the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

adaptive methods for linear programming decoding detectability of failures of linear programming lp decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying lp problem in this paper we make a first step in studying this method and show that by starting from a simple lp problem and adaptively adding the necessary constraints the complexity of lp decoding can be significantly reduced in particular we observe that with adaptive lp decoding the sizes of the lp problems that need to be solved become practically independent of the density of the parity check matrix we further show that adaptively adding extra constraints such as constraints based on redundant parity checks can provide large gains in the performance

loop calculus in statistical physics and information science considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series the leading term in the series is the bethe peierls belief propagation bp contribution the rest are expressed as loop contributions on the factor graph and calculated directly using the bp solution the series unveils a small parameter that often makes the bp approximation so successful applications of the loop calculus in statistical physics and information science are discussed

the admm penalized decoder for ldpc codes linear programming lp decoding for low density parity check codes was introduced by feldman et al and has been shown to have theoretical guarantees in several regimes furthermore it has been reported in the literature via simulation and via instanton analysis that lp decoding displays better error rate performance at high signal to noise ratios snr than does belief propagation bp decoding however at low snrs lp decoding is observed to have worse performance than bp in this paper we seek to improve lp decoding at low snrs while maintaining lp decoding s high snr performance our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers admm algorithm to a set of non convex optimization problems these non convex problems are constructed by adding a penalty term to the objective of lp decoding the goal of the penalty is to make pseudocodewords which are non integer vertices of the lp relaxation more costly we name this class of decoders admm penalized decoders for low and moderate snrs we simulate admm penalized decoding with ell 1 and ell 2 penalties we find that these decoders can outperform both bp and lp decoding for high snrs where it is difficult to obtain data via simulation we use an instanton analysis and find that asymptotically admm penalized decoding performs better than bp but not as well as lp unfortunately since admm penalized decoding is not a convex program we have not been successful in developing theoretical guarantees however the non convex program can be approximated using a sequence of linear programs an approach that yields a reweighted lp decoder we show that a two round reweighted lp decoder has an improved theoretical recovery threshold when compared with lp decoding in addition we find via simulation that reweighted lp decoding significantly attains lower error rates than lp decoding at low snrs

adaptive cut generation algorithm for improved linear programming decoding of binary linear codes linear programming lp decoding approximates maximum likelihood ml decoding of a linear block code by relaxing the equivalent ml integer programming ip problem into a more easily solved lp problem the lp problem is defined by a set of box constraints together with a set of linear inequalities called parity inequalities that are derived from the constraints represented by the rows of a parity check matrix of the code and can be added iteratively and adaptively in this paper we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint or cut at a point in the unit hypercube then we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check rpc constraints that can eliminate pseudocodewords produced by the lp decoder often significantly improving the decoder error rate performance the cut generating algorithm is based upon a specific transformation of an initial parity check matrix of the linear block code we also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut generating algorithm simulation results for several low density parity check ldpc codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between lp decoding and ml decoding

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

master polytopes for cycles of binary matroids abstract prior work on the cycle polytopes p m of binary matroids m has almost exclusively concentrated on regular matroids yet almost all binary matroids are nonregular and almost nothing is known about their cycle polytopes in this paper we introduce a class of binary matroids l k k 1 the complete binary matroids of order k we show that the facets of the cycle polytopes p l k have a rather simple description which may be used to deduce easily some and in principle all facets of the cycle polytopes of general binary matroids m for this reason we call the polytopes p l k master polytopes specifically we describe two methods by which facets of p m can be deduced from the facets of certain master polytopes one method produces a complete description of p m but is not computationally efficient the other one produces a subset of the facets of p m by an efficient lifting procedure

efficient linear programming decoding of hdpc codes we propose several improvements for linear programming lp decoding algorithms for high density parity check hdpc codes first we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks second we propose an efficient mixed integer decoder utilizing the branch and bound method we further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations based on simulation results the proposed decoders achieve near ml performance with reasonable complexity

