Lutz Recke

Humboldt-Universität zu Berlin | HU Berlin · Department of Mathematics

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type ∂2tu(t,x)−a(x,λ)2∂2xu(t,x)=b(x,λ,u(t,x),u(t−τ,x),∂tu(t,x),∂xu(t,x)),x∈(0,1) with smooth coefficient functions a and b such that a(x,λ)>0 and b(x,λ,0,0,0,0)=0 for all x and λ. We state conditions ensuring Hopf bifurcation, i.e., existence,...

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable $x$ and $C^\infty$-smooth with respect to the unknown function $u$, then the solution i...

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more diffic...

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more diffic...

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$ ∂ t 2 u ( t , x ) - a ( x , λ ) 2 ∂ x 2 u ( t , x ) = b ( x ,...

The paper concerns boundary value problems for general nonautonomous first-order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right-hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic....

We consider singularly perturbed Dirichlet problems which are, in the simplest nontrivial case, of the typeε2u″(x)=f(x,u(x)) for x∈[0,1],u(0)=u0,u(1)=u1. For small ε>0 we prove existence and local uniqueness of solutions u=uε, which are close to functions of the typeu¯ε(x)=u¯(x)+φ0(x/ε)+φ1((1−x)/ε) with f(x,u¯(x))=0 for x∈[0,1] and withφ0″(ξ)=f(0,u...

We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type \begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the...

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,\lambda)>0$ and $b(x,\lambda,0,0,0,...

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the L2-generalized solutions to the initial-boundary value problems become eventually C2-smooth for any initial L2-data. We investigate small global classica...

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand side is small. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous...

We consider periodic solutions with internal transition and boundary layers (periodic contrast structures) for a singularly perturbed parabolic equation that is referred to in applications as reaction-advection-diffusion equation. An asymptotic approximation to such solutions is constructed and an existence theorem is proved. An efficient algorithm...

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorit...

In this paper, we present a result of implicit function theorem type, which was designed for applications to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to time-periodic boundary la...

We investigate evolution families generated by general linear first-order hyperbolic systems in one space dimension with periodic boundary conditions. We state explicit conditions on the coefficient functions that are sufficient for the existence of exponential dichotomies on $\R$ in the space of continuous periodic functions.

The main objective of the paper is to present a new analytic-numerical approach to singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layers (fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numerical solution of such problems. It is based on a priori information...

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in a small parameter. It turns out that the algorithm...

This article is concerned with general singularly perturbed second order semilinear elliptic equations on bounded domains Ω⊂ Rn with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, prove existence and local uniqueness of exa...

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but, mainly, we do not suppose that $F(\cdot,u)$ is smooth for all $u$. Even so, we state conditions such that for all...

The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coeffi...

The equation Δu+λu+g(λ,u)u=0Δu+λu+g(λ,u)u=0 is considered in a bounded domain in R2R2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ,0)=0g(λ,0)=0 for λ∈Rλ∈R, λλ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same...

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the...

We consider boundary value problems for semilinear hyperbolic systems of the type partial derivative(t)u(j) + a(j)(x,lambda)partial derivative(x)u(j) + b(j)(x, lambda, u)=0, x is an element of (0, 1), j=1,..., n, with smooth coefficient functions a(j) and b(j) such that b(j) (x, lambda, 0)=0 for all x is an element of [0, 1], lambda is an element o...

We consider a singularly perturbed parabolic periodic boundary value problem for a reaction–advection–diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic...

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptoti...

We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous...

We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that $b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$. We state conditions for Hopf bifurcati...

We consider the Cauchy-Dirichlet problem for second order quasilinear non-divergence form parabolic operators with discontinuous data. Fixing a solution u0 ∈ W2,1p (Q), p > n + 2 in the coefficients and taking the Fréchet derivative of the operator at u0 we obtain formally a linear non-degenerative problem. We apply the Implicit Function Theorem in...

We study locking of the modulation frequency of a relative periodic orbit in a general $S^1$-equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modula...

We consider the singularly perturbed parabolic differential equation under the assumption that f is T-periodic in t and that the degenerate equation f(u, x, t, 0) = 0 has two intersecting roots. In a previous paper [V.F. Butuzov, N.N. Nefedov, L. Recke, and K.R. Schneider, On a class of periodic solutions of a singularly perturbed parabolic problem...

This paper concerns n × n linear one-dimensional hyperbolic systems of the type ∂ t u j + a j (x)∂ x u j + n k=1 b jk (x)u k = f j (x, t), j = 1, . . . , n, with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data a j and b jk and the reflection coefficients such that the system is Fred-holm s...

In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. In the physical context, these steady states correspond to configurations of droplets that arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces,...

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearize...

The Neumann boundary value problem for a class of singularly perturbed integro-parabolic equations is considered. An asymptotic expansion of a new class of solutions of moving front type is constructed, and a theorem of existence of such solutions is proved.

This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a kind of dissipativity condition (depending...

