Partial symmetry of initial value problems

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Partial symmetry of initial value problems Zhi-Yong Zhang College of Sciences, North China University of Technology, Beijing 100144, PR China

a r t i c l e

i n f o

Article history: Received 24 February 2016 Available online xxxx Submitted by H.R. Parks Keywords: Partial symmetry Exact symmetry Initial value problems Invariant solution

a b s t r a c t It is generally believed that the symmetry of initial value problems (IVP) must leave both the partial differential equations (PDEs) and initial conditions invariant. In this paper, we propose partial symmetry of IVP which needs less restrictive conditions on the PDEs and initial conditions and only leaves the IVP invariant on some non-empty subset of the whole solution set of the governing PDEs. Thus some symmetries which either only leave the PDEs invariant or only leave initial conditions invariant are partial symmetries of IVP. Then considering from two different starting points, we define two types of partial symmetry and give the corresponding algorithms where one starts with the PDEs and the other is from the initial conditions. Applications to Boussinesq equation and a class of nonlinear heat equation are performed. © 2017 Published by Elsevier Inc.

1. Introduction Symmetry group of differential equations has a huge list of applications in various fields: e.g., generating new solutions from known solutions, classification of invariant equations, construction of conservation laws, linearization of nonlinear differential equations and so on [2,14,16]. Hence, the extended symmetry methods containing both local and nonlocal symmetries are well-developed based on Lie symmetry group theory [1,3,15]. Whilst symmetry analysis method has been particularly successful in the treatment of PDEs, application of the method to solve IVP associated with nonlinear PDEs still remains a great challenge for researchers and is regarded as an open problem [7]. One reason for such situation is the generally accepted view that a symmetry admitted by IVP is obtained by choosing the symmetries of the governing PDEs which leave the initial conditions invariant as well. The well-developed methods make it possible to find symmetries of PDEs, but the invariance of initial conditions impose strict requirements on the symmetries of PDEs and make the majority of symmetries fail to satisfy. Even more, an IVP may have no symmetries in common with the underlying differential equation [11]. Therefore it is of great value to explore rich symmetries of PDEs to study IVP further if we step forward in this way. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.01.054 0022-247X/© 2017 Published by Elsevier Inc.

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Another reason is the fact that the class of initial conditions for PDEs, that can be efficiently handled by the symmetry reduction routine, is too narrow compared with practical needs. Thus it is significant to generalize the selectable scope of initial conditions based on symmetry group theory. The present situation leads to numerous attempts to weaken initial conditions in order to make the prescribed symmetry leaving the IVP invariant. In [9], Goard iterated the classical symmetry method to solve IVP associated with linear or linearizable PDEs. The same author incorporated the imposed initial value as a side condition to find symmetry [10]. Zhdanov [21] utilized higher-order condition symmetry to study IVP. In [19], we gave a general form of infinitesimal operator which leaves IVP invariant via Lie–Bäcklund symmetry. By definition, on the solution set S of PDEs, an admitted symmetry of the PDEs is a transformation that maps any solution to another one. However, we are not interested in the whole solution set S, but only interested in a subset T of S which satisfies the initial conditions. Therefore, a symmetry of IVP should transform any solution in T into another solution in T . Moreover, the PDEs admit more affluent symmetries on the small subset T than on S, which means that a symmetry admitted by IVP may not be the symmetry of the original PDEs or the symmetry of the PDEs only leave IVP invariant in T . Thus T is a key point for further study of symmetry of IVP in such way, then one question arises: (Q1 ): How to find the subset T . In other words, how to find additional conditions to narrow the solution set S of the original PDEs to T . The starting point of question (Q1 ) is the symmetry of PDEs. Since IVP is composed of the PDEs and initial conditions, thus the second question is: (Q2 ): Can we start with the symmetry of initial conditions to search for the symmetry of IVP? Generally speaking, the initial conditions are simpler than the governing PDEs and may possess more rich symmetry information. Thus one can start with these symmetries to search for new symmetry of IVP. Even more, one can directly set the initial conditions as the characteristic of symmetry to test whether such symmetry can generate new solutions of IVP. In this paper, we will use the idea of partial symmetry to study the above two questions. More precisely, we define two types of partial symmetry for IVP and give the corresponding algorithms where one starts with the symmetry of PDEs and the other is from the symmetry of initial conditions. Indeed, the idea of partial symmetry is contained in [12] by Klein in 1926 and followed in [5,15,18]. Especially in [5], the authors characterized the partial Lie point symmetry both in theoretical terms and in operation ways by some concrete examples to obtain solutions of nonlinear PDEs. Partial symmetry, which does not leave the whole solution set invariant, but makes some non-empty subset invariant. That is to say, the invariant solution space induced by the partial symmetry becomes smaller. It should be mentioned that partial symmetry, which looses the restrictions of the symmetry and contains condition symmetry as a particular case, can lead to a class of solutions which cannot be obtained by the traditional symmetry method. Since the determining equations for partial symmetry are nonlinear as the one for condition symmetry, it is difficult to find all partial symmetries and thus we may need some candidate symmetries which may be considered from physical considerations or other practical efforts. To the end of introduction, we sketch the outline of the paper. In Section 2, some basic notions and principles are reviewed. The definitions of two types of partial symmetry of IVP are introduced and the algorithms for computing them are given in Section 3. Section 4 concentrates on the applications of the method to the Boussinesq equation and a class of nonlinear heat equation. Some concluding remarks are discussed in the last section. 2. Preliminaries In this section, we review some basic notions and main principles concerning exact symmetry and partial symmetry of PDEs. Note that we call Lie point symmetry, conditional symmetry and Lie–Bäcklund symmetry as exact symmetry in order to compare with partial symmetry.

