integrable forms of the one dimensional flow equation for unsaturated heterogeneous porous media the equation for the horizontal transport of a liquid in an unsaturated scale heterogeneous porous medium is t x x c x x e x x c e d a systematic search for lie backlund symmetries leads to the requirement that c a b 2 as in the homogeneous 1 case more generally e may be 1 mx 1 3 2 c or exp mx 3c 2 in these cases the transport equation may be linearized and solved exactly examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for burgers equation

first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

scope for further analytical solutions for constant flux infiltration into a semi infinite soil profile or redistribution in a finite soil profile we attempt to generate new solutions for the moisture content form of the one dimensional richards 1931 equation using the lisle 1992 equivalence mapping this mapping is used as no more general set of transformations exists for mapping the one dimensional richards equation into itself starting from a given solution the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions we first seek new analytical solutions satisfying richards equation subject to a constant flux surface boundary condition for a semi infinite dry soil starting with the burgers model the first iteration produces an existing solution while subsequent iterations are shown to endlessly reproduce this same solution next we briefly consider the problem of redistribution in a finite length soil in this case lisle s equivalence mapping is generalized to account for arbitrary initial conditions as was the case for infiltration however it is found that new analytical solutions are not generated using the equivalence mapping although existing solutions are recovered

new exact solutions of heat conduction in metals we consider the way in which a solution to a class of nonlinear partial differential equationss u ut k u ux x approaches the similarity form the problem we solve is chosen for two main reasons first the equation above is of widespread use in modeling physical situations and second it provides a tractable but significant example of a free boundary problem

a new solution for the nonlinear diffusion convection equation the cauchy problem for a one dimensional nonlinear porous media diffusion convection differential equation is considered a self similar solution is derived that after the riccati transformation is described in terms of the airy functions the asymptotic behavior of the solution for a small amount of species introduced into the porous media is investigated

exact transient solutions to nonlinear diffusion convection equations in higher dimensions the complete lie algebra of classical infinitesimal symmetries of the nonlinear diffusion convection equation in two and three dimensions is presented except for some cases involving constant diffusivity a complete reduction to an ordinary differential equation is not possible however closed form solutions are obtained for special forms of both the 2d and 3d nonlinear diffusion convection equations using a symmetry reduction and an additional physical constraint this extends the small list of closed form transient solutions already known

on the group classification of variable coefficient nonlinear diffusion convection equations we consider the variable coefficient diffusion convection equation of the form f x ut g x d u ux x h x k u ux which has considerable interest in mathematical physics biology and chemistry we present a complete group classification for this class of equations also we derive equivalence transformations between equations that admit lie symmetries furthermore we obtain mappings that connect variable and constant coefficient equations exact solutions of special forms of this equations are constructed using lie symmetries and equivalence transformations

analysis of hybrid selection maximal ratio combining in correlated nakagami fading we present an analysis of a hybrid selection maximal ratio combining diversity system over an evenly correlated slow frequency nonselective nakagami fading channel where the correlation coefficient between any pair of the diversity branch gain amplitudes is the same and all average branch signal to noise ratios snrs are equal in this system the l branches with the largest instantaneous snr out of n available branches are selected and combined using maximal ratio combining from the joint characteristic function cf of the instantaneous branch snrs we obtain an expression for the cf of the combiner output snr as a series of elementary cfs the expression can be conveniently used to obtain the symbol error probability of coherent detection of different m ary modulation schemes we illustrate our methodology using m ary phase shift keying as an example

potential conservation laws we prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws therefore characteristics of pure potential conservation laws have to essentially depend on potential variables this statement provides a significant generalization of results of the recent paper by bluman et al j math phys 47 113505 2006 moreover we present extensions to gauged potential systems abelian and general coverings and general foliated systems of differential equations an example illustrating possible applications of these results is given a special version of the hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

analytical and numerical investigations of spontaneous imbibition in porous media we present semi analytical solutions for co current displacements with some degree of counter current flow the solution assumes a one dimensional horizontal displacement of two immiscible incompressible fluids with arbitrary viscosities and saturation dependent relative permeability and capillary pressures we address the impact of the system length on the degree of counter current flow when there is no pressure drop in the non wetting phase across the system assuming negligible capillary back pressure at the inlet boundary of the system it is shown that in such displacements the fractional flow can be used to determine a critical water saturation from which regions of both co current and counter current flow are identified this critical saturation changes with time as the saturation front moves into the porous medium furthermore the saturation profile in the approach presented here is not necessarily a function of distance divided by the square root of time we also present approximate solutions using a perturbative approach which is valid for a wide range of flow conditions this approach requires less computational power and is much easier to implement than the implicit integral solutions used in previous work finally a comprehensive comparison between analytical and numerical solutions is presented numerical computations are performed using traditional finite difference formulations and convergence analysis shows a generally slow convergence rate for water imbibition rates and saturation profiles this suggests that most coarsely gridded simulations give a poor estimate of imbibition rates while demonstrating the value of these analytical solutions as benchmarks for numerical studies complementing buckley leverett analysis this article is protected by copyright all rights reserved

