SOLVABILITY OF A FREE-BOUNDARY PROBLEM DESCRIBING THE TRAFFIC FLOWS

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1 Electronic Journal of Differential Equations, Vol. 25 (25), No. 74, pp. 2. ISSN: URL: or ftp ejde.math.txstate.edu SOLVABILITY OF A FREE-BOUNDARY PROBLEM DESCRIBING THE TRAFFIC FLOWS ANVARBEK MEIRMANOV, SERGEY SHMAREV, AKBOTA SENKEBAEVA Abstract. We study a mathematical model of the vehicle traffic on straight freeways, which describes the traffic flow by means of equations of one-dimensional motion of the isobaric viscous gas. The corresponding free boundary problem is studied by means of introduction of Lagrangian coordinates, which render the free boundary stationary. It is proved that the equivalent problem posed in a time-independent domain admits unique local and global in time classical solutions. The proof of the local in time existence is performed with standard methods, to prove the global in time existence the system is reduced to a system of two second-order quasilinear parabolic equations.. Introduction This article is devoted to study one of the mathematical models of the vehicle traffic on straight freeways. This is a phenomenological macroscopic model which describes the traffic flow by means of equations of motion of a viscous gas. The first model of this type was proposed in [9, 2] where the flow of vehicles was considered as the one-dimensional flow of a compressible fluid. This model is often called LWR model. The underlying assumptions of this approach are () a bijective relation between the velocity v(x, t) and density ρ(x, t) of the fluid expressed by the condition v(x, t) = V (ρ(x, t)), (.) (2) the mass conservation law (the number of vehicles does not change with time). It is assumed that the function V satisfies the condition V (ρ) <. Let us denote by (ρ) = ρv (ρ) the intensity of the flow of vehicles (the number of vehicles passing through a given cross-section per unit time) and claim that (ρ) < for the single-lane traffic. The assumption of mass conservation is expressed by the equality b a ρ(x, t + )dx b a ρ(x, x)dx = t+ t (ρ(b, τ))dτ + t+ t (ρ(a, τ))dτ. 2 Mathematics Subject Classification. 35B27, 46E35, 76R99. Key words and phrases. Traffic flows; gas dynamics; free boundary problem. c 25 Texas State University - San Marcos. Submitted February 9, 25. Published March 24, 25.

2 2 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 It follows that for every rectangular contour Γ in the half-plane t, x R with the sides parallel to the coordinate axes one has ρ(x, t) (ρ(x, t))dt =. (.2) At every point where ρ(t, x) is smooth Γ t + (vρ) = t (V (ρ)ρ) + ; that is, t + ((ρ)) =. (.3) Equation (.3) is endowed with the initial conditions of Riemann s type: ρ, x < x, ρ(, x) = ρ (x), x x x +, (.4) ρ +, x x +, where ρ ± is a constant. The Cauchy problem (.3), (.4) arises, for example, in the mathematical description of traffic congestion. A number of model problems for the conservation law (.3), such as the problem of traffic lights or evolution of local congestions, is considered in [8]. It turns out that equation (.3) always has a solution that satisfies equation (.2) and the initial condition (.4) in a suitable weak sense, but this solution need not be unique. In 963, Tanaka proposed another definition of V (ρ) for the singlelane traffic (see [4]). Let us assume that the velocity of vehicles can not exceed a threshold value v max and represent the density by the formula ρ(v) = d(v), where d(v) = L + c v + c 2 v 2 is the average (safe) distance between the vehicles at a predetermined velocity of the flow v, L is the average length of the vehicle, c is the time that expresses the driver reaction, c 2 is the factor of proportionality for the stopping distance. From the formulas for d(v) and ρ(v) one can derive the dependence (.) for V (ρ), which satisfies the condition V (ρ) <. The model of Tanaka is a LWR model with the state equation (.) of a special form, this model plays a very important role in the contemporary studies of the traffic flows [4]. It was mentioned yet in 955 but rigorously formulated only in 974 by J. Whitham, [3], that the farsightedness of the drivers can be taken into account in the following way: v(t, x) = V (ρ(x, t)) D(ρ(x, t)) (x, t) ρ(x, t) with D(ρ) >. Substituting this expression into the conservation law for the number of vehicles, t + (ρv) =, (.5) we arrive at the Burgers equation t + (ρ) = ( ) D(ρ), (.6)