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2.1. Exact symmetry Consider the general nonlinear PDEs given in the form of a system of l differential equations, and briefly denoted by E(x, u(k) ) = 0,

(1)

where E = (E1 , E2 , . . . , El ) are smooth functions involving p independent variables x = (x1 , . . . , xp ) ∈ Rp and q dependent variables u = (u1 , . . . , uq ) ∈ Rq , together with the derivatives of uα with respect to the xi (α = 1, . . . , q; i = 1, . . . , p) up to some order k, denoted by u(k) . The summation convention for repeated indices is used if no special notations added. The classical method for obtaining Lie point symmetry admitted by PDEs (1) is to find a one-parameter local transformation group x∗i = xi +  ξi (x, u) + O(2 ), u∗α = uα +  ηα (x, u) + O(2 ),

(2)

which leaves Eq. (1) invariant. Lie’s fundamental theorems showed that such group is completely characterized by the infinitesimal operator X = ξi (x, u)∂xi + ηα (x, u)∂uα , and thus required X satisfying pr(k) X(E) = 0,

when E = 0,

(3)

where pr(k) X stands for k-order prolongation of X calculated by the well-known prolongation formulae [2,14]. Such prolongation reduces the intractable nonlinear conditions (3) to an over-determined linear homogeneous system of PDEs about the infinitesimals ξi and ηα , and the system can be tackled by the symbolic manipulation programs. We refer to [2,14] and references therein for details. Conditional (nonclassical) symmetry method proposed by Bluman and Cole [3] requires that Eq. (1) augmented with the invariant surface conditions ξi (x, u)

∂uα − ηα (x, u) = 0, ∂xi

α = 1, . . . , q,

are invariant under the transformation (2), then one obtains an over-determined nonlinear system of PDEs for the infinitesimals ξi and ηα which contains all solutions of Lie point symmetry, and thus the solution set in the nonclassical case may be larger. It is common knowledge that further generalization of Lie point symmetry is Lie–Bäcklund symmetry which is expressed in the evolutional form V = Qα (x, u(s) )∂uα for some integer s, where Qα (x, u(s) ) is called the characteristic of symmetry. Such representation of Lie–Bäcklund symmetry is more convenient for the computations. For example, for an unknown function u = u(x, t), the k-order prolongation formula of Ve = Q(x, t, ux , ut , uxx , . . . )∂u is the simplified form pr(k) Ve = Q∂u + Dx Q ∂ux + Dt Q ∂ut + (Dx )2 Q ∂uxx + . . . , where the dots denote the prolongation of the operator to other variables used in the given PDEs, Dx and Dt are total derivative operators with respect to x and t respectively, and ux = ∂u/∂x, ut = ∂u/∂t, etc. In α particular, when Q(x, u(1) ) = ξi (x, u) ∂u ∂xi − ηα (x, u), then it is equivalent to a Lie point symmetry. More extensively, a generalization of conditional symmetry and Lie–Bäcklund symmetry is conditional Lie–Bäcklund symmetry which has exerted significant roles in solving PDEs [8,17,20]. Since a symmetry is completely characterized by the corresponding infinitesimal operators, thus we will not differentiate them.

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2.2. Partial symmetry We briefly recall the definition and algorithm of partial symmetry [5]. It arises under the condition that the infinitesimal operator X does not satisfy condition (3), hence pr(k) X(E)|E=0 = 0, then let E (1) := pr(k) X(E)|E=0 , which is a new system of PDEs whose order, denoted by k1 , is in general not greater than the order k of the original Eq. (1). Suppose that the system, consisting of E = 0 and E (1) = 0, has common solutions and denote the set of such solutions by S (1) . Obviously, the set S (1) is smaller than S (0) due to the additional constraint conditions E (1) = 0. Again, applying the infinitesimal criterion for k1 -order PDEs E (1) = 0 on the solution set S (1) , one has E (2) := pr(k1 ) X(E (1) )|S (1) . Up to now, two different cases come into being. One says that, if pr(k1 ) X(E (1) )|S (1) = 0, then X is a symmetry on the subset S (1) of solution set of Eq. (1). The other is that, if pr(k1 ) X(E (1) )|S (1) = 0, then one can look for the solutions of the system E = E (1) = E (2) = 0 and repeat the procedure as before. The ultimate results may generate two cases which are, up to some step kr , either the set of common solutions E = E (1) = · · · = E (kr ) = 0 is empty and the procedure terminates, or pr(kr ) X(E (kr ) )|S (kr ) = 0 and X is a symmetry on the subset S (kr ) . Clearly, the above procedure can be summarized as follows. Proposition 2.1. (Algorithm [5]) Given the general PDEs (1) and an infinitesimal operator X, define E (0) := E and E (r+1) := pr(kr ) X(E (r) )|S (kr ) . Denote by S (r) the set of the simultaneous solutions of the system E (0) = E (1) = · · · = E (r) = 0 and assume that this is not empty for r ≤ s. Assume that pr(kr ) X(E (r) )|S (r) = 0,

for r = 0, 1, . . . , s − 1;

pr(ks ) X(E (s) )|S (s) = 0. Then the set S (s) provides a family of solutions to Eq. (1) which is mapped into itself by the transformations generated by X. By means of the above procedure and algorithm, we state the definition of partial symmetry admitted by Eq. (1). Definition 2.2. (Partial symmetry [5]) As stated in the Proposition 2.1, X is a partial symmetry of Eq. (1) which is globally invariant on the non-empty solution subset S (s) . The number s is the order of partial symmetry. Partial symmetries do not require invariance of any solution under the vector field, but only global invariance of a family of solutions. Once a partial symmetry is determined, we call Eq. (1) partial invariant

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under the partial symmetry. Partial symmetry also has a nice geometrical interpretation [5]. Acting the proper prolongation X ∗ of X on E = 0 yields ∗

eX E = E +  X ∗ E +

∞

 k 2 ∗ 2 (X ) E + · · · = (X ∗ )k E, 2! k!