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

extension of the fujita solution to air and water movement in soils for specific soil properties the fundamental solution of fujita 1952b is extended to include the effect of air movement on horizontal absorption this exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by parlange et al 1982 and sander and parlange 1984 the optimal sorptivity of parlange et al has very good accuracy while the surface water content s is near saturation but decreases in accuracy as s decreases the approximate sorptivity of sander and parlange however proves to be extremely accurate for all s with a maximum error of only 0 5 sander and parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0 5 the aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above alternatively the solution may be used for checking the accuracy of numerical models

eigenvalues of the barenblatt pattle similarity solution in nonlinear diffusion we consider the large time behaviour of the nonlinear diffusion equation u t r 1 r r 1 u u r u 0 0 for certain types of compact initial data we show that the solution approaches the barenblatt pattle similarity solution through an infinite sequence of negative real powers of t which can be found in explicit form these together with their interactive product terms determine the power series development of u r t as t

virtual branch analysis of symbol error probability for hybrid selection maximal ratio combining in rayleigh fading we derive analytical expressions for the symbol error probability sep for a hybrid selection maximal ratio combining h s mrc diversity system in multipath fading wireless environments with h s mrc l out of n diversity branches are selected and combined using maximal ratio combining mrc we consider coherent detection of m ary phase shift keying mpsk and quadrature amplitude modulation mqam using h s mrc for the case of independent rayleigh fading with equal signal to noise ratio averaged over the fading the proposed problem is made analytically tractable by transforming the ordered physical diversity branches which are correlated into independent and identically distributed i i d virtual branches which results in a simple derivation of the sep for arbitrary l and n we further obtain a canonical structure for the sep of h s mrc as a weighted sum of the elementary seps which are the seps using mrc with i i d diversity branches in rayleigh fading or equivalently the seps of the nondiversity single branch system in nakagami fading whose closed form expressions are well known we present numerical examples illustrating that h s mrc even with l spl lt n can achieve a performance close to that of n branch mrc

analysis of hybrid selection maximal ratio combining in rayleigh fading we use a novel virtual branch technique to succinctly derive the mean and variance of the combiner output signal to noise ratio for hybrid selection maximal ratio combining in a multipath fading environment

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

on the multiple microphone method for measuring in duct acoustic properties in the presence of mean flow nowadays the two microphone method is accepted as the standard as specified in astm e1050 90 for measuring in duct acoustic properties however research results on using the least square method with multiple measurement points and broadband excitation have been reported for enhancing the frequency response of the two microphone method in this paper the effects of varying the relative measurement positions on errors in the estimation of the acoustic quantities is studied for the multiple microphone method both the theoretical and experimental results show that among possible sensor positioning configurations the equidistant positioning of sensors yields the smallest error within the effective measurement frequency range in addition it is noted that the measurement accuracy can be increased and the effective frequency range can be widened by increasing the number of equidistant sensors measurement examples are shown and the results support the findings

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

symmetries and integrability of generalized diffusion reaction equations using lie group methods and the painleve test the authors analyse nonlinear diffusion reaction equations ut d u u f u with power law diffusion coefficients d approximately uv and arbitrary nonlinear reaction terms f u which have a wide spectrum of applications in many areas of science the lie group based similarity method leads to a classification of the reaction terms according to its symmetry properties with the help of the adjoint representation the optimal system of similarity reductions is calculated to check the integrability of the partial differential equation the existence of generalized lie backlund symmetries is investigated apart from three known cases no further cases with third order symmetries exist examining the integrability of the second order ordinary differential equations resulting from the reductions only a few parameter combinations can be found for which the painleve property is given however they are able to construct unknown integrals of motion for a much larger range of parameter values from the integrals exact similarity solutions may be derived this is demonstrated by examples corresponding to the important moving wave reduction

backlund transformations for diffusion equations abstract a class of backlund transformations is deduced for the diffusion equation s u t c u u x x by taking advantage of the conservative form of the equation the transformations are used for obtaining new relations connecting various equations of the type u t u a u x x and u t u a u xx

inverse transformations and the reduction of nonlinear dirichlet problems involutory transformations are introduced which allow the solution of a class on nonlinear dirichlet problems the application of coupled involutory and backlund transformations is also described

