a numerical study of the small dispersion limit of the korteweg de vries equation and asymptotic solutions abstract we study numerically the small dispersion limit for the korteweg de vries kdv equation u t 6 u u x 2 u x x x 0 for 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small in the whole x t plane the matching of the asymptotic solutions is studied numerically

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

on critical behaviour in generalized kadomtsev petviashvili equations an asymptotic description of the formation of dispersive shock waves in solutions to the generalized kadomtsev petviashvili kp equation is conjectured the asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the painleve i hierarchy several examples are discussed numerically to provide strong evidence for the validity of the conjecture the numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the subcritical kp equations while in the supercritical kp equations a blow up occurs after the formation of the dispersive shock waves

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

bihamiltonian cohomologies and integrable hierarchies i a special case we present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type and compute the third cohomology for the bihamiltonian structure of the dispersionless kdv hierarchy the result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless kdv hierarchy starting from any of its infinitesimal deformations

deformations of the monge riemann hierarchy and approximately integrable systems dispersive deformations of the monge equation ut uux are studied using ideas originating from topological quantum field theory and the deformation quantization program it is shown that to a high order the symmetries of the monge equation may also be appropriately deformed and that if they exist at all orders they are uniquely determined by the original deformation this leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system quasi miura transformations are also constructed for such deformed equations

universal unification of life death evolution post evolution and extinction abstract a general law of universal maximally rapid entropy decrease unifies metabolism mortality and evolution yields their quantitatively accurate universal laws and singularities proves that accidental premature mortality is relatively small and quantifies it predicts that evolution may be continuous or followed by weak and strong spurts leads to a definition of post evolution and considers its probability suggests an absolute instability of technological terrestrial and extra terrestrial civilizations and allows for computer simulations of mortality and evolution

on integrable conservation laws we study normal forms of scalar integrable dispersive not necessarily hamiltonian conservation laws via the dubrovin zhang perturbative scheme our computations support the conjecture that such normal forms are parametrized by infinitely many arbitrary functions that can be identified with the coefficients of the quasi linear part of the equation moreover in general we conjecture that two scalar integrable evolutionary partial differential equations having the same quasi linear part are miura equivalent this conjecture is also consistent with the tensorial behaviour of these coefficients under general miura transformations

poisson bracket on 1 forms and evolutionary partial differential equations we introduce a bracket on 1 forms defined on i e the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a poisson bracket using this bracket we show that certain hierarchies appearing in the framework of f manifolds with a compatible flat connection m are hamiltonian in a generalized sense moreover we show that if a metric g compatible with is also invariant with respect to then this generalized hamiltonian setup reduces to the standard one

on parametric gevrey asymptotics for some nonlinear initial value cauchy problems abstract we study a nonlinear initial value cauchy problem depending upon a complex perturbation parameter with vanishing initial data at complex time t 0 and whose coefficients depend analytically on t near the origin in c 2 and are bounded holomorphic on some horizontal strip in c w r t the space variable this problem is assumed to be non kowalevskian in time t therefore analytic solutions at t 0 cannot be expected in general nevertheless we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space when the complex parameter belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood of 0 in c these solutions are achieved by means of laplace and fourier inverse transforms of some common depending function on c r analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second when the perturbation parameter belongs to moreover these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w r t with the help of the classical ramis sibuya theorem we obtain the existence of a formal series generally divergent in which is the common gevrey asymptotic expansion of the built up actual solutions considered above

on tronquee solutions of the first painleve hierarchy abstract it is well known that due to boutroux the first painleve equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane in this paper we show that such solutions exist for higher order analogues of the first painleve equation the first painleve hierarchy as well

on deformation of poisson manifolds of hydrodynamic type we study a class of deformations of infinite dimensional poisson manifolds of hydrodynamic type which are of interest in the theory of frobenius manifolds we prove two results first we show that the second cohomology group of these manifolds in the poisson lichnerowicz cohomology is essentially trivial then we prove a conjecture of b dubrovin about the triviality of homogeneous formal deformations of the above manifolds

numerical solution of the small dispersion limit of the camassa holm and whitham equations and multiscale expansions the small dispersion limit of solutions to the camassa holm ch equation is characterized by the appearance of a zone of rapid modulated oscillations an asymptotic description of these oscillations is given for short times by the one phase solution to the ch equation where the branch points of the corresponding elliptic curve depend on the physical coordinates via the whitham equations we present a conjecture for the phase of the asymptotic solution a numerical study of this limit for smooth hump like initial data provides strong evidence for the validity of this conjecture we present a quantitative numerical comparison between the solution to the ch equation and the asymptotic solution the dependence on the small dispersion parameter epsilon is studied in the interior and at the boundaries of the whitham zone in the interior of the zone the difference between the solution to the ch equation and the asymptotic solution is of the order epsilon at the trailing edge of the order sqrt eps