(4)

k=0

where  is the group parameter. If X ∗ E = 0, then X is an exact symmetry while for some s, (X ∗ )s E = 0, then X is a s-order partial symmetry. Then the common solutions (X ∗ )i E = 0 (i = 0, . . . , s) is the required solutions. 3. Partial symmetry of IVP We utilize the idea of partial symmetry to study the IVP composed of the PDEs E(t, x, u(k) ) = 0,

(5)

and the initial conditions p(t, x, u(r) )|t=t0 = κ(x),

(6)

where t is time-variable separated from the set x, the order r is no greater than k of Eq. (5), p(t, x, u(r) ) and κ(x) are smooth functions of their arguments. Note that we still use x to denote remainder variables after deleting t if no confusions occur. Generally speaking, the symmetry admitted by the governing equation (5) may not leave the initial conditions invariant. Indeed, the symmetry in question must leave invariant the initial conditions at t = t0 , and this imposes very strong limitations on the utilization of symmetry reduction in the classical setting. For example, a Lie point symmetry Xe = ξi (t, x, u)∂xi + τ (t, x, u)∂t + ηα (t, x, u)∂uα does not leave t = t0 invariant unless τ (t, x, u) satisfies τ (t, x, u)|t=t0 = 0 [19]. Therefore, we will use the properties of Lie–Bäcklund symmetry to overcome this problem. The main idea is to rewrite the infinitesimal operator in the canonical representation of Lie–Bäcklund vector field V = Qα (t, x, u(1) , . . . )∂uα ,

(7)

q  which leaves t = t0 invariant identically. Then the above operator Xe is equivalent to Ve = α=1 ηα (t, x, u)−  ξi (t, x, u)∂uα /∂xi + τ (t, x, u)∂uα /∂t ∂uα . Of course, the price for this is that the order of the symmetry operator is no longer equal to one. In what follows, we assume that the symmetries admitted by the IVP possess the form (7). In the context of the above statements, we introduce the definition of partial symmetry of IVP and present algorithms to compute them. 3.1. Motivation and basic idea An initial value problem where the PDEs are imposed by initial conditions, the allowable solutions are effectively contained in a very small set T ⊂ S, even T a single solution alone, where S is the solution set of the PDEs. However, the subset T is just what we need and in T , we may find a very large symmetry group where some symmetry is not admitted by the original PDEs. This leads to the question (Q1 ) and we answer (Q1 ) with the help of partial symmetry. Answer to (Q1 ): The answer starts with the partial symmetry of the PDEs. Following the notations in Section 2.2, a s-order partial symmetry is indeed obtained in a smaller solution set which satisfies E (0) = 0

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and additional sequence of PDEs E (i) = 0 (i = 1, . . . , s). Thus one can define T = S (s) , then determine the range of the initial conditions that can be handled within the framework of the partial symmetry approach. As stated in the introduction, since partial symmetry is difficult to specify explicitly, we will choose some ansatz guided by the forms of the infinitesimals found by the direct method or non-classical method [1,5,6]. The other idea is to fully use the symmetries of the PDEs not leaving the initial conditions invariant which account for a big part of the symmetries of PDEs and are invalid in dealing with IVP via traditional techniques. But here such symmetries are selected as the candidates of partial symmetry which leave the initial conditions invariant in a certain small solution set, then to explore the extended form of initial conditions. The above two ideas concentrate on the symmetry of the governing PDEs and we call them Type-I partial symmetry of IVP (see Section 3.2). Now we turn to the other starting point, i.e., symmetry or partial symmetry of the initial conditions, which is called Type-II partial symmetry of IVP. This type of partial symmetry replies to the second question (Q2 ) as follows. Answer to (Q2 ): This method puts the PDEs asides and first considers the symmetry or partial symmetry of initial conditions. Specifically, we first compute symmetry or partial symmetry of initial conditions, then discriminate whether the obtained symmetry leaves the PDEs invariant in some small solution set. In practice, one can choose exact symmetry of initial conditions as candidate symmetry to simplify the computations (see Section 3.2). 3.2. Two types of partial symmetry of IVP In this section, the answers for the two questions (Q1 ) and (Q2 ) are described in mathematical language. Two different types of partial symmetry of IVP are introduced and the corresponding algorithms are given respectively. 3.2.1. Type-I partial symmetry Type-I partial symmetry originates from two aspects. The first aspect is to construct partial symmetry of the PDEs (5) and then check which one is a partial symmetry of the IVP. Since this procedure is closely similar as in Proposition 2.1, thus it will be exemplified by the Boussinesq equation. The second one focuses on the symmetries admitted by the PDEs but not by initial conditions. Specifically, suppose a symmetry with the infinitesimal operator form (7) leaving Eq. (5) invariant, and let I (0) := p(t, x, u(r) )|t=t0 − κ(x) = 0, then acting k0 (= r) order prolongation of V on I (0) yields   I (1) := pr(k0 ) V I (0)

|I (0) =0

,

whose order is denoted by k1 . Then two different cases arise. If I (1) = 0, then V is an exact symmetry of IVP and can be used to construct invariant solution of the IVP. If I (1) = 0, let S (1) := {I (0) = I (1) = 0} and S (1) is nonempty, then   I (2) := pr(k1 ) V I (1)

|S (1)

.

