Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics

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Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion. Developed in the s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail.

This work unfolds systematically in four parts, interweaving theory and applications. The case studies examined in Part III illustrate the impact of reduction techniques, and may serve as prototypes for future new applications. In the same spirit, most chapters include a problem section. Background results and solutions to selected problems close the volume. This book can be used as a text in graduate courses in pure or applied analysis, or as a resource for researchers working with singularities in geometry and mathematical physics.

The book under review provides a careful and instructive introduction into this method. At the end of most of the chapters some problems are posed, which are solved in an appendix. In total this is a highly interesting book containing a lot of original ideas and which suggests new developments. Steinbauer, Monatshefte fur Mathematik, Vol. Help Centre. Track My Order. Stress, stability, and chaos in structural engineering: an energy approach. El Naschie. Dynamic System Identification.

Experiment Design and Data Analysis. Mathematics in Science and Engineering, Volume English. Goodwin Editor Robert L. Payne Editor. Barnett , Michael R. Ziegler , Karl E. Steel , James H. Problems 7. Study in particular the following cases: 7. For background information on general relativity, see Chap. We focus on the mathematical issues addressed by reduction methods in applications to general relativity.

The following is by no means a complete review of cosmological models current today among physicists, nor a suggestion that this particular class of cosmological models is better or worse than others. This would establish the stability of the big bang. The pj are called Kasner exponents. The question whether singularities could be destroyed by perturbation remained unanswered. The situation changed with the Hawking—Penrose singularity theorems [78], which assert that some kind of singular behavior geodesic incompleteness must occur in cosmological solutions under a sign condition on the Ricci tensor.

However, the singularity theorems, by their very generality, give no analytic information on what actually happens at this singularity. For this reason, the techniques developed in proving that result have not been readily extended to more general families. It is not always easy to be sure, in numerical computations, that the constraint equations do hold. In particular, one can identify the location of the spikes, after discounting those related to a bad choice of coordinates, with the critical points of the function X0 x ; see Sect.

We now present these results.

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Gowdy T3 space-times [70] have been extensively studied over the years [89, 48, 47, 18, ]. The system is equivalent to the vanishing of the Ricci tensor of this metric. The last two equations arise respectively from the momentum and Hamiltonian constraints. For extensive references on Gowdy space-times, see [47, 49, 18, 74]. If X vanishes identically, the equation for Z reduces to a linear Euler— Poisson—Darboux equation, which can be solved explicitly. Both sets of equations 8. Numerical computations suggest more complicated behavior in the full nonlinearsystem for X and Z [18].

Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics

They may persist longer at higher resolutions. Solutions such that v 0, given that numerical computations do not give information on the sign of k? We therefore need to perform a reduction. We have four arbitrary functions in these asymptotics, which is reasonable for a set of two equations of second order.

We prove in the next section that u and v solve a Fuchsian system. The equations will involve variable powers of t. Thus, we achieved the desired reduction; it turns out that we directly obtain the second reduced equation in this manner. The results, Theorems 8. It includes both the x-independent solutions and the polarized solutions; this explains why these cases do not lead to a restriction on k.

In fact, one can generate solutions with negative k from solutions with positive k. In such cases, one can patch local solutions obtained from several local charts in hyperbolic space. As is clear from the above equations, these terms disappear precisely if X0 is a constant i.

We obtain solutions of 8. Let us check that these solutions generate solutions u0 and v0 of the original Gowdy system. The same argument applies to v. The computations for the case k 8. Then there exists a unique solution of the form 8. By taking m large enough, we may therefore assume that we have a system to which Theorem 4. These spaces admit an abelian isometry group with spacelike generators, in which, unlike the Gowdy case, the Killing vectors have a nonvanishing twist. One then proves that the remaining constraints hold everywhere if they hold asymptotically at the singularity.

Once these three functions are known, the other equations can be solved easily. Equations 8. In contrast to the Gowdy case, the wave equation 8. We therefore take 8. Also, since equation 8. This is not an immediate consequence of standard results because we are not using any of the standard setups for the initial-value problem.