3 EJDE-25/74 SOLVABILITY OF A FREE-BOUNDARY PROBLEM 3 which expresses the conservation law. The novelty of equation (.6) consists in the fact that the driver reduces the velocity with the increment of the traffic density in front of his vehicle and increases the velocity otherwise. The hydrodynamic model (.), (.4), (.6) is called the Whitham model. Another model was proposed by Payne in 97 []. The model relies on the conservation law (.5) with an independent of ρ velocity v, which means that the desired speed of the vehicle is not attained immediately. The following relation between the desired and the real velocity is accepted: d dt v = v t + v v = ( ( v τ where v is a real speed, while V (ρ) D(ρ) ρ V (ρ) D(ρ) ρ is the desired speed. The parameter τ is of order sec., it expresses the rate of convergence. The resulting system of equations reads ( ) ( ) ( ) ρ v ρ ρ + = ( ). (.7) t v D/(τρ) v v τ V v The system is strictly hyperbolic because the matrix of has different real eigenvalues. In 995, Daganzo [] pointed out several shortcomings of Payne s model, as well as of some models proposed later. It was shown, in particular, that the strong spatial inhomogeneity of the initial density may lead to negative velocities. These drawbacks were corrected in the recent modifications of the model. In conclusion, let us mention the Helbing-Euler-Navier-Stokes third-order model proposed in 995, [5, 6]. In this model, the Payne system is complemented by the energy conservation law, which is represented by an equation for a new unknown θ that describes the dispersion of velocity about some mean value. The second equation of system (.7), understood as an equation for the mean velocity, includes an additional term which depends on θ. 2. Formulation of the problem 2.. Euler variables. Our model of the traffic of vehicles relies on the hypotheses of continuum mechanics, that is, it is assumed that the traffic flow is continuous and possesses the principal characteristics of continuous media such as density, pressure and velocity. It is to be noted here that if the initial velocity of a gas equals zero, the motion may be caused by the inhomogenuity of density. Unlike gas dynamics, the initially motionless vehicle can start moving only if an exterior force is applied. Since in the system of equations of a viscous gas the gradient of the pressure is the only component that makes the gas moving, to avoid the vehicle motion in the absence of exterior forces one has to assume that the pressure is constant, i.e., the gas is isobaric. For this reason we regard the traffic flow as the one-dimensional flow of an isobaric viscous gas. The flow is described by the system of two differential equations for the velocity u(x, t) and density ρ(x, t) )), t + (ρu) =, (2.)

4 4 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 t (ρu) + (ρu2 ) = ( u) µρ + ρf (2.2) on the interval L < x < L for t >. Here µ = const > is the viscosity of the flow, F (u) denotes a given external force (acceleration), which is assumed to satisfy the following conditions: F C 2 (, ), F (u), F (u) = F = positive const. for < u u δ, F (u) for u δ u u, and F (u) = for u > u = positive const.. System (2.) (2.2) is complemented by the initial and boundary conditions where u(±l, t) =, (2.3) u(x, ) = u (x), ρ(x, ) = ρ (x), (2.4) u (x) u, ρ (x) ρ + = const. (2.5) The study will be confined to the special situation when the initial density ρ has the form ρ (x) for L x <, < ρ ρ (x) ρ +, ρ = const. for x, ρ (x) for < x L. The aim of the work is to find a weak solution of system (2.) (2.6) such that (2.6) ρ(x, t) for L x < X (t), < ρ(x, t) < for X (t) x X (t), ρ(x, t) for X (t) < x L for all < t < T, (2.7) where x = X (t), x = X (t) are a priori unknown boundaries of supp ρ(x, t). Conditions (2.7) imply discontinuity of the density ρ(x, t) across the boundaries x = X i (t), for this reason the motion of the medium should be understood in a generalized sense, see, e.g., []. Since equations (2.), (2.2) have the form of conservation laws, the Rankine-Hugoniot jump conditions on the surfaces of discontinuity x = X (t), x = X (t) read Applying (2.7) we find that [ρ(u Ẋi)] =, i =,, (2.8) [ρu(u Ẋi) µρ u ] =, i =,. (2.9) u(x i (t), t) = Ẋi(t), i =,, (2.) u (X i(t), t) =, i =,. (2.) The problem consists in finding a solution of system (2.)-(2.2) satisfying the initial conditions (2.4), (2.6) and the boundary conditions (2.)-(2.). This is a free boundary problem that describes the motion of a group of vehicles initially located within the interval (, ).