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type $(\varepsilon(x)^2 u'(x))'=f(x,u(x))+ g(x,u(x),\varepsilon(x) u'(x)), 0< x

The Neumann boundary value problem for a class of singularly perturbed integro-parabolic equations is considered. An asymptotic expansion of a new class of solutions of moving front type is constructed, and a theorem of existence of such solutions is proved.

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type $\varepsilon^2 u''=f(x,u,\varepsilon u',\varepsilon), 0< x

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness ε undergo a slow-time coarsen...

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is sup...

We study a bifurcation problem for the equation Δu+λu+g(λ,u)u=0 on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. Here λ∈R is the bifurcation parameter, and g is a small perturbation. We prove, under certain assumptions concerning an eigenf...

This paper concerns a linear first-order hyperbolic system in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; 1\le j\le n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the coefficients $a_j$ and $b_{jk}$ and the r...

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.

We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, whic...

We prove that the solutions to a 2D Poisson equation with unilateral boundary conditions of Signorini type as well as their contact intervals depend smoothly on the data. The result is based on a certain local equivalence of the unilateral boundary value problem to a smooth abstract equation in a Hilbert space and on an application of the Implicit...

Ein Laser ist eine Lichtquelle, bei der durch Zufuhr von elektrischer Energie Licht besonderer Güte oder Reinheit (einfarbig, kohärent, polarisiert) erzeugt werden kann. Das dynamische Verhalten des Lasers hängt von der Energiezufuhr ab. Sie bestimmt, ob der Laser zum Beispiel dauerhaft Licht konstanter Intensität liefert, ob er dauerhaft flackert,...

The direction of bifurcation of nontrivial solutions to the elliptic boundary value prob- lem involving unilateral nonlocal boundary conditions is shown in a neighbourhood of bifurcation points of a certain type. Moreover, the stability and instability of bifurcat- ing solutions as well as of the trivial solution is described in the sense of minima...

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type ε 2 u '' =f(x,u,εu ' ,ε), 0<x<1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary con...

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type (ε2(x)u′′(x))=f(x,u(x))+g(x,u(x),ε(x)u′(x))(ε(x)2u′(x))′=f(x,u(x))+g(x,u(x),ε(x)u′(x)), 0<x<10<x<1, with Dirichlet and Neumann boundary conditions. Here the functions ε and g are small and, hence, regarded as sing...

We consider the periodic parabolic differential equation ε2(∂2u∂x2−∂u∂t)=f(u,x,t,ε) under the assumption that ε is a small positive parameter and that the degenerate equation f(u,x,t,0)=0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x,t,ε) which is T-periodic in t, satisfies no-flu...

In this paper we describe linear stability properties for the special type of thin film equation corresponding to a presence both destabilizing van der Waals and stabilizing Born forces in the intermolecular interactions. The final stage of the evolution described by such type equation is characterized by the slow–time coarsening dynamics after for...

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type "2u00 = f(x,u,"u0,"), 0 < x < 1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the dierential equation and the boundary condi...

We analyze the final stages of the dewetting process of nanoscopic thin polymer films on hydrophobized substrates using a lubrication model that captures the large slippage at the liquid-substrate interface. The final stages of this process are characterized by the slow-time coarsening dynamics of the remaining droplets. For this situation we deriv...

This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (...

This paper concerns Crandall–Rabinowitz type bifurcation for abstract variational inequalities on nonconvex sets and with multidimensional bifurcation parameter. We derive formulae which determine the bifurcation direction and, in the case of potential operators, the stability of all solutions close to the bifurcation point. In particular, it follo...

We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a...

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type $$ \begin{aligned} \partial _{j} {\left( {a^{{ij}}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } + b^{j}_{\beta } {\left( {u,\lambda } \right)}} \right)} + c^{i}_{{\alpha \beta }} {\left( {u,\lambd...

We prove a bifurcation theorem of Crandall-Rabinowitz type (local bifurcation of smooth families of nontrivial solutions) for general variational inequalities on possibly nonconvex sets with infinite-dimensional bifurcation parameter. The result is based on local equivalence of the inequality to a smooth equation with Lagrange multipliers, on scali...

The Dirichlet boundary value problem for a class of singularly perturbed quasilinear integro-differential equations is considered. The asymptotic expansion for a new class of solutions, which have internal layers, is constructed. Theorems on existence, local uniqueness and asymptotic stability of such internal layer solutions are proved.

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u 0 such that the linearized at u 0 problem is non-degenerate,...

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is invest...

We deal with Dirichlet's problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution $u_0$ such that the linearized in $u_0$ problem is non-degenerate,...

In this paper linear elliptic boundary value problems of second order with non-smooth data (L∞-coefficients, Lipschitz domain, mixed boundary conditions) are considered. It is shown that the weak solutions are Hölder continuous and that they depend smoothly - in the sense of Hölder spaces - on the coefficients of the equation.

Variational inequalities on closed convex sets of a certain type (not cones in general) with a multidimensional parameter are considered, and local Ck-smooth dependence of their solutions on the parameter is proved. The basic idea is to show that under some assumptions, the variational inequality is in a neighbourhood of a given solution equivalent...