on bi hamiltonian deformations of exact pencils of hydrodynamic type in this paper we are interested in nontrivial bi hamiltonian deformations of the poisson pencil deformations are generated by a sequence of vector fields x2 x3 x4 where each xk is homogeneous of degree k with respect to a grading induced by rescaling constructing recursively the vector fields xk one obtains two types of relations involving their unknown coefficients one set of linear relations and an other one which involves quadratic relations we prove that the set of linear relations has a geometric meaning using miura quasitriviality the set of linear relations expresses the tangency of the vector fields xk to the symplectic leaves of 1 and this tangency condition is equivalent to the exactness of the pencil moreover extending the results of lorenzoni p 2002 j geom phys 44 331 75 we construct the nontrivial deformations of the poisson pencil up to the eighth order in the deformation parameter showing therefore that deformations are unobstructed and that both poisson structures are polynomial in the derivatives of u up to that order

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

the law of invariant mortality abstract a concept of invariance is applied to mortality data it predicts an exact law of invariant mortality consequences of this law are presented and some are tested the most intriguing consequence is that mortality might be predominantly related to a thermodynamic mechanism and may be reversed simultaneously for all ages to its value at a much younger age

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

hamiltonian pdes deformations integrability solutions we review recent classification results on the theory of systems of nonlinear hamiltonian partial differential equations with one spatial dimension including a perturbative approach to the integrability theory of such systems and discuss universality conjectures describing critical behaviour of solutions to such systems

dispersionless 3 1 dimensional integrable hierarchies in this paper we introduce a multi dimensional version of the r matrix approach to the construction of integrable hierarchies applying this method to the case of the lie algebra of functions with respect to the contact bracket we construct integrable hierarchies of 3 1 dimensional dispersionless systems of the type recently introduced in sergyeyev 2014 http arxiv org abs 1401 2122

dispersive approximations in fluid dynamics we have compared the results of a von neumann richtmyer calculation with the weak limit of calculations performed by von neumann s original method although there are discernible differences they are surprisingly small we analyze von newmann s algorithm for solving the initial value problem for the lagrangian equations of compressible flow

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

evolution on trees a brief review about the extinctions of species and the case of genocide in a darwinistic punctuated equilibrium evolution model abstract numerical and theoretical investigations of species evolution in presence of extinction rules are recalled in the framework of punctuated equilibrium evolution models previous results on the bak sneppen model are mentioned together with extensions taking into account a variable number of species conditions for limited growth due to species interactions external fields or intrinsic population density conditions have already been considered an original genocide model is presented as a step towards describing effects due to the extinction of species critical exponents for i the avalanche distribution ii the steady state approach and iii the tree size are given a mean field approximation is shown to be quite accurate for describing the critical state the screening of species from the evolution process is not numerically studied

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

exact poisson pencils τ structures and topological hierarchies abstract we discuss in the framework of the dubrovin zhang perturbative approach to integrable evolutionary pdes in 1 1 dimensions the role of a special class of poisson pencils called exact poisson pencils in particular we show that in the semisimple case exactness of the pencil is equivalent to the constancy of the so called central invariants of the theory that were introduced by dubrovin liu and zhang

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

phenomenological theory of mortality evolution its singularities universality and superuniversality the probability to survive to the age x universally increases with the mean lifespan x for species as remote as humans and flies for a given x the rate of its evolution with x is constant except for the narrow vicinity of a certain x x which equals 75 years for humans and 32 days for flies and which is independent of age population and living conditions at x x the evolution rate jumps to a different value its next jump is predicted at x 87 years for humans and 59 days for flies such singularities are well known in physics and mathematics as phase transitions in the considered case different population phases have significantly different survival evolution rates the evolution is rapid life expectancy may double within a lifespan of a single generation survival probability depends on age x and mean longevity x only for instance survival curves of 1896 swedes and 1947 japanese with approximately equal x are very close although they are related to different races in different countries at different periods in their different history with no adjustable parameters its presented universal law quantitatively agrees with all lifetable data according to this law the impact of all factors but age reduces to the mean lifespan only in advanced and old age this law is superuniversal it is approximately the same for species as remote as humans and flies it yields survival probability that linearly depends on the mean lifespan x as a result when human x almost doubles from 35 5 to 69 3 years life expectancy at 70 years increases from 8 to 9 5 years only other implications of the universal law are also considered

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

on the generation of dispersive shock waves abstract we review various methods for the analysis of initial value problems for integrable dispersive equations in the weak dispersion or semiclassical regime some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains so called dispersive shock waves as the result of shock formation in a limiting dispersionless system they also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves revealing fascinating universal wave patterns