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

decoding binary linear block codes using local search this paper presents a novel iterative hard decision decoding algorithm for binary linear block codes over a binary symmetric channel bsc the problem is formulated as a 0 1 integer programming problem which is known to be np hard when the crossover probability c of the channel is known the solution space of the decoding problem can be decreased to a sphere whose radius is related to c using the penalty function method the problem is reformulated on this reduced solution space then an iterative multi flip local search algorithm is designed to find the global solution of this decoding problem for a code with minimum distance d when the radius of the sphere is not greater than d 1 2 this algorithm has the maximum likelihood ml certificate property i e if the decoder outputs a codeword it is guaranteed to be the ml codeword compared to the probabilistic suboptimal iterative belief propagation bp decoder this approach has lower complexity and better performance numerical results show that in terms of speed and performance the proposed decoding method outperforms bp decoding in the error floor region

guessing facets polytope structure and improved lp decoder we investigate the structure of the polytope underlying the linear programming lp decoder introduced by feldman karger and wainwright we first show that for expander codes every fractional pseudocodeword always has at least a constant fraction of nonintegral bits we then prove that for expander codes the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword we further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as lp decoding that provably performs better the method is very simple it first applies ordinary lp decoding and when it fails it proceeds by guessing facets of the polytope and then resolving the linear program on these facets while the lp decoder succeeds only if the ml codeword has the highest likelihood over all pseudocodewords we prove that the proposed algorithm when applied to suitable expander codes succeeds unless there exists a certain number of pseudocodewords all adjacent to the ml codeword on the lp decoding polytope and with higher likelihood than the ml codeword we then describe an extended algorithm still with polynomial complexity that succeeds as long as there are at most polynomially many pseudocodewords above the ml codeword

smoothing of bump functions abstract let x be a separable banach space with a schauder basis admitting a continuous bump which depends locally on finitely many coordinates then x admits also a c smooth bump which depends locally on finitely many coordinates

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

fully localised solitary wave solutions of the three dimensional gravity capillary water wave problem a model equation derived by kadomtsev petviashvili sov phys dokl 15 539 541 1970 suggests that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction this prediction is rigorously confirmed for the full water wave problem in the present paper the theory is variational in nature a simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties the reduced functional is related to the functional associated with the kadomtsev petviashvili equation and a nontrivial critical point is detected using the direct methods of the calculus of variations

capillary gravity waves on the free surface of an inviscid fluid of infinite depth existence of solitary waves permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated an application of the hodograph transform converts the free boundary value problem into a boundary value problem for the cauchy riemann equations in the lower halfplane with nonlinear differential boundary conditions this can be converted to an integro differential equation with symbol k2 4 k 4 1 where is a bifurcation parameter a normal form analysis is presented which shows that the boundary value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for small this normal form system has been studied thoroughly by several authors iooss kirchgassner 8 iooss peroueme 10 dias iooss 5 it admits a pair of solitary wave solutions which are reversible in the sense ofkirchgassner 11 by applying a method introduced in 11 it is shown that this pair of reversible solitary waves persists for the boundary value problem and that the decay at infinity of these solitary waves is at least like 1 x

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

adaptive linear programming decoding of nonbinary linear codes over prime fields in this work we consider adaptive linear programming alp decoding of linear codes over prime fields i e the finite fields mathbb f p of size p where p is a prime when used over a p ary input memoryless channel in particular we provide a general construction of valid inequalities using no auxiliary variables for the codeword polytope or the convex hull of the so called constant weight embedding of a single parity check spc code over any prime field the construction is based on sets of vectors called building block classes that are assembled to form the left hand side of an inequality according to several rules in the case of almost doubly symmetric valid classes we prove that the resulting inequalities are all facet defining while we conjecture this to be true if and only if the class is valid and symmetric valid symmetric classes impose certain symmetry conditions on the elements of the vectors from the class while valid doubly symmetric classes impose further technical symmetry conditions for p 3 there is only a single valid symmetric class and we prove that the resulting inequalities together with the so called simplex constraints give a complete and irredundant description of the codeword polytope of the embedded spc code for p 5 we show that there are additional facets beyond those from the proposed construction as an example for p 7 we provide additional inequalities that all define facets of the embedded codeword polytope the resulting overall set of linear in equalities is conjectured to be irredundant and complete such sets of linear in equalities have not appeared in the literature before have a strong theoretical interest and we use them to develop an efficient relaxed alp decoder for general non spc linear codes over prime fields the key ingredient is an efficient separation algorithm based on the principle of dynamic programming furthermore we construct a decoder for linear codes over arbitrary fields mathbb f q with q p m and m 1 by a factor graph representation that reduces to several instances of the case m 1 which results in general in a relaxation of the original decoding polytope finally we present an efficient cut generating algorithm to search for redundant parity checks to further improve the performance towards maximum likelihood decoding for short to medium block lengths numerical experiments confirm that our new decoder is very efficient compared to a static lp decoder for various field sizes check node degrees and block lengths

