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Math-Net.Ru All Russian mathematical portal V. A. Galaktionov, Ordered invariant sets for KdV-type nonlinear evolution equations, Zh. Vychisl. Mat. Mat. Fiz., 1999, Volume 39, Number 9, 1564 1570 Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use http://www.mathnet.ru/eng/agreement Download details: IP: 46.3.193.253 November 29, 2017, 10:34:02

ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ, 1999, том 39, 9, с. 1564-1570 УДК 517.958:53 ORDERED INVARIANT SETS FOR NONLINEAR EVOLUTION EQUATIONS OF KdV-TYPE 1999 г. У о A, Galaktionov Keldysh Institute of Applied Mathematics, Miusskaya Sq.4,125047 Moscow, Russia and School of Mathematical-Sciences, University of Bath, Bath BA2 7AY, UK Received September 9, 1998 Рассматривается квазилинейное эволюционное уравнение третьего порядка где - достаточно гладкая функция. При \\г(и) = и это уравнение является уравнением Кортевега-де Вриза (КдВ). Для функций \ /, удовлетворяющих нелинейному обыкновенному ' дифференциальному уравнению четвертого порядка, доказано, что уравнение (I) допускает точные решения, принадлежащие так называемому упорядоченному инвариантному множеству S 0. ЭТИ решения являются новыми и, хотя они не являются инвариантными относительно групп преобразований Ли, обсуждается их связь с инвариантно-групповыми решениями. Именно, для рассматриваемых \ / данное уравнение (I) на S 0 становится обыкновенным дифференциальным уравнением первого порядка. Решения и(-, t) е S 0 упорядочены в том смысле, что они удовлетворяют стандартной теореме сравнения. Показано, что уравнение КдВ является предельным случаем рассматриваемых уравнений, допускающих инвариантные множества. Получены некоторые асимптотические свойства инвариантных решений. Обсуждается возможность обобщения на случай квазилинейных уравнений.. 1. INTRODUCTION We construct invariant sets and new exact solutions of the generalized KdV equation which is a semilinear third-order evolution equation of the form u t - u xxx + y{ u ) u xi (x, i)euxu, (I) where \\f(u) is a sufficiently smooth function. If \\f(u) = и then (I) becomes the KdV equation from the theory of water waves (a model for long wave propagation in a channel), while for \\f(u) = u 2 it is the modified KdV equation. The KdV-type equations and their generalizations are well known in the mathematical literature concerning soliton theory, inverse scattering, symmetries of evolution equations and application of group theory (see e.g. [1,2]) as well as in problems of well-posedness of (I) with sufficiently arbitrary functions \\f(u) in suitable functional classes (see e.g. [3-5] and references therein). The KdV and the modified KdV equation are special integrable equations and admit a countable set of exact solutions called solitons. The soliton solutions are known to play a fundamental role in the description of nonlinear interaction of structures in the nonlinear media under consideration. Much less is known about exact solutions of the generalized KdV-type equations (I) with a general nonlinear smooth non-power coefficient \ /. In this paper we show that there exists a class of equations (I), with a specially chosen nonlinear coefficient V /(w), and any such equation admits a one-dimensional invariant set S 0. These new exact solutions are presented in Theorem 1. The solutions in S 0 are not invariant under Lie groups of transformations. Nevertheless, they can be treated as those constructed by means of a certain nonlinear generalization of the classical approaches based on groups of scaling transformation. See the last section. The special generalized scaling structure of these solutions essentially differs from the soliton structure for the KdV equations. Moreover, a curious features of S 0 is that the evolution equation (I) restricted to S 0 generates an order preserving group, where the order is understood in the standard comparison sense. It is well-known that the solutions of the third-order equation (I) do not satisfy the usual comparison with respect to initial data. The Comparison Principle (CP), which is a natural property of general first-order ordinary differential equations (ODE) (as well as of the second-order parabolic equations), is not true for higher-order equations. 1564