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

analytical model of infiltration under constant concentration boundary conditions 1 known integrable models for 1d flow in unsaturated soil have a rescaled soil water diffusivity that is either constant or proportional to c c 1 c 2 where is the degree of saturation and c 1 is constant with a wider more realistic range of hydraulic conductivity functions than has been used in this context before a formal series solution is developed for infiltration subject to constant concentration boundary conditions a readily programmed iteration algorithm applicable for any value of c is used to construct many coefficients of the infiltration series without requiring any numerical integration in particular for either c 1 small or 1 c small several infiltration series coefficients are constructed as formal power series in c 1 or in 1 c for which we construct a number of terms explicitly in the limit as the diffusivity approaches a delta function the infiltration coefficients are obtained in simpler closed form all but the sorptivity depend on the form of the conductivity function

error analysis of spectral estimates with application to the measurement of acoustic parameters using random sound fields in ducts several methods have been proposed using multiple point pressure measurement of random sound fields in ducts to determine acoustic properties of materials and systems this paper presents an error analysis of the spectral estimates used in these techniques expressions for the normal acoustic absorption coefficient and impedance are derived for a random sound field in a duct theory is developed to determine the bias and random errors in estimating the spectral density function for plane wave propagation in the duct a bivariate stochastic process has been employed to model the acoustic system experimental and theoretical calculations show that minimum bias error can be achieved by using a small bandwidth in estimating the spectra and by locating the microphones close to the sample furthermore random error can be minimized by maintaining a high coherence between microphone signals this implies that the microphones should have a small spacing however high coherence may not be realized when a microphone location coincides with a node point in the sound field

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

application of linked backlund transformations to nonlinear boundary value problems a class of nonlinear boundary value problems is reduced to linear canonical form by a combination of backlund transformations

large time solution of a nonlinear diffusion equation in this paper we consider the way in which the solution to a class of initial boundary value problems for the plane nonlinear diffusion equation frac partial partial t frac partial partial x u beta frac partial u partial x t geqslant 0 x geqslant 0 beta 0 approaches the similarity form as t rightarrow infty the problem we solve is chosen for two main reasons firstly the equation above is of widespread use in modelling physical situations and secondly it provides a tractable but significant example of a free boundary problem

symmetry groups and separation of variables of a class of nonlinear diffusion convection equations in this paper a class of nonlinear diffusion convection equations ut d u uxn x p u ux which has quite a large number of physical applications is analysed by using symmetry group methods which include the classical method the potential symmetry method and the generalized conditional symmetry method a complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits lie s point symmetry groups and potential symmetry groups the separation of variables for the equation is investigated using the generalized conditional symmetry approach for some interesting cases exact solutions using the method of separation of variables are discussed in detail

exact solutions of the richards equation with nonlinear plant root extraction the richards equation commonly used to model water flow in unsaturated soils is highly nonlinear thus making it very challenging to solve analytically for situations meaningful in practical applications the inclusion of realistic forms of root water uptake rates in this equation adds complications in deriving exact solutions this study provides for the first time analytical solutions of the richards equation with a sink term nonlinearly dependent on soil water content these solutions are applied to irrigation furrows using cartesian coordinates and irrigation from a circular plate in cylindrical coordinates

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

characterization of the optimum transmitter correlation matrix for mimo with antenna subset selection we give a simple and complete characterization of the optimum transmitter correlation matrix for multiple input multiple output mimo systems when antenna subset selection is used it is assumed that the receiver feeds back to the transmitter only the selected set of antennas without providing any kind of rank ordering of the selected antennas it turns out that the optimal correlation matrix in this case must be a convex combination of an identity matrix and a matrix whose elements are all ones this reduces the problem of finding the actual optimum to a one dimensional search in any given scenario

a classification of pseudospherical surface equations of type ut uxxx g u ux uxx the equations of type ut uxxx g u ux uxx which describe pseudospherical surfaces are characterized a new class of such equations is obtained this class together with the korteweg de vries kdv equation the modified korteweg de vries mkdv equation and the linear equation completes the classification

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

the equivalence problem for the heat equation in this note we ask for the classes of equations of the second order which can be transformed into the heat equation ut uxx to give a partial answer to the question we express the heat equation by differential forms and prolong it by the estabrook wahlquist method this is motivated by the fact that our analysis is based upon conservation laws for which ideals of differential forms are a very suitable framework necessary conditions are derived for deciding whether a given equation can be transformed by some invertible point transformation into the heat equation or into its prolongation in particular the prolongation method enables us to understand the connection of various equations to the heat equation

potential equivalence transformations for nonlinear diffusion convection equations potential equivalence transformations pets are effectively applied to a class of nonlinear diffusion convection equations for this class all possible potential symmetries are classified and a theorem on their connection with point symmetries via pets is also proved it is shown that the known nonlocal transformations between equations under consideration are nothing but pets the action of pets on sets of exact solutions of a fast diffusion equation is investigated

approximate lie group analysis and solutions of 2d nonlinear diffusion convection equations approximate lie symmetries of the 2 1 dimensional nonlinear diffusion equation with a small convection are completely classified it is known that the invariance principle furnishes a systematic method of solving initial value problems the solutions of instantaneous source type of the 2d diffusion convection equation are obtained for the case of power law diffusivity using a symmetry reduction