If I (2) = 0 and S (2) := {I (0) = I (1) = I (2) = 0} is nonempty, then repeat the above procedure for I (2) . Iterating similar steps for r times, the general form is   I (r+1) := pr(kr ) V I (r)

|S (r)

,

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where S (r) := {I (0) = · · · = I (r) = 0}. Since the order of I (r) (r ≥ 1) is smaller than the one of I (r−1) , thus the procedure of partial symmetry performing for initial condition I (0) = 0 will terminate for some s if S (s) is nonempty. Definition 3.1. (Type-I partial symmetry) An exact (a partial) symmetry admitted by Eq. (5) is a s-order partial symmetry of IVP if pr(kr ) V (I (r) )|S (r) = 0,

for r = 0, 1, . . . , s − 1;

pr(ks ) V (I (s) )|S (s) = 0, hold for nonempty S (s) . Type-I partial symmetry, as a new class of symmetry of IVP, extends the scopes of the candidate symmetry for IVP by partial symmetry of PDEs, and also points out a new applying orientation for the symmetries admitted by the governing PDEs but not by the initial conditions. It may generate new solutions of IVP via symmetry technique and extends the selectable scope of initial conditions. In particular, we formulate the procedure for computing Type-I partial symmetry of IVP as the following three steps. 1. Finding out rich symmetry information of PDEs (5) such as Lie point symmetry, conditional symmetry, Lie–Bäcklund symmetry, partial symmetry and etc. 2. Using Definition 3.1 to choose partial symmetry of IVP from the obtained symmetries. This step divides into two parts: ① Exact symmetry: If an exact symmetry of PDEs leaves the initial conditions invariant, then it is an exact symmetry of IVP. Otherwise, one applies partial symmetry method in Definition 3.1 for the initial conditions to check whether the exact symmetry is a partial symmetry of IVP or not. ② Partial symmetry: If a partial symmetry of PDEs satisfies Definition 3.1, then it is a partial symmetry of IVP, otherwise it is not. 3. Using the obtained symmetry or partial symmetry to construct exact solutions of IVP. In [9,10], Goard found the symmetries admitted by the governing equation do not leave the initial conditions invariant and used them to construct exact solutions of the corresponding IVP. For example, the PDE E (0) = ut − uxx − u2x − 2ux /x = 0 associated with initial condition u(x, 0) = ln(1 + x) is admitted by   the symmetry Ve = (2t + x2 + x)ut + (2tx + x3 + x2 )ux + 2t − x2 − 2 ∂u [10]. In fact, it is a third-order Type-I partial symmetry of IVP. Specifically, one has E (1) = pr(2) Ve (E (0) )|E (0) =0 = 0, E (2) = pr(3) Ve (E (1) )|E (0) =E (1) =0 = 0, and E (3) = pr(2) Ve (E (2) )|E (0) =E (1) =E (2) =0 = μ(x, t)(x3 ux + x2 ux + 2txux + 2t − x2 ), where μ(x, t) is a function of x and t, the finial step E (4) = pr(1) Ve (E (3) )|E (0) =E (1) =E (2) =E (3) =0 = 0,

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thus Ve is a third-order Type-I partial symmetry of E (0) = 0. Solving E (0) = E (1) = E (2) = E (3) = 0 together with u(x, 0) = ln(1 + x) gives u = ln(x2 + x + 2t) − ln x. Note that the order of prolongation in E (2) is 3, greater than 2 in E (1) , because in the computation of E (1) on the solution set of E (0) = 0, we solve E (0) = 0 for ut and substitute it into pr(2) Ve (E (0) ), not for uxx . 3.2.2. Type-II partial symmetry Type-II partial symmetry starts from initial conditions. We take exact symmetry or partial symmetry of initial conditions (6) as the candidate to search for partial symmetry of IVP. Since the main idea is similar as Type-I partial symmetry except for different starting point, thus we only state the concrete procedure of the case where the partial symmetry of initial conditions (6) without leaving PDEs (5) invariant is used to find Type-II partial symmetry of IVP. Specifically, let V = Q(t, x, u(1) , . . . )∂u be a partial symmetry of initial conditions (6). Suppose V does not leave the governing PDEs (5) invariant, which means E (1) := pr(l0 ) V (E (0) )|E (0) =0 = 0,

(8)

where E (0) = E and l0 = k in order to facilitate notations. Then, following the procedure of finding partial symmetry of PDEs, a s-order partial symmetry of IVP is well-defined if there exists a nonnegative integer s such that pr(lr ) V (E (r) )|T (r) = 0,

for r = 0, 1, . . . , s − 1;

pr(ls ) V (E (s) )|T (s) = 0,

(9)

 hold, where T (r) := { 0≤r≤s E (r) = 0} is nonempty. Summarizing the above whole process, we give the definition for Type-II partial symmetry of IVP. Definition 3.2. (Type-II partial symmetry) An exact (a partial) symmetry in the form V = Q(t, x, u(1) , . . . )∂u of initial conditions (6) is a s-order partial symmetry of IVP if it is a s-order partial symmetry of Eq. (5), i.e., it satisfies conditions (9).   For simplicity, one can directly set Vf = p(t, x, u(r) ) − α(x, t) ∂u with α(x, t)|t=t0 = κ(x). Obviously, operator Vf leaves initial conditions invariant identically. Then one can perform the above procedure to find the required partial symmetry of IVP. 4. Applications 4.1. Example I: Boussinesq equation Consider the Boussinesq equation utt + uuxx + u2x + uxxxx = 0,

(10)

associated with the initial conditions α(x, t)ux + β(x, t)u|t=t0 = κ(x),

α(x, t) = 0.