1 ORDERED INVARIANT SETS 1565 We call such invariant sets S 0 ordered. Equation (I) on S 0 becomes a first-order in time t nonlinear ODE, and the spatial variable x then playsi the role of a parameter. Under natural assumptions, the solutions on S Q satisfy the standard comparison thebrem: if u x (-, t) and u 2 (-, i) are local in time solutions of (I) on S 0 then l! U (; 0) < W 2 (-, 0) t)<u 2 (', t) for t>0. (1.1) Moreover, we show that it is possible to compare different solutions on S 0 = S 0 (\ /) corresponding to different nonlinear coefficients \ /, thus establishing an order solution property of a family of nonlinear equations (I). It is convenient to determine thej invariant set via a first-order differential constraint of the form ij 5Q- = j и = u(x, t): u x = -F(w) j, (1.2) with an unknown smooth function F(u) Ф 0 depending on the nonlinear coefficient of the equation under consideration. The differential structure of the constraint makes it possible to reduce the spatial differentiation on S 0 to a simple algebraic procedure (the algebraic differentiation on S 0 ), and finally reduce (I) to an ODE. We thus apply this algebraic technique to the third-order evolution equation. As for the second-order ones, this constraint was proved in [6, Sect. 3] to be efficient for a general quasilinear second-order parabolic equation of the form j; I u t = (k(u)u x ) x + F(u), (1.3) and for the corresponding N-dimenkional equations in the radial spatial geometry. It gives a class of new exact solutions of (1.3), which are riot of a group-invariant nature. Such new exact solutions exist even for the corresponding semilinear heat equation with k(u)=\ \ щ = u xx + F(u), (1,4) which contains a single nonlinear coefficient F. These exact solutions are constructed for F(u) satisfying the ODE I I FF" - IF + 2 = 0, *i - see [6, Example 3.2]. Moreover, foif the parabolic equation (1.3) the corresponding first-order operator j; I Ж(и) = u -^F(u) (1.5) is proved [6, Prop. 3.3] to be a signunvariant, i.e. Ж(и) preserves signs on the evolution orbits of (1.3) in the sense of a formal application of the Maximum Principle (MP) for linear parabolic equations. See details in [6, Sect. 1, 2]. i For the higher-order equation (I) the MP is not valid, and the operator like (1.5) cannot be a sign-invariant (a sign preserving operator).. Nevertheless, we show that due to the special structure of the constraint (1.2) we can get a certain order preserving property. In Section 2 we consider the KdV-type equation (I). Section 3 is devoted to some generalizations of the method. In particular we show that an ordered invariant set and the corresponding exact solutions are constructed for a quasilinear equation of the form I u t = <P("K^ + y(wk,. (1.6) which contains an extra arbitrary smpoth nonlinear coefficient (p(w). It turns out that the invariant set can be constructed for arbitrary fixed (p with a special choice of \\f depending on ф. Therefore we construct invariant sets for an infinite-dimensional family of equations (1.6). 2. MAIN RESULT 1. levarlaet set and exact solutions. We state the main result of the paper. Theorem 1. Let F{u) be a solutiqn of the following third-order ordinary differential equation D(F) = F\FFT-6FF x -3(F) 2 + 9 F - 6 = 0. (2.1) ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999 I

1566 GALAKTIONOV We set in (I) V /( M) = exp^ J - ^ h (2-2) {\F(z)\- i Then: (i) Equation (I) и>г7/г у given &)>.(2.2) admits the invariant set S 0, (1.2), of exact solutions Г 1 * = log*+ #(?), (2.3) F(z) where the function H solves the ordinary differential equation Я' = de 3H + e H, te R, (2.4) IR ftemg г/ге constant d = [(FF)'-3F + 2](1). (2.5) (ii) Equation (I) б>л S 0 w equivalent to a nonlinear first-order ODE for the function и = w(-, 0 и, = i[f(ff, ), -3FF + F](ii) + -(\ TF)(ii). (2.6) The ODE (2.6) is the evolution equation (I) on the invariant set S 0. We will show that the KdV equation represents a particular trivial case of 5 0. 