INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM∗

Lu YAN（闫璐）

Xingzhi College，Xi’an University of Finance and Economics，Xi’an 710038，ChinaE-mail:xiaolu 4002@163.com

Zhenhua SHI（时振华） Hao WANG（王昊）

School of Mathematics，Northwest University，Xi’an 710069，ChinaE-mail:andy szh@163.com;610191181@qq.com

Jing KANG（康静）†

Center for Nonlinear Studies and School of Mathematics，Northwest University，Xi’an 710069，ChinaE-mail:jingkang@nwu.edu.cn

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM∗

Lu YAN（闫璐）

Xingzhi College，Xi’an University of Finance and Economics，Xi’an 710038，China
E-mail:xiaolu 4002@163.com

Zhenhua SHI（时振华） Hao WANG（王昊）

School of Mathematics，Northwest University，Xi’an 710069，China
E-mail:andy szh@163.com;610191181@qq.com

Jing KANG（康静）†

Center for Nonlinear Studies and School of Mathematics，Northwest University，Xi’an 710069，China
E-mail:jingkang@nwu.edu.cn

Invariant subspace method is exploited to obtain exact solutions of the twocomponent b-family system.It is shown that the two-component b-family system admits the generalized functional separable solutions.Furthermore，blow up and behavior of those exact solutions are also investigated.

invariant subspace;generalized conditional symmetry;generalized functional separable solution;Camassa-Holm equation;two-component b-family system 2010 MR Subject Classification37K05;37K35;35Q35

1　Introduction

In this paper，we apply the invariant subspace method to construct solutions of the following two-component b-family system

Furthermore，if one sets ρ=0 in system（1.2），it reduces to the CH equation

which was derived by Camassa and Holm［3］as a model for describing unidirectional propagation of the shallow water waves over a flat bottom（see also［5］）.Remarkably，the peaked solitons of the CH equation were discovered［3］.Indeed，the CH equation can be derived by using the recursion operator of the KdV equation［4］.In an intriguing paper by Olver，Rosenau［1］，they proposed the so-called tri-Hamiltonian duality approach，which was used to recover the CH equation from the bi-Hamiltonian structure of the KdV equation.The CH equation（1.3）can also be derived from the shallow water wave equation by using the asymptotic methods［11-13］through the Kodama transformation.Because of the several nontrivial properties，the CH equation was studied in a huge number of literatures（see for example［14-19］and the references therein）.Similarly，if one applies the tri-Hamiltonian duality approach［1］to the Ito equation［20］

the resulting equation is the two-component CH equation（1.2）.If ρ=0，and k1=3 in（1.1），it becomes Degasperis-Procesi equation［21，22］

Its well-posedness and blow-up phenomena were also discussed［23］.Note that the two-component CH equation（1.2）and the two-component b-family system （1.1）admit the symmetry v= u∂u+x∂x.So they possess the particular similarity solution of the form

Furthermore，the perturbational method was used by Yuen［10，24，25］to construct exact solutions of the form

to the two-component Camassa-Holm equation（1.2）.In this paper，we shall prove that such solutions are associated with conditional symmetries of the system.The blow-up phenomena and behavior of solutions（1.6）and（1.7）were also discussed.

The invariant subspace method is an effective method to construct exact solutions of nonlinear partial differential equations［26-28］.Indeed，there were many examples of nonlinear evolution equations，whose exact solutions can be constructed by the invariant subspace method［26］.In particular，the generalized functional separable solutions can be derived by using the invariant subspace method［29］.The invariant subspace method is related to the generalized conditional symmetry（GCS）method［30，31］.A key point for the invariant subspace method admitted by the evolutionary partial differential equations is the dimensional estimate［24，32］.

The object of this paper is to derive generalized functional separable solutions of（1.1）by using the invariant subspace method.The outline of the paper is as follows.In Section 2，we provide a brief account of the invariant subspace and the generalized conditional symmetry methods.The main results are presented and proved in Sections 3 and 4.

2　Invariant Subspace Method

Consider the systems of kth-order nonlinear PDEs

are linearly independent.If a vector operator F satisfies［32］

then the vector field F is said to admit the invariant subspace，which means that there existsuch that

If the operator F［U］admits the subspace W，then system（2.1）possesses solutions of the form

Notice that the invariant subspace W has the dimensionthen the system reduces to the-dimensional dynamical system.

It was shown that the invariant subspace method can be explored by using the GCS method［29］.The GCS method was introduced by Zhdanov［30］and Fokas and Liu［31］，which was developed to deal with various nonlinear evolution equations，and a number of results were obtained（see［33-38］and the references therein）.