5 EJDE-25/74 SOLVABILITY OF A FREE-BOUNDARY PROBLEM Lagrangian variables. Let the pair ρ, u be a classical solution of problem (2.)-(2.). Let us introduce the new independent space variable y = x X (t) ρ(s, t)ds. (2.2) On the plane of variables (y, t) the unknown domain Ω(t) = {x : X (t) < x < X (t)} for t > transforms into the time-independent domain = {y : < y < y } with Indeed: by (2.), dy dt = X (t) X (t) X(t) = y = X(t) X (t) ρ(x, t)dx = ρ (x)dx. t dx + Ẋ(t)ρ ( X (t), t ) Ẋ(t)ρ ( X (t), t ) X (t) (ρ u)dx + Ẋ(t)ρ ( X (t), t ) Ẋ(t)ρ ( X (t), t ) = (ρ u) ( X (t), t ) (ρ u) ( X (t), t ) + Ẋ(t)ρ ( X (t), t ) Ẋ(t)ρ ( X (t), t ) =. The function x is considered as a function of the variables (y, t). Differentiating the second equality of (2.2) with respect to y we find that x y (y, t)ρ(x(y, t)) =. This equality defines the first sought function J = x y (y, t). The second unknown is the velocity of the flow: v(y, t) = u(x, t), J(y, t) = ρ(x(y, t), t). It is easy to calculate that x = y x y = ρ y. Let us take two arbitrary points y, y 2 (, ). By (2.2) y 2 y = x(y2,t) x(y,t) ρ(s, t)ds = y2 y ρ(x(y, t), t)j(y, t) dy = const. Differentiating this equality in t and using (2.) we find that = d ( y 2 ) ρ(x(y, t), t)j dy dt = = = y2 y y2 y y2 y y ( (ρt + ρ x x t + ρu x )J + (ρj t ρu x J) ) dy ( (ρt + (ρu) x )J + (ρj t ρu x J) ) dy ρ(j t v y ) dy. Since y, y 2 (, ) are arbitrary, it is necessary that J t = v y. Now we may compose the system of equations for defining J and v as functions of Lagrangian coordinates (y, t): J t = v, (2.3)

6 6 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 v t = F + ( µ v ) J 2 The initial and boundary conditions transform into in T = (, T ). (2.4) v v (, t) = (y, t) =, (2.5) v(y, ) = v (y) u (x), (2.6) J(y, ) = J (y) ρ (x). (2.7) It is clear that every classical solution of problem (2.3)-(2.7) generates a classical solution to the problem (2.)-(2.) and vice versa, which means the equivalence of the problems in Euler and Lagrange formulations. Let us denote 3. Main results Ω T = <t<t Ω(t). Throughout the text we use the traditional notation from [8] for the function spaces and the norms. Theorem 3.. Let u, ρ H 2+α [, ], < α <. Assume that u, ρ satisfy conditions (2.5) and (2.6). Then there exists a maximal time interval [, t ) such that problem (2.)- (2.) has a unique solution X, X H + α 2 [, t ], u, ρ 2+α 2+α, H 2 (Ω t ) which possesses the properties u(x, t) u, < ρ(x, t) < for x Ω(t), t < t. Theorem 3.2. Under the conditions of Theorem 3., for every T > there exists a unique solution X, X H + α 2+α 2 2+α, [, T ] and u, ρ H 2 (Ω T ) of problem (2.)- (2.) such that u(x, t) u, < ρ < ρ(x, t) < ρ for x Ω(t), t < T. 4. Proof of Theorem 3. To prove Theorem 3. we rewrite equations (2.3), (2.4) as a system of two parabolic equations and apply the fixed point theorem, [7]. We establish the existence of a classical solution on some small interval (, t ), where u(x, t) u, 2 ρ ρ(x, t) 2ρ +. Starting at the initial moment t we prove the existence of a classical solution on some interval (t, t 2 ) wherein u(x, t) u, 4 ρ ρ(x, t) 4ρ +. Continuing this process we obtain a sequence < t < t 2 <... < t n <..., such that u(x, t) u, 2 n ρ ρ(x, t) 2 n ρ + for x Ω(t), t n t t n. Without loss of generality we may assume that t < and t n t n <. There are two possibilities: () lim n t n =,