2. Proof of Theorem 1. (i) Let us show that under the given hypotheses the set of smooth functions S 0 = M(JC,0:. Ж(и) = и х -^(и) = oj (2.7) is invariant under the flow generated by the equation (I). Indeed, calculating the time derivative we obtain that ^ * = o = u tx -hf\u)u t = u xxxx + (\\fu x ) x -^F(u xxx + \\fu x ). (2.8) Using the following algebraic rule of differentiation on S 0 : and u xx =- 2 (F-l), u xxx = 4[(FF)'-3F + 2], (2.9) u xxxx = ^[(F(FF)T-6(FF)'+ 11F-6] x and substituting into (2.8) we conclude that the invariance condition Ж,\ ж = 0 = 1(и) + В(Р) = 0, (2.10) is true provided that the coefficients of the linearly independent functions x~ 2 and r 4 vanish identically: /(w) = F\ /'-\ / = 0 and D(w) = 0. (2.11) This dynamical system on the coefficients coincides with (2.2) and (2.1). Substituting F = \ /'/\ / into (2.1) we obtain a fourth-order ODE for the coefficient \ / of (I). Equation (2.11) is the criterion of the invariance of the whole set (2.7) containing nonstationary solutions u(x, t) which essentially depend on the time variable t. ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999

]'! ORDERED INVARIANT SETS 1567 In order to construct the exact solutions (2.3), we integrate in x equation Ж{и) = 0 and conclude that the function I! м I - satisfies ^ ^ \ 4 r \ = i g* + #(0 < 2-1 2 ) ' I. j 1 v t = u t /F(u) = #', v x = u x /F(u) = l/x. ; i Substituting щ = Fv = F/JK and from (2.9) into (I) and using that \ /(w) = e v by (2.2), (2.12), we arrive at the following equation for the function v: v t \= I[(FF')'-3F' + 2] +V. (2.13) x j x ' ' ' I " Substituting (2.12) into (2.13) we otftain il H\t) = ^{v)fe H, з with O(v) = (FF')'(w) - 3F(u) + 2. (2.14) The function Ф satisfies 0>HO»[(FF'r-3F"]F = ЗФ, (2.15) which is exactly the ODE (2.1). Integrating (2.15) we obtain O(v) = de 3v where d is calculated from (2.14) by setting in the last formula v= 0, i.e. и = 1. Substituting <D(v) into (2.14) we arrive at the ODE (2.4) for the function H(t). \ (ii) Let w(-, 0 6 S 0. Substituting the third derivative from (2.9) and u x = Fix into (I) we obtain the ODE (2.6). This completes the proof of Theorem 1. Remark: a parabolic representation. Using the differentiation rule on S 0 given by (2.9), one can derive that ue S 0 also solves the second-or,der evolution equation with the coefficients i; u t = Au xx + Bu x, (2.16) A = -F', В = ^[FF"-2(F'-1)] + \ /, (2.17) 2 x l which is of parabolic type provided that A > 0. Then the comparison principle follows from the application of the MP for linear parabolic equations. Obviously, this parabolic representation of (I) on S 0 is not unique. 3, Order properties. Let us discuss order preserving properties of the evolution groups restricted to the invariant sets. For a fixed function i.e., the coefficient \ /, the order (comparison) property of different solutions on S 0 in terms of the function v(x, 0 presented in (2.12) follows from the first-order ODE (2.4) which admits monotone trajectoriesjjonly. On the other hand, consider two different functions F x {u) and F 2 (u) which solve (2.1) so that the coiresponding exact solutions щ e S 0 (\ /,-), /=1,2, satisfy different equations (I). Assume now that the functions F t are chosen so that the constants d t given by (2.5) satisfy d x > d 2. Then by standard comparison results from the ODE's theory we conclude that v { (-, t) < v 2 (-, t)fort>0 provided that this inequality holds at t Ц0. One can translate that to the comparison of the solutions щ under a suitable hypotheses on the functions jjf,-. 4. KdV as the limit equation. Wfe first construct trivial invariant set for the KdV equation. Let us find a linear solution F{u) = Си of the ODE (2.1). Substituting we arrive at the quadratic equation С 2-3C + 2 = 0 whence two linear solutions i, F { (M) = и and F 2 (u) = 2u. (2.18) The first one gives ^{(u) = и and coitesponds to the KdV equation u j- t = Uxxx + UU x. (2.1.9) It then follows from (2.3) that u{x, t) = e m x = h{t)x on S 0. Substituting into (2.19) we get К = h 2 whence the separate-variables solution u(x, t).= -^x/(t + c). It is clear that these solutions are ordered. ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999