Eq. (10) was introduced by Boussinesq to describe the propagation of long waves in shallow water [4] and later also arises in several other physical applications [6]. Classical and nonclassical symmetries are well-known in [6,13].

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4.1.1. Type-I partial symmetry Case I. Lie point symmetry We first consider Lie point symmetry in the evolutional form, X = Qcs ∂u with Qcs = (c1 x+c2 )ux +(2c1 t+ c3 )ut + 2c1 u, as candidate symmetry to search for partial symmetry of IVP. Following the procedure for Type-I partial symmetry, applying first-order prolongation pr(1) X = X + (Dx Qcs )∂ux on initial conditions I (0) = α ux + β u − γ = 0 with γ(x, t)|t=t0 = κ(x), one obtains I (1) := pr(1) X(I (0) )|I (0) =0 =

1 (χ1 u + χ2 ) , α

where χ1 = (c1 x + c2 )(αβx − βαx ) + (2c1 t + c3 )(αβt − βαt ) + c1 αβ, χ2 = (c1 x + c2 )(αγx − γαx ) + (2c1 t + c3 )(αγt − γαt ) + 3c1 αγ. Iterating similar step, one obtains I (2) := X(I (1) )|I (0) =I (1) =0 = Qcs χ1 /α and I (3) := X(α I (2) /Qcs ) ≡ 0, then I (1) = I (2) = 0 yields χ1 = 0 and χ2 = 0. Then solving χ1 = χ2 = 0 to find α, β and γ, we obtain Eq. (10) together with initial conditions ux +

c1 c2 c31 c32 f (z)u = g(z) c1 x + c2 (c1 x + c2 )3

(11)

is admitted by the second-order Type-I partial symmetry X = Qcs ∂u , where f (z), g(z) are arbitrary functions of z = (x + c2 /c1 )/(2t + c3 /c1 )1/2 . Then by means of partial symmetry X = Qcs ∂u , Eq. (10) subject to initial conditions (11) can be reduced to 2z 3 U  + 2c2 z 2 f U − c32 g = 0, U (4) + z 2 U  + 7(zU ) + 2(U U  ) + U = 0,

(12)

where U (z) = tu + c3 u/(2c1 ). Case II. Conditional symmetry

Consider the conditional symmetry X = Qns ∂u with Qns = 2t2 + x ux + 2tut + 2u + 8t2 + 4x. Note that it is a first-order partial symmetry of Eq. (10). Similarly, acting first-order prolongation pr(1) X = X + (Dx Qns )∂ux on I (0) = α ux + β u − γ = 0 with γ(x, t)|t=t0 = κ(x), we have I (1) := pr(1) X(α ux + β u − γ)|I (0) =0 =

1 (ψ1 u + ψ2 ) , α

where ψ1 = (2t2 + x)(αβx − αx β) + 2t(αβt − αt β) + αβ, ψ2 = (2t2 + x)(αγx − γαx + 4αβ) + 2t(αγt − γαt ) + 3αγ + 4α2 .

(13)

Then I (2) := X(I (1) )|I (0) =I (1) =0 = Qns ψ1 /α, and I (3) := X(α I (2) /Qns ) ≡ 0, which implies ψ1 = ψ2 = 0. It is difficult to find the general solutions of ψ1 = ψ2 = 0, but one can construct some particular solutions. √ √ For example, a particular solution is α = t, β = f (z), γ = −8/9 t2 f (z) − 4/3 xf (z) + g(z)/t − 4/3 t with z = (x − 2t2 /3)t−1/2 , then Eq. (10) together with initial conditions

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[m3L; v1.198; Prn:26/01/2017; 15:03] P.10 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

√

√ 1 4 t 8 2 4 tux + f (z)u = − t f (z) − xf (z) + g(z) − 9 3 t 3

is admitted by X = Qns ∂u , where f (z) and g(z) are two arbitrary functions, and the IVP is reduced to 4U  + 4f U − z 2 f − 4g − 2z = 0, 8U (4) + 4U (2U  + 3) + 6zU  + 8U  2 − 9z 2 = 0,

(14)

2

where U (z) = tu + t [x/(2t) + t] . Further consideration for Eq. (10) is a conditional symmetry X = ∂t +t∂x −2t∂u . With similar procedure, Eq. (10) together with initial condition ux + f (z)u = g(z) − 2xf (z) with z = t2 − 2x is admitted by the second-order Type-I partial symmetry X = (ut + tux + 2t)∂u . Then the reduced equations are 2U  − f U + zf + g = 0, 8U (3) + 2U U  + U − z + c1 = 0,

(15)

where U (z) = u + t2 and c1 is the integrated constant. Observe that the reduced equations (12), (14) and (15) contain two arbitrary functions respectively, which extend selectable scope of initial conditions from the point of view of symmetry. For example, in Eqs. (15), for f (z) = 0, g(z) = 2, one obtains u = 2x − 2t2 is a solution of Eq. (10) subject to (ux )|t=t0 = 2. 4.1.2. Type-II partial symmetry Suppose the candidate symmetry is in the form V = Q ∂u , where Q = α(x, t) ux + β(x, t) u − γ(x, t) with γ(x, t)|t=t0 = κ(x). Obviously, V = Q ∂u leaves the initial conditions invariant identically. Let E (0) := utt + uuxx + u2x + uxxxx , then one has E (1) := pr(4) V E (0) |E (0) =0

= Dt2 Q + uDx2 Q + uxx Q + 2ux Dx Q + Dx4 Q |E (0) =0 = 4αx uxxxx + (6αxx + 4βx )uxxx + (2uαx + uβ + 4αxxx + 6βxx − γ)uxx + (2αx + β)u2x + (uαxx + 4uβx + αtt + αxxxx + 4βxxx − 2γx )ux + 2αt uxt + 2βt ut + u2 βxx + uβtt + uβxxxx − uγxx − γtt − γxxxx .