7 EJDE-25/74 SOLVABILITY OF A FREE-BOUNDARY PROBLEM 7 (2) lim n t n = t <. In the first case t might be any positive number, and in the second case t = t. 4.. Auxiliary equations. We begin by deriving some auxiliary equations for the solutions of problem (2.3)-(2.7). To derive the first auxiliary equation we substitute v into equation (2.4) for v and make use of equation (2.3): v t ( ) µ J J 2 = F, t or v ( t + 2 µ ) = F. t J Integration in time gives the equality v + µ ( J ) t = Φ F ( v(y, τ) ) dτ + v + µ ( J ). (4.) To get the second auxiliary equation we differentiate (4.) with respect to y and once again apply (2.3): J t + µ 2 ( ) Φ = 2 J. (4.2) The boundary conditions (2.5) for v and equation (2.3) yield the boundary conditions for J: J J (, t) = t t (y, t) =, or J(, t) = J (), J(y, t) = J (y ). (4.3) 4.2. Reduction to a fixed point theorem. Now we rewrite equations (2.4) and (4.2) in the form where ρ = /J, v t µρ 2 2 v 2 = F (ρ, v), t µρ 2 2 ρ 2 = F 2(ρ, v), F (ρ, v) = F + 2µ ρ v, F 2(ρ, v) = ρ 2 Φ. Let = (, y ), t = (, t ), ρ (y) = J, a = max{ v (y) (+α), ρ (+α) } and { M = (ṽ, ρ) : ṽ, ρ Hα, α 2 (t ), ṽ(y, t) u, 2 ρ ρ(y, t) 2ρ +, max{ ṽ (α) t, ρ (α) t } 2a For every (ṽ, ρ) M, the linear problem constituted by the equations }. v t µ ρ 2 2 v 2 = F ( ρ, ṽ), (4.4) t µ ρ 2 2 ρ 2 = F 2( ρ, ṽ), (4.5)

8 8 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 the initial and boundary conditions (2.5), (2.6), and the conditions defines the operator ρ(, t) = ρ (), ρ(y, t) = ρ (y ), ρ(y, ) = ρ (y) (4.6) (ρ, v) = Ψ( ρ, ṽ) = ( Ψ ( ρ, ṽ), Ψ 2 ( ρ, ṽ) ). Every fixed point of the operator Ψ in M is the sought solution of the problem (2.3)-(2.7) on an interval (, t ) Correctness of the linear problem. A straightforward calculations show that for every ( ρ, ṽ) M there is the inclusion F i ( ρ, ṽ) H α, α 2 (Ω t ), i =, 2, and for every ( ρ, ṽ ), ( ρ 2, ṽ 2 ) M F i ( ρ, ṽ) (α) t C max { ṽ (α) t, ρ } (α) t, F i ( ρ, ṽ ) F i( ρ 2, ṽ 2 ) (α) t C max { ṽ ṽ 2 (α) t, ρ ρ 2 } (α) t, i =, 2, where the constant C is independent of ( ρ, ṽ ), ( ρ 2, ṽ 2 ) M. It follows from the well-known results in [8] that the linear problem (2.5), (2.6), (4.4)-(4.6) has a 2+α 2+α, unique solution (ρ, v) = Ψ( ρ, ṽ) H 2 (Ω t ) and max { v (2+α) t Moreover, for every ( ρ, ṽ ), ( ρ 2, ṽ 2 ) M, max { v v 2 (2+α) t, ρ ρ 2 (2+α) } t, ρ (2+α) } { ṽ t C max (α) t, ρ } (α) t. (4.7) C max { ṽ ṽ 2 (α) t, ρ ρ 2 } (4.8) (α) t with C depending only on the constant a Existence of the fixed point. The existence of at least one fixed point of the operator Ψ follows from we the Schauder fixed point theorem [7]. To apply this theorem one has to prove that (a) the operator Ψ is completely continuous on M, (b) Ψ transforms the set M into itself. Assertion (a) follows from estimates (4.7), (4.8). The former estimate and the 2+α +α 2+α, imbedding H 2 +α, (Ω t ) H 2 (Ω t ) (see [8]) yield compactness of the operator Ψ. The latter estimate implies continuity of Ψ. Assertion (b) follows from the maximum principle (5.3) proved in Subsection 5. k+α k+α, below and the properties of the norms in H 2 (Ω t ), k =,, 2,..., [8]. For example, ρ(y, t) ρ (y) ρ (2+α) t t, t t +α 2, (y, t) (y) (2+α) ρ (y, t + τ) τ α 2 (y, t) ρ (2+α) t t /2,