1568 GALAKTIONOV Similarly, \ / 2 (w) = Ju and the equation takes the form with the invariant set consisting of the solutions u{x, t) = x 2 /(c -1) 2. u t - u xxx + +Juu x, (2.20) Let us now show that the linear solutions (2.18) are the limit cases of more general solution of the ODE (2.1). By a standard local analysis of the ODE at и = it can be shown that it admits other solutions satisfying up to higher-order terms F(u)'= u + 0(e~ 6u ), and F(u) = 2u + 0(e~ 3u ) as и. (2.21) The first asymptotic expansion means that the KdV equation (2.19) plays a role of the limit state (as и оо that means rapidly growing solutions) of the family of KdV-type equations (I) with ordered invariant sets under a special choice of the coefficients i /(w). 5. On the blow-up structure of the maximal solution. First of all, let us mention that the solutions of the ODE (2.4) are essentially local in time. Let d > 0. Then for any initial value #(0) the function H(i) blows up in finite time T < and as t T~ H(t) = - log[3d(r-0][l + O(l)] oo. (2.22) Second, it can be shown that there exists a solution of (2.1), which exhibits the maximal growth as и the following expansion for и > 1: with F m {u) = nlog-n[l + 0(1)]. (2.23) Substituting both expansions (2.22) and (2.23) into (2.3), we arrive at the following asymptotic blow-up behaviour of the solutions on S 0 (\\f m ): u(x, t) ~ txp{x 3,2 /j3d(t-1)}. Observe that this blow-up pattern is an asymptotic solution of the equation Remark: solutions on an invariant subspace. Let us present another example of exact solutions to (I) with a logarithmic nonlinearity Щ = u xxx + (\ogu)u x. (2.24) Setting и - e v we obtain the equation 7 r = B ( v ) E v m + 3 v ^ + (vj 3 +vv r (2.25) One can see that the linear span W 2 = Span{ 1, x) is invariant under the cubic operator B: B(W 2 ) с W 2. Therefore the evolution equation (2.25) on W 2 reduces to a two-dimensional dynamical system for the coefficients of the expansion v(x, t) = C 0 (t) + C x (f)x e W 2, which has the form and is easily integrated. This gives the following exact solution of (2.24): u(x, t) = exp{(l + At-xt)t }, where A is an arbitrary constant. This solution blows up as t - 0". Several examples of linear subspaces and sets invariant under nonlinear operators of the third and higher orders can be found in [7] and [8]. Remark: on an operator representation. It can be shown that the set of solutions (2.3), (2.4) can be treated as an one-dimensional subspace invariant under a certain nonlinear operator. The concept of linear subspaces admitted by nonlinear operators is a useful one in the study of the one-dimensional and multidimensional nonlinear evolution equations, see [6-9] and references therein. It was shown that given a nonlinear operator, the problem of the determination of linear invariant subspaces reduces to an extremely difficult nonlinear eigenvalue problem. On the other hand, for an ODE operators of finite order, given a linear subspace, the class of operators admitted it can be described in terms of Lie-Backlund symmetries of the ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999

ORDERED INVARIANT SETS 1569 linear ODE describing the subspace ^[10], In the present paper we thus give a new representation of nonlinear operators admitting one-dimensional invariant subspaces of a special structure, i! - 3. QUASILINEAR KdV-TYPE EQUATIONS In this section we show that the technique from Section 2 applies to quasilinear equations of the KdVtype. 1, We begin with the equation j j щ = (p(u)u xxx + \y(u)u x, (3.1) where both ф(и) and Щи) are arbitrary smooth functions. Let us briefly discuss some new features in the application of the same technique.! (i) The coefficient \ / is still given by the expression (2.2) where F solves the following governing ODE! 0(F) + ^F[(FF')' - 3F' + 2] = 0, (3.2) "\ Ф " - with the operator В from (2.1). The exact solution is given by (2,3) where H(t) solves the ODE (2,4) with the constant \ \d = {9[(FF7-3F' + 2]}(1). (ii) Equation (3,1) on S 0 is equivalent to the ODE ^[(FFy~W + 2] + ly. (3,3) I x * The proof goes along the same lines. In order to derive from (3,3) the ODE for the function Hit) we again note that on the left-hand side щ = F(u)H\ and on the right-hand side the function Ф(У) = ф(w)[(ff, ), -ЗF, Ф2](w) (3.4) is the exponential one, Ф(у) = de 3v, which follows from the equation <f>' v = ЗФ or in view of (3,4) that This is the ODE (3.2). ^[(FF')V3F + 2]}'F = 39[(FF7-3F' + 2], (3.5) Observe the invariant set and the exact solutions exist for arbitrary nonlinear coefficient ф(и). On the other hand, this means that for an arbitrary fixed second coefficient \ /(w), which determines the function F by (2.2), there exists the first coefficient <$(u) 'such that (3.1) admits the invariant sjpt, 2 Let us finally show how the "Scaling order" of the nonlinear operator affects the ODE structure of the invariant set SQ, If we add to the equation a quasilinear second order term u t fc <?