(16)

If E (1) = 0, then V is an exact symmetry of Eq. (10) given by Ve = ∂x . Otherwise, we search for first-order partial symmetry of the IVP about (10), i.e., E (2) := pr(4) V E (1) |E (0) =E (1) =0 = 0. We don’t give the special expression of E (2) due to the limited space. However the method for dealing with E (2) = 0 is similar to obtain symmetry determining equations, which is to separate it into an over-determined nonlinear system with respect to u and its derivatives. Then solving the system to look for the required partial symmetries. In what follows, we give a classification of first-order Type-II partial symmetry of IVP associated with Eq. (10), which include classical and nonclassical symmetries in the given form. Note that a, b and c are arbitrary constants if no specialization added.

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AID:21080 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.198; Prn:26/01/2017; 15:03] P.11 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

11

The key point to solve the determining system of first-order Type-II partial symmetry of IVP is 3αt αx2 + ααx αxt − ααt αxx = 0.

(17)

For ααx αt = 0, multiplying Eq. (17) by 1/(ααx αt ), then one can integrate it once as αx = ω(t)α3 αt with nonzero arbitrary function ω(t). However, no solutions for this case exist, thus in what follows, we consider ααx αt = 0 and find first-order partial symmetries, then use them to construct reductions and exact solutions of the corresponding IVP associated with Eq. (10). Case I: α = α(t)(α = 0)

  In this case, the partial symmetry of Eq. (10) takes the form V = αux + 2x (α )2 /α − f (t) ∂u , where f (t) is an arbitrary function and α(t) satisfies αα α − 2(α )3 = c1 with an integrated constant c1 . Then E (1) becomes 

   2α2 (α )3 x − α3 α f uxx + 2(α )2 α3 uxt + α2 6(α )3 + c1 ux + 10c1 (α )2 x + 24(α )5 x − α3 α f  = 0.

(18)

  Thus the IVP of Eq. (10) imposed by initial conditions αux + 2x (α )2 /α |t=t = f (t0 ) is admitted by the 0 first-order partial symmetry V and its solution satisfies Eq. (18) which is an additional constraint condition that narrows the solution set of Eq. (10). The general solutions of αα α − 2(α )3 = c1 are expressed by Hypergeometric functions, but it also has particular elementary function solutions.   (1). V1 = (at + b)ux + 2a2 x/(at + b) − f (t) ∂u This case corresponds to α(t) = a t +b (a = 0), then the required solution of the IVP composed of Eq. (10) and initial conditions [(at + b)ux ]|t=t0 = −2a2 x/(at0 + b) + f (t0 ) also satisfies 

 2a2 x − (at + b)f uxx + 2a(at + b)uxt + 4a2 ux − (at + b)f  +

4a4 x = 0. (at + b)2

(19)

The general solutions of Eq. (19) is not easy to find though it can be integrated once, thus we consider a particular case of Eq. (19), i.e., b = f (t) = 0, then after been integrated once Eq. (19) becomes xux + tut + u +

x2 = c2 , t2

(20)

where c2 is the integrated constant. Eq. (20) gives u = c2 − x2 /t2 − 1/x φ(x/t) where φ(x/t) is a function of x/t, then substituting it into Eq. (10) yields u = −x2 /t2 + c3 /t with an arbitrary constant c3 . This solution contains u = −x2 /t2 induced by the weak symmetry X = x∂x + t∂t [15]. Thus Eq. (10) together with (tux )|t=t0 = −2x/t0 is admitted by the first-order partial symmetry V11 = (tux + 2x/t)∂u and has the solution in the form u = −x2 /t2 + c3 /t.   (2). V2 = a/(t + b)ux + 2ax/(t + b)3 − f (t) ∂u The integrated constant c1 = 0 leads to α(t) = a/(t + b) (a = 0). Then Eq. (10) with initial conditions [a/(t + b)ux ]|t=t0 = −2ax/(t0 + b)3 + f (t0 ) is admitted by first-order partial symmetry V2 and the solution satisfies   24ax (b + t)3 f − 2ax uxx + 2a(b + t)uxt − 6aux + (b + t)3 f  − = 0, (b + t)2 which has the same form as Eq. (19) and thus it can be tackled with similar ways.

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Doctopic: Partial Differential Equations

[m3L; v1.198; Prn:26/01/2017; 15:03] P.12 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

Case II: α = α(x)(α = 0) In this case, the general form of first-order partial symmetry is V = [α(x) ux + β(x) u − γ(x, t)] with γ(x, t)|t=t0 = κ(x). The existence of such partial symmetry requires αβα − 4(α )3 − αα β  + β 2 α = 0.

(21)

The solutions of Eq. (21) are β = −2(α4 − 1)α /(α4 + 1) and β = ±2α . The former cannot be solved in general while the latter is considered and divided into four cases.

(3). V3 = xux − 2u − 48/x2 ∂u It is a conditional symmetry admitted by Eq. (10) and gives the Weierstrass elliptic function solutions [15] u = ℘(t)x2 −

12 , x2

(22)

where Weierstrass elliptic function ℘ satisfies 6℘2 − ℘ = 0. Thus Eq. (10) together with initial condition (xux − 2u)|t=t0 = 48/x2 is admitted by the first-order partial symmetry V3 and has solutions in the form (22). (4). V4 = (xux − 2u) ∂u It is also a conditional symmetry admitted by Eq. (10). Here, performing partial symmetry method to Eq. (10) gives u = f (t)x2 + h(t),

(23)

where 6f 2 + f  = 2hf + h = 0 and h(t)|t=t0 = 0. When h = 0, the solution is u = −x2 ℘ (a − t; 0, b) which is obtained by conditional symmetry reduction, thus we obtain more general solutions via partial symmetry. In summary, Eq. (10) subject to initial condition (xux − 2u)|t=t0 = 0 is admitted by the first-order partial symmetry V4 and has solutions in the form (23).   (5). V5 = xux + 2u − f (t)x2 ∂u , where 2f  + 3f 2 = 0.