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed

immiscible two phase flow in a porous medium utilization of a laplace transform boost a boost operator is defined for the laplace transform it is used in conjunction with the laplace transform to solve the diffusion equation on a domain with multiple moving boundaries this problem arises in the study of immiscible multiphase flow in a finite medium including end effects extensions of this method to the solution of other linear differential equations are conceivable

uncertainty analysis of the two microphone method accurate duct acoustic propagation models are required to characterize and subsequently reduce aircraft engine noise the boundary conditions for these models ultimately rely on accurate measurements of the acoustic impedance for candidate nacelle liners the standard two microphone method is widely used to estimate acoustic impedance but does not provide uncertainty estimates that are essential for data quality assessment and model validation a systematic framework is presented in this paper to propagate the experimental 95 confidence interval uncertainty estimates in the measured quantities to the calculated acoustic impedance data using the two microphone method for small and large input uncertainties one method is an analytical small perturbation technique using multivariate statistics this method provides useful analytical scaling information a second monte carlo method is utilized to validate the analytical multivariate method both methods are applied to the two microphone method for simulated data representative of sound hard and sound soft acoustic materials the results indicate that for realistic experimental parameters the uncertainty in the input measurements results in nonlinear perturbations therefore the small perturbation technique fails and the monte carlo method should be used to obtain uncertainty estimates furthermore the nonlinear perturbations also distort the output statistical distributions of the calculated acoustic impedance and reflection coefficient which results in significant deviation from gaussian behavior the distortion is a function of the acoustic impedance of the specimen itself finally the monte carlo method is applied to experimental data for a ceramic tubular specimen and compared to the multivariate results to demonstrate the implementation and usefulness of the technique

a two phase free boundary problem for a nonlinear diffusion convection equation a two phase free boundary problem associated with a diffusion convection equation is considered the problem is reduced to a system of nonlinear integral equations which admits a unique solution for small times the system admits an explicit two component solution corresponding to a two component shock wave of the burgers equation the stability of such a solution is also discussed

non integrability of non linear diffusion convection equations in two spatial dimensions it is generally accepted that integrable partial differential time evolution equations possess lie backlund symmetries of arbitrarily high finite order it has already been established that in 1 1 dimensions the full class of nonlinear fokker planck convection diffusion equations having this property consists of the known integrable examples these are closely related either to the burgers equations or to the fokas yortsos rosen equation both of which have been applied to unsaturated flow in porous media here the author shows that higher order lie backlund symmetries do not exist for any examples of the nonlinear fokker planck equation in 1 2 one time and two space dimensions

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

universal scaling of spontaneous imbibition for water wet systems 1 spontaneous counter current imbibition si is a key mechanism in many multiphase flow processes such as cleanup of nonaqueous phase liquids napls bioremediation or co2 storage for interpreting and upscaling laboratory si data and modeling and prediction purposes scaling groups are an essential tool the question of how to formulate a general scaling group has been debated for over 90 years here we propose the first scaling group that incorporates the influence of all parameters on si that are present in the two phase darcy model the group is derived rigorously from the only known exact analytical solution for spontaneous imbibition by relating the cumulative water phase imbibed to the normalized pore volume we show the validity of the group by applying it to 42 published si studies for water oil and water air experiments for a wide range of viscosity ratios different materials different initial water saturations and different length scales in all cases water was the wetting phase our group serves as a master equation whose generality allows the rigorous prediction of the validity of a large number of specialized scaling groups proposed during the last 90 years furthermore our results give strong evidence that the darcy model is suitable for describing si and that including dynamic effects in capillary pressure is not necessary for counter current si contrary to what has been hypothesized two key applications of the group are discussed first the group can serve as the long sought after general transfer rate for imbibition used in dual porosity models second it is the so far missing proportionality constant in imbibition germination models for plant seeds

new explicit solutions of quasilinear heat equations with general first order sign invariants abstract for the one dimensional quasilinear heat equations ut u uxx f u with smooth functions u 0 and f u we derive conditions under which a general quadratic hamilton jacobi operator h u ut h2 u ux 2 h1 u ux h0 u with the functions h2 h1 h0 to be determined preserves both the signs 0 and 0 on the solutions u x t h u is then called a sign invariant of the given parabolic equation several types of different sign invariants are established which are governed by a linear second order ordinary differential equations new explicit solutions via finite dimensional dynamical systems corresponding to zero preserving of the first order operator are constructed