Let’s give a brief account of the generalized conditional symmetry（GCS）method［30，31］. Letbe an evolutionary vector field with the characteristic η（a smooth function of t，r，u，ur，···）and

be a nonlinear evolution equation，where we use the following notations

Definition 2.1The evolutionary vector field（2.2）is said to be a generalized symmetry of（2.3）if and only ifwhere L is the set of all differential consequences of the equation，that is

Definition 2.2（see［30，31］）The evolutionary vector field（2.2）is said to be a GCS of（2.3）if and only if V（ut-E）|L∩M=0，where L is given as in Definition 2.1，and M denotes the set of all differential consequences of equation η=0 with respect to r，that is

Theorem 2.3（see［30，31］）Equation（2.3）admits the CLBS（2.2）if there exists a function W（t，r，u，η）such that

On the GCS of（2.5），we have the following result.

Theorem 2.4System（2.5）admits the generalized conditional symmetry

ProofWe can prove this theorem in terms of Theorem 2.4.A direct calculation，the details which we omit，verifies that η satisfies the following system

It follows from Theorem 2.4 that system（2.5）admits the following formal exact solutions

3　Solutions of System （1.1）

In view of Theorem 2.1，we first have the following result.

Theorem 3.1For the two-component b-family system （2.5），there exists a family of solutions

In the following，we are concerned with the special case ofThe form of solutions and their blow-up phenomena are given in the next two theorems.

Theorem 3.2Assume that the function a（s）is a solution of the Emden equation

1）σ＜0.

2）σ＞0;

2.1）0＜k＜1.Solution（3.4）blows up if and only ifOtherwise，the solution exists globally.

2.2）k≥1.Solution（3.4）exists globally.

Theorem 3.3For the two-component b-family system（1.1）withAssume thatsatisfies（3.4）withThen the two-component b-family system（2.5）admits a family of solutions

We now consider two cases regarding to the sign of σ.

1）σ＜0.

1.1）0＜k＜1.Solution（3.5）blows up if and only ifIn the contrary case，it exists globally.

1.2）k≥1.Solution exists globally.

2）σ＞0.

2.1）0＜k≤1.Solution（3.5）blows up in a finite time.

2.2）k＞1.Solution（3.5）blows up if and only ifOtherwise，it exists globally.

Proof of Theorem 3.1We prove the theorem in three steps.

Step 1Note that the velocity u is linear.The momentum equation（2.5）becomes

Substituting the expression for u（t，x）in（2.6）into（3.6），we get

which leads to

Integrating（3.7）from 0 to x，we have

and finally we get

Step 2Next，we consider the mass equation in（2.5）

Substituting（3.8）into（3.9），we arrive at

which yields the equations involving

Step 3As the third step，we solve the above system（3.10）.First we consider the third equation in（3.10）.Applying the Hubble’s transformation，

where r is some constant to be determined later.Then the third equation in（3.10）is transformed to

For simplicity，we set r=1+k1，so the above equation becomes

Integrating the above equation，we find thatsatisfies the Emden equation（3.3）withare arbitrary constants.Next，for equation（3.10）aboutit can be further simplified in terms of the functionThanks to（3.11）and（3.3），the second equation in（3.10）reduces to the second equation in（3.2）.

Then the first equation in（3.10）becomes

Let ρ（0，0）=β.Then the solution of（3.14）is

It is inferred from（3.8）that the density function is given by

4　The Generalized Functional Separable Solutions of the b-Family System

To prove Theorems 3.2 and 3.3，we need the following lemmas.

Lemma 4.1For the two-component b-family system（1.1），there exist the solutions

Remark 4.2The solution constructed here depends on the auxiliary functionwhich satisfies the Emden equation and varies with choices of the four parameters

The following three lemmas demonstrate the properties ofby which the corresponding blow-up and global existence of the analytical solution can be established.

Lemma 4.3For the Emden equation

（1）If λ＜0，there exists a finite time s，such thatOtherwise the solution a（s）exists globally，and

（2）If λ＞0，the solution a（s）exists globally，and

（3）If λ=0，a1＜0，the solution a（s）vanishes at s=-a0/a1.Otherwise it exists globally and

Lemma 4.4（see［10］）For the Emden equation（4.1）with 0＜k＜1.

（1）If λ＜0，there exists a finite time s，such tha t

（2）If λ＞0，there exists a finite time s such that，if and only ifOtherwise，the solution exists globally and

Lemma 4.5In the case of k=1，the solution of（4.1）satisfies

（1）If λ＜0，there exists a finite time s such that

（2）If λ＞0，the solution a（s）exists globally and there holds

Proof of Lemma 4.3（k＞1）

（1）λ＞0.The assumption λ＞0 implies that the curve a（s）is convex.The existence of the solution guarantees that a（s）exists at least in some neighborhood of s=0.Multiplying equation（4.1）byand integrating the resulting equation leads to the energy conservation equation

Therefore for any constant a1，the solution a（s）must increase after some finite time.Then there are two possibilities to consider:

（1.1）a（s）exists only in some finite internal［0，s0］such that

（1.2）a（s）exists globally，and

We now claim that the first possibility does not exist.Because the time for traveling the intervalcan be estimated as

Collecting the above analysis，we show that in this subcase the solutionexists globally， and

（2）λ＜0.First，the assumption λ＜0 implies that the curveis concave upwards. Furthermore，in view of the energy conservation equation（4.1），we need to distinguish two subcase:E0≥0 and E0＜0.