(u)u xxx + p(u)u xx + y(u)u x9 (3.6) then it follows from (2,9) that the corresponding ODE on S 0 takes the form (compare with (3.3)) u t mfh ш F \q>[(ff<y - 3F + 2] 4- - 2 p(f'- 1) + V (3.7) \x x Therefore in order to get the exact solution (2,11) we need to introduce an extra equation in addition to (2.2) and (3.2), Namely, we deduce that tlie function Ф(у) = p{u)[f\u) «1] must satisfy = 2Ф, i.e. we obtain the third equation ; which is equivalent to Г j! [p(f^l)]'f.= 2p(F'-l), From (3.7) we then obtain the ОРЙ FF m^2f + 2 + 6-F(F 1-1) = 0. H s de H + d x e 1H + e H, (3.8). 1(Q) ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999

1570 GALAKTIONOV where d x = Ф(0) = [p(f-l)](l). 3. Comparison with solutions invariant under a group of scalings. The number of distinct exponential terms on the right-hand side of the ODE (3.8) (which is (3.6) on S 0 ) thus depends on the total number of the distinct scaling orders of the operators in (3.6), which contains three operators of the scaling orders 3, 2 and 1. For instance, in the model equation u t = y(u)u xxx + \i(u)(u x ) 3 + v(u)u x u xx, the scaling orders of all the three operators is equal to 3. Then the ODE on S 0 takes the simplest one-term form Я' = de H, whence the solution H(t) = -log(3d(-0)/3. Substituting this into the exact solution from (2.3) we obtain the self-similar solution of the form u(x,t) = O(^), % = xlt ll \ (3.9) which is invariant under a Lie group of scaling transformations with the infinitesimal generator = х^- + ox з. ot Substituting (3.9) into (3.1) one obtains the ODE for the function 0 Ф(в)в т + p(0)(0') 3 + v(e)e, e n + e f S = o. Thus in the case of a single scaling order operator, the ordered invariant set S 0 is generated by the invariant of a Lie group of scalings. The idea of the first order differential constraint (1.2) applies to more general evolution equations with rather arbitrary algebraic scaling structure. ACKNOWLEDGMENTS This paper was finished while the author visited Department of Mathematics, University Autonoma of Madrid as "Profesor Visitante Iberdrola de Ciencia у Tecnologia", Iberdrola, Bilbao. He is grateful to both institutions for their hospitality and support. REFERENCES 1. Bluman G.W. and Kumei S. Symmetries and Differential Equations, New York: Springer, 1989. 2. Ibragimov N.H. Transformations Groups in Mathematical Physics, Dordrecht: D. Reidel, 1985. 3. Bona L. and Smith R. The Initial Value Problem for the Korteveg-de Vries Equation, Proc. Roy. Soc. London, 1978. V.A278. pp. 555-601. 4. Christ M. and Weinstein M. Dispersion of Small Amplitude Solutions of the Generalized Korteveg-de Vries Equation, J. Funct. Analys., 1991. V. 100, pp. 87-109. 5. Kruzhkov S.N. and Faminskii A.V. Generalized Solutions of the Korteveg-de Vries Equation, Dokl. Acad. Nauk USSR, Ser. Math., 1981. V. 261, pp. 1296-1298 (English transl.: Soviet Math. Dokl., 1981. V. 24). 6. Galaktionov VA. Quasilinear Heat Equations with First-Order Sign-Invariants and New Explicit Solutions, Nonlinear Analys., Theory, Meth. and Appl., 1994. V. 23, pp. 1595-1621. 7. Galaktionov VA. On New Exact Blow-up Solutions for Nonlinear Heat Conduction Equations with Source and Applications, Differ. Integr. Equat., 1990. V. 3, pp. 863-874. 8. Galaktionov VA., Posashkov SA. and Svirshchevskii S.R. On Invariant Sets and Explicit Solutions of Nonlinear Equations with Quadratic Nonlinearities, Differ. Integr. Equat., 1995. V. 8, pp. 1997-2024. 9. Galaktionov VA. Invariant Subspaces and New Explicit Solutions to Evolution Equations with Quadratic Nonlinearities, Proc. Roy. Soc. Edinburgh, 1995. V. 125A, pp. 225-246. 10. Svirshchevskii S.R. Invariant Linear Subspaces and Exact Solutions of Nonlinear Evolutions Equations, Nonlinear Math. Phys., 1996. V. 3, pp. 164-169. ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ том 39 9 1999