It nevertheless does hold; see Problem 8. As far as the number of singularity data is concerned, observe that the initial data for 8. Similarly, we will obtain a family of singular solutions of 8. Step B: Leading-order asymptotics. For 8. It is natural that there should be four singularity data, given that equations 8. Constraint 8. These asymptotics may be compared with those of the solutions obtained in the Gowdy case. The asymptotics 8. Step G: Asymptotics determine a unique solution.

The main result is the following Theorem: Theorem 8. Each of these solutions generates space-times with AVD asymptotics. We conclude from Theorem 4. Indeed, 8.

We achieve this using This means, using This completes the proof. Problems 8. Solve the Gowdy equations for nonanalytic data using the methods of Chap. Apply the argument of Section Solve the initial-value problem for 8. Obtain solutions to 8. The equations need to be split in a way that does not correspond to the one suggested by the standard approach to the nonanalytic Cauchy problem in general relativity for the latter, see [].

Most results are taken from []; Theorem 9. After various choices of local coordinates, they were led to the following PDE problem: Let xi denote local coordinates on M , where indices i, j, etc. We solve this problem by determining the degree of nonuniqueness of the series, and by proving its convergence in all cases. Theorem 9. More precisely [], we have the following result: Theorem 9.

Equation 9. Equations 9. These results have the following consequences. Despite the presence of logarithms, the metric admits of local uniformization by the introduction of variables of the form r ln r k. Remark 9. The observation that 9. The question solved here goes back further, in particular to the work of Schouten and Haantjes [], who investigated formal solutions in integer powers when the determinant of h is normalized to unity.

Logarithms, or convergence issues, were not considered by them. We now describe the strategy carried out in the following sections. As a result, there is a unique local solution that is holomorphic in r and r ln r. The conclusions of the theorem follow. For background information on Riemannian geometry, see []. Local coordinates on G We collect some information on g x0 ,.

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Recall that indices i, j, etc. Indices a, b, etc. Greek indices take only the values 0 and m. The usual conventions of Riemannian geometry, including the summation convention on repeated indices, are used throughout. The special form of the metric ensures that 1. This will be an easy consequence of a reduction to a Fuchsian system, 9. It is not convenient to work with variables of the form rp ln r, because it makes keeping track of the degree of homogeneity of the terms in the series more cumbersome. If desired, one could, after the solution has been obtained, see whether it can be rearranged as a series in r and rp ln r for some p.

The structure of formal solutions to 9. Equating the righthand side of 9. As a consequence of 9. Using the decomposition 9. One then applies Theorem 2. Coming back to the proof of Theorem 9. Next, substitute 9. This produces a new generalized Fuchsian system in which the indices all have positive real parts. This achieves the second reduction. This system falls within the scope of Theorem 4. It yields the desired u k. We now implement this program. Substituting into 9. System 9. Therefore the above system has a unique holomorphic solution v. It remains to check that this solution does provide a solution to the original problem.

We have therefore proved that the formal solutions do converge, QED. More precisely, it follows from 9. It is in fact somewhat simpler here, see Section 9. Using Remark 9. As was pointed out in [60], in case n is odd and no fractional powers are allowed, equation 9.

This would generate a logarithmic term in 9. Lemma 9. In one space dimension, the solutions of 9. Furthermore, G Proof. The solution is determined by a single function u x, r , which represents the only component of gij. The form of u follows. One then checks directly that 9.

Now, G its curvature tensor vanishes as well. Adding the two gives us a particular solution. The general solution is obtained by adding to it a solution of the Cauchy—Riemann equations. For the corresponding result in two dimensions, see Problem 9. The main result is the following: Theorem 9. It follows from Theorem 9. The number n plays a double role: it determines the space dimension as well as the nonlinearity. It is the latter that is essential here. Furthermore, 9. This was the motivation of Loewner and Nirenberg [].

Motivated by this, Bandle and Flucher conjectured Theorem 9. It follows from [7, pp. From this information, equation 9. Thus, earlier results give the leading-order behavior of u. It follows from Chap. A key step is the inversion of the analogue of L in the half-space, which plays the role of the Laplacian in the usual Schauder theory. In order to use Theorem 6.