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

conservation laws of metabolism and mortality abstract quantitatively accurate laws of metabolism and mortality which are the same for species with different evolutionary history e g humans and flies are derived and extensively verified such conservation with respect to evolution i e in non stationary and heterogeneous conditions laws are unique for self organized live systems they impose restrictions on biological heterogeneity establish two universal ways of its evolution predict a new kind of rapid compared to lifespan adaptation and reversible mortality which may be directed predictions agree with experimental data exact criterion unravels universal conservation law in non universal experimental dependences and accurately divides a heterogeneous population into non universal groups which yield the law with 5 mean accuracy this allows to approximate and forecast mortality in different countries and resolves a two century old controversy with the universal law of mortality

pole free solutions of the first painleve hierarchy and non generic critical behavior for the kdv equation we establish the existence of real pole free solutions to all even members of the painleve i hierarchy we also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the kdv equation in the small dispersion limit this was understood in the case of a generic critical point and we generalize it here to the case of non generic critical points

numerical solution of the small dispersion limit of korteweg de vries and whitham equations the cauchy problem for the korteweg de vries kdv equation with small dispersion of order 2 1 is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order these oscillations are approximately described by the elliptic solution of kdv where the amplitude wave number and frequency are not constant but evolve according to the whitham equations in this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the kdv equation in the small dispersion limit and the corresponding approximate solution for values of between 10 1 and 10 3 the numerical results are compatible with a difference of order close to the center within the interior of the whitham oscillatory zone of order 1 3 at the left boundary outside the whitham zone and of order at the right boundary outside the whitham zone 2007 wiley periodicals inc

asymptotics for a special solution to the second member of the painleve i hierarchy we study the asymptotic behavior of a special smooth solution y x t to the second member of the painleve i hierarchy this solution arises in random matrix theory and in the study of the hamiltonian perturbations of hyperbolic equations the asymptotic behavior of y x t if x for fixed t is known and relatively simple but it turns out to be more subtle when x and t tend to infinity simultaneously we distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior and we obtain rigorous asymptotics in both regions we also discuss two critical transitional asymptotic regimes

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

painleve ii asymptotics near the leading edge of the oscillatory zone for the korteweg de vries equation in the small dispersion limit in the small dispersion limit solutions to the korteweg de vries equation develop an interval of fast oscillations after a certain time we obtain a universal asymptotic expansion for the korteweg de vries solution near the leading edge of the oscillatory zone up to second order corrections this expansion involves the hastings mcleod solution of the painleve ii equation we prove our results using the riemann hilbert approach c 2009 wiley periodicals inc

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

numerical study of shock formation in the dispersionless kadomtsev petviashvili equation and dispersive regularizations abstract the formation of singularities in solutions to the dispersionless kadomtsev petviashvili dkp equation is studied numerically for different classes of initial data the asymptotic behavior of the fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity the approach is first tested in detail in 1 1 dimensions for the known case of the hopf equation where it is shown that the break up of the solution can be identified with prescribed accuracy for dissipative regularizations of this shock formation as the burgers equation and for dispersive regularizations as the korteweg de vries equation the fourier coefficients indicate as expected global regularity of the solutions the kadomtsev petviashvili kp equation can be seen as a dispersive regularization of the dkp equation the behavior of kp solutions for small dispersion parameter 1 near a break up of corresponding dkp solutions is studied it is found that the difference between kp and dkp solutions for the same initial data at the critical point scales roughly as 2 7 as for the korteweg de vries equation

integrable equations in 2 1 dimensions deformations of dispersionless limits we classify integrable third order equations in 2 1 dimensions which generalize the examples of kadomtsev petviashvili veselov novikov and harry dym equations our approach is based on the observation that dispersionless limits of integrable systems in 2 1 dimensions possess infinitely many multi phase solutions coming from the so called hydrodynamic reductions in this paper we adopt a novel perturbative approach to the classification problem based on the method of hydrodynamic reductions we first classify integrable quasilinear systems which may potentially occur as dispersionless limits of soliton equations in 2 1 dimensions to reconstruct dispersive deformations we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart this procedure leads to a complete list of integrable third order equations some of which are apparently new

critical behavior for scalar nonlinear waves abstract in the long wave regime nonlinear waves may undergo a phase transition from a smooth behavior to a fast oscillatory behavior in this study we consider this phenomenon which is commonly known as dispersive shock in the light of dubrovin s universality conjecture dubrovin 2006 dubrovin and elaeva 2012 and we argue that the transition can be described by a special solution of a model universal partial differential equation this universal solution is constructed using the string equation we provide a classification of universality classes and an explicit description of the transition with special functions thereby extending dubrovin s universality conjecture to a wider class of equations in particular we show that the benjamin ono equation belongs to a novel universality class with respect to those known previously and we compute its string equation exactly we describe our results using the language of statistical mechanics where we show that dispersive shocks share many of the features of the tricritical point in statistical systems and we also build a dictionary of the relations between nonlinear waves and statistical mechanics