It is a new partial symmetry of IVP about Eq. (10). Then Eq. (10) with initial conditions xux +2u |t=t = 0 f (t0 )x2 satisfies

8utt + 2f 4xux + x2 uxx + 2xu − 3x2 f 2 = 0. Then this IVP has solutions in the form u = f (t)x2 /4 − 12/x2 , where f (t) satisfies 2f  + 3f 2 = 0. (6). V6 = (xux + 2u) ∂u The operator V6 is a first-order partial symmetries of IVP associated with Eq. (10) but only gives trivial solution u = a t + b. However it may be used for other aspects though now we have no knowledge about it. Case III: α(= 0) is a constant Without loss of generality, set α = 1. Then this case is separated into two subcases. One is β = 0, we find β(x, t) satisfies 

βx βt

x

 βx = β+ = 0. β x

Then we obtain a first-order partial symmetry V = (ux + b u + c)∂u (b = 0) which only gives trivial constant solutions. The other is β = 0 which gives more affluent first-order partial symmetries. (7). β = 0

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Doctopic: Partial Differential Equations

[m3L; v1.198; Prn:26/01/2017; 15:03] P.13 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

13

We obtain V7 = [ux − γ(x, t)] ∂u , where γ = γ(x, t) satisfies 

γxx γ



 =

x

γtt + 4γx γ

= 0,

γ = 0.

x

This case contains several interesting subcases. For example, γxx = 0 yields γ = f (t)x + g(t) with f g = g  f , then we obtain V = [ux − f (t)x − g(t)] ∂u which is a conditional symmetry [15]. Thus Eq. (10) with the initial conditions (ux )|t=t0 = f (t0 )x + g(t0 ) has solutions 

u=

1 f (t)x2 + g(t)x + h(t), 2

f (t) = 0,

where f  + 3f 2 = g  + 3f g = h + f h + g 2 = 0. In particular, for f = −2/t2 , g = t3 , then V = (ux + 2x/t2 − t3 )∂u is a conditional symmetry and gives the same solution as in [15]. Another particular case is V = [ux − g(t)] ∂u (g(t) = 0), which requires g(t) = c1 t + c2 , then the solutions take the form u = x (c1 t + c2 ) −

1 4 (c1 t + c2 ) + c4 t + c3 , 12c21

where ci (i = 1, . . . , 4) are arbitrary constants which make the solutions significant. To the end of this example, we give two remarks about partial symmetry of Eq. (10). Remark 4.1. The obtained Type-II partial symmetries of Eq. (10) contain four conditional symmetries in the case τ = 0 [15]. Moreover, we obtain new partial symmetries of IVP associated with Eq. (10) and consequently find new solutions of the IVPs. Remark 4.2. It is worthy of saying that one can also compute second-order partial symmetry of IVP. For example, a particular case is V = (xux − 2u − f (t)) ∂u (f (t) = 0) which is a second-order partial symmetry of Eq. (10) together with the initial condition (xux − 2u)|t=t0 = f (t0 ), then one can show that the solutions take the form u = ℘(t) x2 − f (t)/2, where ℘ + 6℘2 = f  + ℘f = 0. 4.2. Example II: nonlinear Heat equation Consider the nonlinear heat conductivity equation ut = uxx + F (u),

(24)

where F (u) is an arbitrary function and makes Eq. (24) nonlinear. We study Eq. (24) with partial generalized symmetry in the form Vg = Qg ∂u , where Qg = uxx + A(u)ux + B(u) (A = 0). Let E (0) := ut − uxx − F (u), then E (1) := pr(2) Vg E (0) |E (0) =0

= Dt Qg + Dx2 Qg + QF  |E (0) =0

= F (ux A + B  ) − ux u2x A + 2uxx A + ux B  − ux F  − BF  . The conditions for first-order partial generalized symmetry of Eq. (24) require E (2) := (1) pr Vg E|E (0) =E (1) =0 = 0, then separating E (2) = 0 with respect to different powers of ux yields an over-determined nonlinear ordinary differential equation system about A(u), B(u) and F (u). One simple (2)

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Doctopic: Partial Differential Equations

[m3L; v1.198; Prn:26/01/2017; 15:03] P.14 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

14

constraint is A (F B  −BF  ) = 0, thus solving F B  −BF  = 0 gives B = λ1 F . Then substitute it into the system, one has A A = 0, i.e., A = λ2 u + λ3 (λ2 = 0), then F = −λ22 u3 /[2(λ1 − 1)2 ] + μ3 u2 + μ2 u + μ1 (λ1 = 1). Note that if λ1 = 1, then λ2 = 0 which is beyond the scope of this case. Under these conditions,

λ32 u3  3λ2 uux − E (1) := ux λ2 (μ1 + μ2 u + μ3 u2 − 2uxx ) − 2c3 (λ1 − 1)ux + 2 λ1 − 1 2(λ1 − 1)2 and E (2) = 0, thus we obtain VgI = QgI ∂u with QgI = uxx + (λ2 u + λ3 )ux −

λ1 λ22 u3 + λ1 (μ3 u2 + μ2 u + μ1 ) 2(λ1 − 1)2

is a first-order partial generalized symmetry of the PDEs ut = uxx −

λ22 u3 + μ3 u2 + μ2 u + μ1 . 2(λ1 − 1)2

(25)