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Postponing the proofs of Theorems 9. We now appeal to Theorem 6. This completes the proof of Theorem 9. It remains to prove Theorems 9. Problems 9. The system may be viewed as a PDE in which s plays the role of space variable, but in which no x-derivatives occur. After a general introduction to the issue of blowup, stressing the emergence of the notions of blowup pattern and blowup stability [], we prove the most detailed result to date on the correspondence between singularity data and Cauchy data []. We then present the applications of reduction to laser collapse [34, 35], the weak detonation problem, and the WTC problem in soliton theory [, , ].

The results on the detonation problem are new. These works showed that there are singularities near which the linear part 10 Applications to Nonlinear Waves does not dominate. This new phenomenon generally came to be called blowup; of course, whether the solution or one of its derivatives becomes unbounded depends on the problem.

The results of type ii were strongly stimulated by the discovery of Lp estimates for the wave equation Strichartz. Other authors have tried to construct augmented systems in which an eikonal-type equation is added to the problem; this leads to problems with double characteristics that, unlike Fuchsian PDEs, are ill-posed for nonanalytic data in general. One can obtain C 1 regularity for special classes of data in this manner but apparently not the higher regularity results we obtain via the use of the Nash—Moser inverse function theorem.

The upshot of these studies is that 1 It would be interesting to extend these results to Fuchsian equations. The so-called blowup time, i. First, the blowup time is not a Lorentz invariant. It should therefore be possible to embed a singular solution in a family of solutions with the maximum number of free functions or parameters. The parameters should be directly related to the asymptotics of the solution at its singularity.

If we can establish this stability property, we will have established in particular an explicit description of singularities for all solutions in this class. Thus, in the case of the Korteweg—de Vries KdV equation, an initial condition close to a one-soliton leads to a solution that is not close in the sup-norm to the unperturbed soliton, but does remain close to the set of all translates of this soliton see Strauss [, ], Bona, Souganidis, and Strauss [20] for the KdV case, and their references; further results for KdV-like equations are also found in [].

The problem can now be decomposed into two separate issues: 1. Find a reference blowup pattern.

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Prove its stability under general perturbations. If we establish stability, we conclude that the blowup mechanism is the same as in the reference solution: We will know that the blowup corresponds to a regime in which the equation is close to a linear Fuchsian equation, for which the singular set is characteristic, even though it may not be characteristic for the problem one started from. For a possible scenario leading to lack of stability, see Sect.

Reduction therefore establishes a very detailed asymptotic representation of the solution at the outset. This result provides the prototype for applications of reduction to nonlinear waves, and shows that the blowup mechanism predicted by reduction is generic: perturbation of Cauchy data corresponds to perturbation of the singularity data, in particular, of the blowup surface.

It also shows that the more regular the data, the more regular the blowup surface. The goal of this section is the following result: Theorem Logarithmic terms are absent if and only if the blowup surface has vanishing scalar curvature. The upshot of the theorem is a complete description of blowup, with an expansion that enables one to compute which functions of the solution and its derivatives blow up or not. Solutions are positive near blowup. Only at the end of the argument will one know that solutions generated by the Cauchy data do correspond to a regular blowup surface.

General references on blowup include [, , 96, 86, , ], which also give information on the complementary issue of global existence. There are related results for other power nonlinearities and other dimensions that can be found in []. Several other results for power nonlinearities can be found in []. Our objective is to understand more precisely the behavior of solutions with a nonempty blowup set. Reduction shows that blowup is regulated by a degenerate hyperbolic model, which is not the leading part of the equation, and for which the blowup surface is characteristic, although it is not characteristic for the wave operator.

The notation and basic reductions have been written out in Sect. It is often convenient to use the same letter to denote a function in the x, t or the X, T coordinates. The same convention applies to other functions. The system for w is a Fuchsian symmetric system; see Sect.