9 EJDE-25/74 SOLVABILITY OF A FREE-BOUNDARY PROBLEM 9 (y + h, t) h α (y, t) max t 2 ρ (y, t), 2 2 ρ 2 (y, t) 2 ρ 2 (y) ρ (2+α) t t α 2. These relations entail the inequalities 2 ρ ρ ρ (2+α) t t ρ(y, t) ρ + + ρ (2+α) t t 2ρ +, (y, t) (y) + ρ (2+α) t t +α 2 a + Cat +α 2, (y, t + τ) (y, t) α Cat 2 (y + h, t) h α τ α 2 (y, t) max, 2 ρ (y) + ρ (2+α) t 2 t t α 2 a + Cat α 2. It follows that for the sufficiently small t operator Ψ transforms the convex set M into itself. 5. Proof of Theorem 3.2 We will rely on the already established existence of a classical solution to problem (2.3)-(2.7) on the interval [, t ). The estimates v(y, t) u, < ρ < ρ(y, t) < ρ for x t with ρ = ξ(t ), and < ξ(t ) < for t <, are the main ingredients of the proof of the global in time existence. The proof is split into several steps. 5.. The maximum principle. The proof of the estimate v(y, t) u (5.) is quite standard. Let us introduce the function w by the relation v = we αt, α >. The function w satisfies in T the equation w t + α w ( µ w J 2 ) = F e αt. (5.2) Let us assume that w attains its negative minimum at a point (y, t ) T. Then the left-hand side of equation (5.2) is strictly negative because w t (y w, t ), (y 2 w, t ) =, 2 (y, t ), α w(y, t ) <, while the right-hand side remains strictly positive. This contradiction means that w is nonnegative in T. If w attains its local positive maximum at a point (y, t ), it is necessary that at this point and by equation (5.2), w t (y, t ), w (y, t ) =, α w(y, t ) e αt F ( w(y, t ) e αt). Since F (v) = for v > u by assumption, we have w(y, t ) e αt u, 2 w 2 (y, t )

10 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 whence v(y, t) = w(y, t) e αt w(y, t ) e αt u e α (t t) in T. It remains to notice that by Hopf s principle [2, 3] and the boundary conditions (2.6) v cannot attain its maximal and minimal values on the lateral boundaries of T. Since α > is arbitrary, estimate (5.) follows Corollaries of the maximum principle. In what follows we choose T < t. The first corollary is the estimate = ( ) = J J 2 J C, (y, t) T, (5.3) which follows after applying estimate (5.) to (4.). The second corollary is the estimate ρ = J C, (y, t) T, (5.4) which follows from (5.3) and the boundary condition (4.3) The basic integral estimate. Let us multiply equation (4.2) by J t and integrate by parts over the domain : J 2 J t 2 dy + µ J 2 t dy = Φ J t dy. The integrals over the boundaries y = and y = y equal zero due to the boundary conditions (4.3). The second term on the left-hand side of the last equality can be written in the form Thus, where µ J J 2 2 J t dy = µ 2 J = µ Φ = t 2 dy + µ 2 t Let us estimate J and J 2 : J (y, t) µ J 2 (y, t) d dt d dt J 2 J 2 dy + µ J 2 J 2 dy = J + J 2, J J 3 t J 2 dy, J 2 = J 3 J t J 2 dy. Φ J t dy, F ( v(y, τ) ) v (y, τ)dτ + v (y) + ( J ) (y). Cµ ( C C 4 t J J J 2 J dy t J J dy ) /2 ( t 2 dy t 2 dy + C ) /2 J 2 J 2 dy J 2 J 2 dy,

