let μ satisfy and choose . Due to the fact that , we may choose such that holds. From , and the fact that are decreasing, we obtain that for all and all for some sufficiently large . Having chosen sufficiently small, the inequality implies that
Hence, the maximum principle implies that for any given , the function satisfies on . Indeed, dominates on the boundary of due to the fact that
Sending R to infinity, we obtain that
which, together with the a priori bounds from the first step, yields a contradiction to the assumption (A.1). This proves the uniform exponential decay.
Step 3. Conclusion. Given the uniform exponential decay of , we obtain a uniform bound on , which, using the differential equation (1.1), gives a uniform bound on . This finishes the proof. ∎
Let us mention that in view of Proposition 3.4, the a priori bounds from the above lemma cannot be extended to the interval . Furthermore, positive solutions of (1.1) are not uniformly bounded for parameters s belonging to neighbourhoods of when , see Remark 3.6. Notice that the assumption in the proof of the above lemma only becomes important when we apply [14, Theorem 8.4].
B A Functional Analytic Setting for
In this section, we show that in the one-dimensional case, the function given by (3.1) is a compact perturbation of the identity for an appropriately chosen Banach space X such that are continuous curves in . Let be fixed and set to be the Hilbert space given by
where and . One may check that is a Hilbert space and the subspace consisting of smooth even functions having compact support is dense in X. We will use the formula
for all and , where
Proof of Well-Definedness. First, let us prove for all the estimate
It suffices to prove these inequalities for . For such functions, we have
Next, using that and the fact that have compact support, we obtain
Then, performing the analogous rearrangements for v, yields for all that
Applying this inequality to for and a suitable family of cut-off functions converging to 1, we obtain
and using the estimate (B.2), we find that there is a positive number C depending on but not on such that
By the density of in X, this inequality also holds for . If now is a sequence in converging to , then similar estimates based on (B.2) show that
for some , implying that is well defined and that (B.4) also holds for .
Proof of Compactness of . Let now be a bounded sequence in X. Then, without loss of generality, we can assume that and pointwise almost everywhere. We set
where and . Then, we have and pointwise almost everywhere and the estimate (B.2) implies that
for some positive number . Using the estimate from above, we therefore obtain that
which is all we had to show.
C A Spectral Theoretic Result
Finally, we prove a spectral theoretical result which we used in the proof of Proposition 3.5 and for which we could not find a reference in the literature. The key ingredient of this result is the min-max principle for eigenvalues of semibounded self-adjoint Schrödinger operators, see, for instance, [16, Theorem XIII.2]. As in Proposition 3.5, we denote by , , the k-th eigenvalue of the compact self-adjoint operator
for potentials vanishing at infinity, that is, as .
Let , , and let be a family of radially symmetric potentials vanishing at infinity and satisfying
Then, we have as for all .
The min-max principle and (i) imply that
So, it remains to show the corresponding estimate from below. Given the assumptions and (ii), we find that it is sufficient to show that as , where denotes the k-th eigenvalue of the compact self-adjoint operator defined by . Here, denotes the indicator function of the ball in centered at the origin with radius . Since is continuous on with respect to the operator norm, the min-max characterization of the eigenvalues implies that the mapping
is continuous on , where
By the definition of , the boundary value problem
for has a nontrivial solution. Testing the differential equation on with ϕ, we obtain that . Hence, ϕ is given by
for some . Here, K denotes the modified Bessel function of the second kind and J represents the Bessel function of the first kind. From we get the conditions
on c and . Due to the continuity of on and due to the fact that K is positive whereas J has infinitely many zeros going off to infinity, we infer that is bounded on . In particular, this gives that and, thus, as , which is all we had to show. ∎
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About the article
Published Online: 2015-12-02
Published in Print: 2016-02-01
Funding Source: Deutsche Forschungsgemeinschaft
Award identifier / Grant number: MA 6290/2-1
This project was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant MA 6290/2-1.