Note that VgI is not a generalized conditional symmetry of Eq. (25) since E (1) = 0 on the solution space of QgI = 0. But for λ1 = −1/2, λ3 = μ3 = 0, VgI becomes a generalized conditional symmetry of Eq. (25). In particular, for λ1 = 2, λ2 = 1, μ1 = μ2 = μ3 = 0, the common solutions of E (0) = E (1) = 0 are u=

12 (c1 + x) , 6c1 x + 18t + 3x2 + c1 (3c1 − 8)

where c1 is an arbitrary constant. If set c1 = 0, then Eq. (25) together with initial conditions 4x u|t=t0 = , 6t0 + x2



12t2 + 44tx2 − 9x4  8 3x4 − 14t0 x2 u or ux − = 2 x (12t2 + 20tx2 + 3x4 ) |t=t0 (x2 + 6t0 ) (3x2 + 2t0 ) has the solution u = 4x/(6t + x2 ). 5. Conclusion We propose the definition of partial symmetry for IVP and analyze it from two aspects, the governing PDEs and initial conditions, to define and compute Type-I and Type-II partial symmetries respectively. Type-I partial symmetry provides a new chance for the symmetries of PDEs without leaving initial conditions invariant which are successfully used to construct new reductions and solutions of IVP. Type-II partial symmetry contains the conditional symmetries of given form as particular cases and also gives new solutions of IVP. Two classes of partial symmetry extend the selectable scope of initial conditions which may exert potential applications in physics and associated fields. The illustrated examples demonstrate the effectiveness of the proposed partial symmetry method for solving IVP. However, the extended applications of symmetry method for IVP still have a long way to go, thus some related problems are worthy to be further studied. For instance, one question is how to effectively choose candidates of partial symmetry from known symmetries such that these candidates can generate new reductions and solutions of IVP. Alternatively, it can be issued as how to find partial symmetry via a unified approach such as classical Lie method for point symmetry. Another interesting one is how to effectively explore the roles of partial symmetry in mathematical physics, engineering and other fields. We will go step further for these problems.

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Doctopic: Partial Differential Equations

[m3L; v1.198; Prn:26/01/2017; 15:03] P.15 (1-15)

Z.-Y. Zhang / J. Math. Anal. Appl. ••• (••••) •••–•••

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Acknowledgments This paper is supported by the National Natural Science Foundation of China (Nos. 11671014 and 11301012), Beijing Natural Science Foundation (No. 1173009), Scientific Research Project of Beijing Educational Committee, the Youth Talent Program (No. XN071009) of North China University of Technology. References [1] G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969) 1025–1042. [2] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. [3] G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806–811. [4] J. Boussinesq, Theory of wave and swells propagated in long horizontal rectangular canal and imparting to the liquid contained in this canal, J. Math. Pures Appl. 17 (1872) 55–108. [5] G. Cicogna, G. Gaetam, Partial Lie-point symmetries of differential equations, J. Phys. A: Math. Gen. 34 (2001) 491–512. [6] P.A. Clarkson, M.D. Kruskal, New similarity solutions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [7] P.A. Clarkson, E.L. Mansfield, Open problems in symmetry analysis, in: J.A. Leslie, T. Robart (Eds.), Geometrical Study of Differential Equations, in: Contemp. Math. Ser., vol. 285, Amer. Math. Soc., Providence, RI, 2001, pp. 195–205. [8] A.S. Fokas, Q.M. Liu, Nonlinear interaction of traveling waves of nonintegrable equations, Phys. Rev. Lett. 72 (1994) 3293–3296. [9] J. Goard, Iterating the classical symmetries method to solve initial value problems, IMA J. Appl. Math. 68 (2003) 313–328. [10] J. Goard, Finding symmetries by incorporating initial conditions as side conditions, European J. Appl. Math. 19 (2008) 1–15. [11] P.E. Hydon, Symmetry analysis of initial-value problems, J. Math. Anal. Appl. 309 (2005) 103–116. [12] F. Klein, Vorlesungen über höhere Geometrie, 3rd edn, Springer-Verlag, 1926. [13] D. Levi, P. Winternitz, Nonclassical symmetry reduction example of the Boussinesq equation, J. Phys. A: Math. Gen. 22 (1989) 2915–2924. [14] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. [15] P.J. Olver, E.M. Vorob’ev, Nonclassical and conditional symmetries, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis, vol. 3, 1996, pp. 291–327. [16] L.V. Ovsiannikov, The Group Analysis of Differential Equations, Academic, New York, 1980. [17] C.Z. Qu, L.N. Ji, L.Z. Wang, Conditional Lie–Bäcklund symmetries and sign-invariants to quasi-linear diffusion equations, Stud. Appl. Math. 119 (2007) 355–391. [18] E.M. Vorob’ev, Partial symmetries of systems of differential equations, Dokl. Akad. Nauk SSSR 287 (1986) 536–539, English transl.: Sov. Math., Dokl. 33 (1986) 408–411. [19] Z.Y. Zhang, Y.F. Chen, Classical and nonclassical symmetries analysis for initial value problems, Phys. Lett. A 374 (2010) 1117–1120. [20] R.Z. Zhdanov, Conditional Lie–Bäcklund symmetries and reductions of evolution equations, J. Phys. A: Math. Gen. 128 (1995) 3841–3850. [21] R.Z. Zhdanov, A.Yu. Andreitsev, Non-classical reductions of initial-value problems for a class of nonlinear evolution equations, J. Phys. A: Math. Gen. 33 (2000) 5763–5781.