11 EJDE-25/74 SOLVABILITY OF A FREE-BOUNDARY PROBLEM t (y, t) F ( v(y, τ) v ( t (y, τ) dτ dy + C ) /2 t (y, t) 2 dy C ( t t (y, t) v ) /2 ( (y, τ) 2 dτ dy + C ) /2 t (y, t) 2 dy ( C ) /2 ( t t (y, t) 2 dy v ) /2 ( (y, τ) 2 dy dτ + C ) /2 t (y, t) 2 dy 4 t (y, t) 2 dy + C t t (y, τ)t 2 dy dτ + C. Here we have used (2.3) and expressed v J through t. Finally we have 4 t (y, t) 2 dy + µ d 2 dt J 2 (y, t) J (y, t) 2 dy ( t C J 2 (y, t) J (y, t) 2 dy + ) t (y, τ) 2 dy dτ +. Set t z(t) = t (y, τ) 2 dy dτ + J 2 (y, t) J (y, t) 2 dy. Then (5.5) is equivalent to dz dy C(z + ), z() = z. By the Gronwall inequality, the last inequality entails the estimate T t (y, t) 2 dy dt + max t T J 2 (y, t) J (y, t) 2 dy C. (5.5) 5.4. Consequences of the basic integral estimate. Let us consider the function w(y, t) = ln J(y, t). By (5.4) we have whence max t T y w(y, t) 2 = w(, t) ( ln J () 2 + y w (y, t) 2 dy C, w(s, t) w (s, t) ds s ) /2 ( w(y, t) 2 dy w ) /2 (y, t) 2 dy ( ( ) /2 ) C + w(y, t) 2 dy. Integrating over and using Hölder s inequality we obtain the estimate w(y, t) 2 dy C. max t T Reverting to (5.6) we conclude that and, therefore, max w(y, t) C, t T J(y, t) = e w(y,t), < e C J(y, t) e C <. (5.6)

12 2 A. MEIRMANOV, S. SHMAREV, A. SENKEBAEVA EJDE-25/74 This estimate means that t = and T might be any bounded number. Acknowledgments. The research of the second author was supported by the Research Grant MINECO-3-MTM P (Spain) and the Program Science Without Borders, CSF-CAPES-PVE, Proceso /24- (Brazil) References [] C. F. Dazango; Fundamentals of transportation and traffic operations, Elsevier Science Inc., 997. [2] A. Friedman; Partial differential equations of parabolic type, Englewood Cliffs, N.J.: Prentice- Hall, 964. [3] A. Friedman; Remarks on the maximum principle for parabolic equations and its applications, Pacific Journal of Mathematics, 8(2), 958, [4] N. H. Garther, C. J. Messer, A. K. Rathi (editors); Traffic flow theory: A state-of-the-art report, Washington DC, Transportation Research Board, 2. [5] H. Helbing; Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73, (2) 67. [6] H. Helbing; Improved fluid-dynamic model for vehicular traffic, Phys. Rev. E, 5, (995) 364. [7] W. A. Kirk, B. Sims; Handbook of Metric Fixed Point Theory, Kluwer Academic, London (2) [8] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva; Linear and uasilinear Equations of Parabolic Type, Providence, Rhode Island, 968. [9] M. J. Lighthill, G. B. Whitham; On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 229, 955, , [] L. V. Ovsiannikov; Introduction to Mechanics of Continuous Media, Parts, 2. Novosibirsk State University, Novosibirsk (977) (Russian) [] H. J. Payne; Models of freeway traffic and control. Math. Models Publ. Sys., 28 () 97, 5-6. [2] P. I. Richards; Shock waves on the highway. Operations Res., 4, 956, 42-5, [3] G. B. Whitham; Linear and nonlinear waves Pure and Applied Mathematics, Wiley- Interscience [John Wiley & Sons], New York-London-Sydney, 974, xvi+636, Anvarbek Meirmanov Kazakh-British Technical University, Tole Bi 59, Almaty, Kazakhstan address: anvarbek@list.ru Akbota Senkebaeva Kazakh-British Technical University, Tole Bi 59, Almaty, Kazakhstan address: akbota.senkebayeva@gmail.com Sergey Shmarev Department of Mathematics, University of Oviedo, c/calvo Sotelo s/n, 337, Oviedo, Spain address: shmarev@uniovi.es