a new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs the theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal in this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid applied to cuts in graphs this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph if the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid specialized to cuts in graphs this result solves a problem posed by lovasz

iterative approximate linear programming decoding of ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of low density parity check ldpc codes is considered an iterative algorithm similar to min sum and belief propagation for efficient approximate solution of this problem was proposed by vontobel and koetter in this paper the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose in particular we are interested in obtaining a feasible vector in the lp decoding problem that is close to optimal in the following sense the distance normalized by the block length between the minimum and the objective function value of this approximate solution can be made arbitrarily small it is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length combined with previous results that have shown that the lp decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity this is achieved by first applying the iterative lp decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits and then correcting the remaining errors using some standard method these conclusions are also extended to generalized ldpc codes

global solutions of the gravity capillary water wave system in three dimensions in this paper we prove global regularity for the full water wave system in three dimensions for small data under the influence of both gravity and surface tension this problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models to construct global solutions we use a combination of energy estimates and matching dispersive estimates there is a significant new difficulty in proving energy estimates in our problem namely the combination of slow pointwise decay of solutions no better than lvert t rvert 5 6 and the presence of a large codimension 1 set of quadratic time resonances to deal with such a situation we propose here a new mechanism which exploits a non degeneracy property of the time resonant hypersurfaces and some special structure of the quadratic part of the non linearity connected to the conserved energy of the system the dispersive estimates rely on analysis of the duhamel formula in the fourier space the main contributions come from the set of space time resonances which is a large set of dimension 1 to control the corresponding bilinear interactions we use harmonic analysis techniques such as orthogonality arguments in the fourier space and atomic decompositions of functions most importantly we construct and use a refined norm which is well adapted to the geometry of the problem

a new linear programming approach to decoding linear block codes in this paper we propose a new linear programming formulation for the decoding of general linear block codes different from the original formulation given by feldman the number of total variables to characterize a parity check constraint in our formulation is less than twice the degree of the corresponding check node the equivalence between our new formulation and the original formulation is proven the new formulation facilitates to characterize the structure of linear block codes and leads to new decoding algorithms in particular we show that any fundamental polytope is simply the intersection of a group of the so called minimum polytopes and this simplified formulation allows us to formulate the problem of calculating the minimum hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables we then propose a branch and bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem in addition we prove that for the family of single parity check spc product codes the fractional distance and the pseudodistance are both equal to the minimum distance finally we propose an efficient algorithm for decoding spc product codes with low complexity and maximum likelihood ml decoding performance

some analytical properties of capillary gravity waves in two fluid flows of infinite depth capillary gravity waves of finite amplitude at the interface of a two fluid flow are considered the fluids of constant densities extended to infinity in each direction are inviscid and incompressible the flow is two dimensional and irrotational in each fluid assume that there is a permanent wave at the interface moving with a constant speed and the wave decays to zero at infinity with a certain rate it is shown that the wave must decay at infinity quadratically and the solution of the exact governing equations corresponding to such a wave must satisfy two integral identities for the net displacement and the energy of the wave while the former implies non existence of waves of pure depression or pure elevation in the flow

improved bounds on the word error probability of ra 2 codes with linear programming based decoding this paper deals with the linear programming based decoding algorithm of feldman and karger for repeat accumulate turbo like codes we present a new structural characterization that captures the event that decoding fails based on this structural characterization we develop polynomial algorithms that given an ra 2 code compute upper and lower bounds on the word error probability p sub w for the binary symmetric and the additive white gaussian noise awgn channels our experiments with an implementation of these algorithms for bounding p sub w demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by feldman et al the experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases the computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

smoothness in weakly compactly generated banach spaces received august 25 1982 revised april 15 1983 if x is a weakly compactly generated wcg banach space then x admits an equivalent c smooth norm if a wcg banach space x admits a ck smooth function with bounded support then x admits ck smooth partitions of unity

a separation algorithm for improved lp decoding of linear block codes maximum likelihood ml decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming ip problem feldman relaxed this ip problem and presented linear programming lp based decoding in this paper we propose a new separation algorithm to improve the error correcting performance of lp decoding for binary linear block codes we use an ip formulation with indicator variables that help in detecting the violated parity checks we derive gomory cuts from the ip and use them in our separation algorithm an efficient method of finding cuts induced by redundant parity checks rpc is also proposed under certain circumstances we can guarantee that these rpc cuts are valid and cut off the fractional optimal solutions of lp decoding it is demonstrated on three ldpc codes and two bch codes that our separation algorithm performs significantly better than lp decoding and belief propagation bp decoding