Using methods of perturbation theory, we investigate the global behavior of trajectories on a toroidal attractor and in its neighborhood for a system of differential equations that arises in the study of synchronization of oscillations in the mathematical model of an optical laser.

The existence of smooth families of solutions bifurcating from the trivial solution for a two-parameter bifurcation problem for a class of variational inequalities is proved. As an example, a model of an elastic beam compressed by a force $\lambda$ and supported by a unilateral connected fixed obstacle at the height $h$ is studied. In the language...

A quasilinear elliptic equation with unilateral nonlocal boundary conditions is used for explanation of our recent results concerning smooth bifurcation branches for variational inequalities, their direction and stability.

We consider a class of variational inequalities with a multidimensional bifurcation parameter under assumptions guaranteeing the existence of smooth families of nontrivial solutions bifurcating from the set of trivial solutions. The direction of bifurcation is shown in a neighborhood of bifurcation points of a certain type. In the case of potential...

We consider a mathematical model (the so-called traveling-wave system) which describes longitudinal dynamical effects in semiconductor lasers. This model consists of a linear hyperbolic system of PDEs, which is nonlinearly coupled with a slow subsystem of ODEs. We prove that a corresponding initial-boundary value problem is well posed and that it g...

We consider two single-mode semiconductor lasers, which are coupled face to face via the injection of the optical field. We describe the symmetries of the coupled rate-equations model, the associated stationary solutions (synchronous and antisynchronous), and the bifurcations between them for the case when the propagation delay of the injected fiel...

We consider two semiconductor lasers coupled face to face under the assumption that the delay time of the injection is small. The model under consideration consists of two coupled rate equations, which approximate the coupled Lang-Kobayashi system as the delay becomes small. We perform a detailed study of the synchronized and antisynchronized solut...

We consider a class of variational inequalities which includes a model of a beam compressed by a force λ and unilaterally supported by a connected obstacle at the height h. A smooth dependence on parameters λ and h of solutions and intervals of their contact with the obstacle is proved in a neighbourhood of a given solution. The basic idea is to sh...

We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which...

The implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a (possibly infinite-dimensional) parameter. In this way, results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions. The abstract results are applied to a l...

A mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers, is investigated. The authors introduce a suitable weak formulation and prove existence, uniqueness and regularity prop...

In this paper we describe two limiting processes for families of Banach spaces closely related to the standard definition of projective and inductive limits. These processes lead again to Banach spaces. Information about linear operators and duality between basic families of spaces is carried over to the corresponding limit spaces.The abstract resu...

In this paper linear elliptic boundary value problems of second order with non-smooth data (L∞-coefficients, Lipschitz domains, regular sets, non-homogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fred- holm operators between appropriate Sobolev-Campanato spaces, that the weak solutions are...

We present definitions of Banach spaces predual to Campanato spaces and Sobolev-Campanato spaces, respectively, and we announce some results on embeddings and isomorphisms between these spaces. Detailed proofs will appear in our paper in Math. Nachr.

We formulate a result of the type of the Implicit Function Theorem for abstract equivariant equations, and we demonstrate by two examples (problems for ordinary and partial dierential equations) how the assumptions can be verified and how the assertions can be interpreted.

Essential features of two-section DFB semiconductor lasers can be described by a boundary value problem for the so-called coupled wave equations, a linear hyperbolic system of first order partial differential equations with piecewise constant coefficients. In this paper we investigate spectral properties of an operator H defined by this boundary va...

WIAS 1991 Mathematics Subject Classiication. 35Q60, 35L50, 35D05. Key words and phrases. Laser dynamics modeling, existence, uniqueness and regularity of weak solutions, discontinuous data.

We describe the frequency locking of an asymptotically orbitally stable rotating wave solution of an autonomous S1-equivariant differential equation under the forcing of a rotating wave.

We consider abstract forced symmetry breaking problems of the typeF(x, λ)=y. It is supposed that for allλthe mapsF(·, λ) are equivariant with respect to the action of a compact Lie group, thatF(x0, λ0)=0 and, hence, thatF(x, λ0)=0 for all elementsxof the group orbit (x0) ofx0. We look for solutionsxwhich bifurcate from the solution family (x0) asλa...

A method is developed which allows for the calculation of locking regions of self-pulsating multi-section lasers which are exposed to external optical data sequences. In particular, resonant locking is investigated where both wavelength detuning and detuning of the modulation frequency are important.

this paper linear elliptic boundary value problems of second order with non-smooth data (L

In this paper we consider boundary value problems for systems of quasilinear elliptic equations of the type -â{sub j}[a{sub ijaβ}(x, u, λ)â{sub i}u{sub a} + b{sub jβ}(x, u, λ)]++c{sub iaβ}(x, u, λ)â{sub i}u{sub a} + d{sub β}(x, u, λ) = â«Î². In (1.1) (and in the sequel) the summation over the repeated subscripts i,j = 1, ..., N and a = 1,.....