Note that w is, as a function of X, T, T ln T , as smooth as the data permit, even on the blowup surface. We wish now to invert this process, constructing singularity data from Cauchy data. Other nearly constant data can be handled in a similar fashion. This setup suggests the use of an implicit function theorem. We use the Nash— Moser theorem, in a form recalled in Chap. The main point is the proof of the invertibility of the linearization of the map K from singularity data to Cauchy data. The inverse of this linearization is computed by comparing two expansions of a solution to the linearization of We also introduce notation for the linearizations of the various maps used in the proof.

Solving 1. The situation is summarized in Fig. Solution u may be determined by Cauchy data or singularity data, by solving the Cauchy problem, or the Fuchsian initial-value problem IVP. The two data are related by an inverse function theorem IFT The goal is to invert K. The regularity of these operators is studied next. Properties of S To study S, we begin by recasting We use the reduction to a system developed in Sect. We summarize the results in the following theorem: Theorem We now solve the equation for w. We must check that Q, A, Aj , and f satisfy the assumptions of Theorem 5.

The following existence theorem is a simple consequence of Theorem 5. The statement on the domain of existence is proved in the following section. Theorem Similarly, the second variation is computed by linearizing again. The If we allow for a loss of 3 derivatives, we see that we can achieve S of class C 2. Since there is an asymmetry between the x and t variables here, we provide the details in a form convenient for the rest of the argument.

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There are now two cases. We must now characterize solutions of the linearization of K in order to be able to identify its inverse in the following section. Compute the solution of the linearization of 1. As usual, U Proof. Since S is the composition of the solution operator associated with the Fuchsian equation 1. This proves 1. Finally, E is linear. These systems contain derivatives of w.

The same considerations apply to the linearization of 1. U By Note also the absence of a pure ln T term. We next show that a solution of a linear Fuchsian equation cannot have a singularity worse than a power of T.

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We then prove iteratively that the solution of the linearized equation has in fact an expansion in powers of T and T ln T to all orders. Finally, by comparing this expansion with We provide below the expansion of eu in terms of T ; its existence is a consequence of the representation 1. Let us therefore start with a general Fuchsian system, and show that its solutions have only power singularities in T.

We then apply the argument to U. In our case, we need in addition to track the number of derivatives involved carefully. The 10 Applications to Nonlinear Waves dependence of f on space and time coordinates is suppressed. In fact, from Chap. Existence of an expansion for U We now apply these general facts to the Fuchsian equation for U. The process can be iterated. The existence of a logarithmic series for U follows; the source of the logarithmic terms is to be found already in the logarithms in the expansion of eu.

Comparing with In fact, this operator is also a right inverse, as we proceed to show. Let us apply our inverse to a given pair U0 , U1. Since this system has, by Theorem 5. Application of the Nash—Moser theorem, end of proof We wish to use the Nash—Moser theorem with smoothing to invert the mapping K. We use the form given in Sect. This proves the announced result. Also, because the Fuchsian equation 1. One therefore reaches the conclusion that singular solutions have a meaningful continuation after blowup. Indeed, our argument for the invertibility of the linearization of K did not use in any essential way the properties of this reference solution.

This means that the set of data leading to blowup is open. It would be interesting to implement this numerically. The above method is not limited in scope to the particular example treated here because the exact form of the reference solution is not used. The equation was viewed as an evolution equation in which z plays the role of evolution variable.

Beam paraxiality seems to arrest singularity formation [61], and time dispersion could allow it [, 63]. Also, NLS is a stationary equation, and therefore does not describe dynamic behavior. Now, for the wave equation, the initial-value problem with z as evolution variable is linearly ill-posed. Therefore, taking z as evolution variables seems more appropriate for stationary rather than dynamic situations.

It was therefore suggested to improve the model and to take into account terms uzz and utt. This equation, written here after scaling variables, so that the leading part is the wave operator with speed one, has been proposed as an envelope equation for laser propagation in Kerr media. The objective of this section is to suggest a mathematical mechanism to account for some of the qualitative features of blowup which, in this context, corresponds to laser breakdown.