mathematical programming decoding of binary linear codes theory and algorithms mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches challenging established but often heuristic algorithms based on iterative message passing concepts from mathematical programming used in the context of decoding include linear integer and nonlinear programming network flows notions of duality as well as matroid and polyhedral theory this paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary input memoryless symmetric channels

efficient admm decoding of ldpc codes using lookup tables linear programming decoding with the alternating direction method of multipliers admm is a promising decoding technique for low density parity check ldpc codes where the computational complexity of euclidean projections onto check polytopes becomes a prominent problem in this paper the problem is circumvented by building lookup tables luts and quantizing the inputs to approach approximate euclidean projections at low computational complexities to challenge the huge memory cost of luts we first propose two commutative compositions of euclidean projection and self map and show the existence of a small quantization range which does not alter the euclidean projection then we investigate the design and simplification of the luts by exploiting the commutative compositions and check node decomposition techniques an efficient algorithm for the lut based projection is demonstrated by using one simplification method simulation results show that for both the regular and irregular ldpc codes the admm decoding using lut based projection can substantially reduce the decoding time while maintaining the error rate performance at a comparatively large memory cost

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

stopping set elimination by parity check matrix extension via integer linear programming error rate floor phenomenon is known to be a serious impediment to the use of low density parity check ldpc codes for some practical applications that demand high data reliability in the case of binary erasure channels becs certain error prone patterns known as stopping sets are proven to cause this performance degradation a possible approach to diminish this drawback over becs is to eliminate stopping sets by parity check matrix extension given a parity check matrix h and a list mathcal l of its stopping sets we present an integer linear programming ilp formulation to find a parity check equation which eliminates the maximum number of stopping sets in mathcal l one of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as limiting the weight of the new parity check row making the new row redundant or linearly independent 4 cycle avoidance and taking into account the sizes of stopping sets armed with these adjustments the method can provide good performance improvements as evidenced by simulation results furthermore for a given varrho in mathbb n by extending the basic formulation we provide an ilp formulation for finding a set of size varrho of parity check equations which can best eliminate the stopping sets in mathcal l among all such sets

a decomposition theory for binary linear codes the decomposition theory of matroids initiated by paul seymour in the 1980s has had an enormous impact on research in matroid theory this theory when applied to matrices over the binary field yields a powerful decomposition theory for binary linear codes in this paper we give an overview of this code decomposition theory and discuss some of its implications in the context of the recently discovered formulation of maximum likelihood ml decoding of a binary linear code over a binary input discrete memoryless channel as a linear programming problem we translate matroid theoretic results of grotschel and truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ml decoding problem can be solved in time polynomial in the length of the code one such family is that consisting of codes for which the codeword polytope is identical to the koetter vontobel fundamental polytope derived from the entire dual code cperp however we also show that such families of codes are not good in a coding theoretic sense either their dimension or their minimum distance must grow sublinearly with code length as a consequence we have that decoding by linear programming when applied to good codes cannot avoid failing occasionally due to the presence of pseudocode words

existence and conditional energetic stability of three dimensional fully localised solitary gravity capillary water waves in this paper we show that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction the proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy which is given in dimensionless coordinates by epsilon eta phi integral r2 1 2 integral 1 eta 0 phi 2 x phi 2 y phi 2 z dy 1 2 eta 2 beta root 1 eta 2 x eta 2 z 1 dxdz subject to the constraint that the momentum i eta phi integral r2 eta x phi vertical bar y 1 eta dzdz is fixed here x y z x z is an element of r y is an element of 0 1 eta x z is the fluid domain phi is the velocity potential and beta 1 3 is the bond number these functionals are studied locally for eta in a neighbourhood of the origin in h 3 r 2 we prove the existence of a minimiser of epsilon subject to the constraint i 2 mu where 0 mu 1 the existence of a small amplitude solitary wave is thus assured and since epsilon and i are both conserved quantities a standard argument may be used to establish the stability of the set d mu of minimisers as a whole stability is however understood in a qualified sense due to the lack of a global well posedness theory for three dimensional water waves we show that solutions to the evolutionary problem starting near d mu remain close to d mu in a suitably defined energy space over their interval of existence they may however explode in finite time due to higher order derivatives becoming unbounded c 2012 elsevier inc all rights reserved