One would like to account for the possibility that there may be more that one point in space-time at which breakdown might occur. It also shows that i singular solutions are generic: they may be embedded in a family of solutions parameterized by four real-valued functions of three variables, which is also the number of functions needed to encode the Cauchy data for a complex-valued u; ii the expansion of the solution is related to the local geometry of the singular set in space-time. In the process, we justify formal computations by Papanicolaou, Fibich, and Malkin in particular. However, here again, self-similar asymptotics We begin this section with a detailed analysis of the real cubic nonlinear wave equation, which contains most of the salient features of the analysis, and then turn to the general case.

Results are mostly from [35, 34]. We seek real solutions of Remark The form of the solutions may now be determined. Equation We are in the framework of Theorem 2. Second reduction and existence of solutions We prove that the formal solutions correspond to actual solutions. We present two results: one for analytic data, the other for data with limited regularity. We have determined the formal solution up to the last positive resonance, namely 4.

Let z be the column vector with components z1 ,. On Conversely, every solution of Every solution of We now turn to the nonanalytic case. We wish to apply Theorem 5. In particular, u solves Applying Theorem 5. Observe that Let us seek formal solutions of The main result is the following: Theorem The phase of u0 remains arbitrary.

Substitution into In the analytic case, we have the following theorem: Theorem Moreover, f X belongs to the range of A. Thus Conversely, let z be a solution of For the nonanalytic case, the main result is our next theorem: Theorem To prove this result, we reduce the problem to a generalized Fuchsian system: Theorem The argument being similar to the case of the cubic wave equation, we merely give the choices of Q, etc.

If w is solution of Thus u is clearly a solution for Since equation T Replacing u0 , u1 , u2 in This completes the proof of ii and iii. Both classes of examples are obtained by truncating the series We now turn to solutions of limited regularity. As usual, we give a complete expansion of solutions and specify the terms in the expansion that determine all the others. This shows how to relate the variation of the data to the variation of the nonplanar detonation front.

Since the detonation path is not a straight line, the detonation wave is not a steady wave; in this sense, it is quasisteady, because it is nevertheless expected to admit a tangent plane. We now describe the mathematical problem, and solve it by reduction. The system The detonation path is close to the blowup set, because of the expansion of the solution. This suggests logarithmic leading-order asymptotics for the three unknowns. We now show that this is indeed the case. System Substitution of the value of u T into the equation for v gives The determination of solutions from the singularity data follows from the existence theorems in Chap.

The variation of the solution when the singularity data are varied may now be now computed by solving the linearization of the reduced Fuchsian equation. This is possible because this linearization is again Fuchsian; the procedure parallels that of Sect. The theory of solitons is useful because it replaces the search for approximate solutions of an exact model equation by the search for exact solutions of an approximate model with rich mathematical structure.

We prove that reduction techniques justify the so-called ARS-WTC expansions for solutions of integrable systems, and show how to generalize them to nonintegrable systems [,,]. Let us begin with some background information. As such, they admit complex singularities that travel at the same speed as the wave.

Weiss et al. This result will follow from reduction. Since integrable systems generally require only pure powers, it was widely held that expansions with logarithms were not relevant to the theory of solitons. However, as a rule, perturbation of soliton equations does lead to expansions with logarithms. This would prove that the arbitrary functions in the expansion determine only one solution of the corresponding PDE.

Reduction techniques give answers to both questions. In fact, the examples considered by Weiss et al. Take for instance the case of KdV. Let us associate a system to the KdV equation. Performing the second reduction and appealing to Theorem 4. Let us now outline the corresponding results for other integrable equations. Results are in all respects similar to those for KdV, except that the exponent m, and the values j for which uj is arbitrary, vary from one equation to another.

The sine—Gordon equation in the form Other examples may be found in the problems. In integrable cases, it turns out that whenever j is a resonance, the expression fj also happens to be identically zero. What about nonintegrable problems? In other words, even the nonintegrable case corresponds to a function of several complex variables that is free from branching. This provides a resonance whenever the ak are not all zero. Thus, the resonances in the sense of dynamical systems correspond to the possible arbitrary terms in a series of the solution in powers of exponentials.

This model has been studied extensively; see [, ]. It arises as a model for water waves, and as a general model

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