（2.1）E≥0.It follows from（4.2）that

（2.2）E0＜0.Thanks to the energy conservation equation

It follows from the analysis of the above two cases that if λ＜0，there exists a finite time s.Such thatIf and only ifOtherwise，the solution exists globally and

（3）λ=0.In this case，the Emden equation becomesIt is easy to show statement（3）.Thus we complete the proof of Lemma 4.3.

Proof of Lemma 4.5If k=1，it follows from the corresponding Emden equation

（1）λ＜0.By the assumption λ＜0，we get lnwhich results inSo thatIt is clear that a（s）must vanish in some finite time s.Since

（2）λ＞0.First，we claim that if λ＞0，the solution a（s）does not vanish at any s＞0， that is a（s）＞0 for any s＞0.Indeed，if there exists a finite time s such thatthen，this contradicts the fact that the energy E0defined in equation（4.4）is finite.On the other hand，equation（4.4）implies the solution a（s）is uniformly bounded below. Moreover，

Then three cases arise:

（2b）a（s）only exists in some finite interval［0，s0］，such that

First，statement（2a）is not true since it contradicts to the Emden equation by noting that

Next，from equation（4.4）and in view of，the time for traveling the intervalcan be estimated asTherefore statement（2b）is excluded，and we arrive at conclusion（2c）.The proof is then completed.

Now，we consider two cases on systemBy Theorems 3.2 and 3.3，the blow up and global existence for the analytic solutions of the corresponding system is given in the following theorem.

Theorem 4.6For the 2-component b-family system（1.1），we have the following results:

References

［1］Olver P J，Rosenau P.Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support.Phys Rev E，1996，53:1900-1906

［2］Constantin A，Ivanov R I.On an integrable two-component Camassa-Holm shallow water system.Phys Lett A，2008，372:7129-7132

［3］Camassa R，Holm D.An integrable shallow water equation with peaked solitons.Phys Rev Lett，1993，71: 1661-1664

［4］Fuchssteiner B，Fokas A.Symplectic structures，their B¨acklund transformations and hereditary symmetries. Physica D，1981/1982，4:47-66

［5］Constantin A，Lannes D.The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations.Arch Ration Mech Anal，2009，192:165-186

［6］Chen M，Liu S Q，Zhang Y J.A two-component generalization of the Camassa-Holm equation and its solutions.Lett Math Phys，2006，75:1-14

［7］Eschel J，Lechtenfeld O，Yin Z.Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation.Discrete Contin Dyn Syst Ser A，2007，19:493-513

［8］Guan C X，Yin Z Y.Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system.J Diff Equat，2010，248:2003-2014

［9］Gui G L，Liu Y.On the global existence and wave-breaking criteria for the two-component Camassa-Holm system.J Funct Anal，2010，258:4251-4278

［10］Yuen M.Self-similar blow up solutions to the two-component Camassa-Holm equations.J Math Phys，2010，51:093524

［11］Fokas A S，Liu Q M.Asymptotic integrability of water waves.Phys Rev Lett，1996，77:2347-2351

［12］Dullin R，Gottwald G，Holm D D.An integrable shallow water equation with linear and nonlinear dispersion. Phys Rev Lett，2001，87:4501-4504

［13］Johnson R S.Camassa-Holm，Korteweg-de Vries and related models for water waves.J Fluid Mech，2002，455:63-82

［14］Fu Y G，Liu Z R，Tang H.Non-uniform dependence on initial data for the modified Camassa-Holm equation on the line.Acta Math Sci，2014，34B（6）:1781-1794

［15］Constantin A.Existence of permanent and breaking waves for a shallow water equation:a geometric approach.Ann Inst Fourier（Grenoble），2000，50:321-362

［16］Constantin A，Escher J.Wave breaking for nonlinear nonlocal shallow water equations.Acta Math，1998，181:229-243

［17］Constantin A，Escher J.Well-posedness，global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation.Comm Pure Appl Math，1998，51:475-504

［18］Constantin A，Escher J.Global existence and blow-up for a shallow water equation.Ann Scuola Norm Sup Pisa，1998，26:303-328

［19］Constantin A，Strauss W A.Stability of a class of solitary waves in compressible elastic rods.Phys Lett A，2000，270:140-148

［20］Ito M.Symmetries and conservation laws of a coupled nonlinear wave equation.Phys Lett A，1982，91: 335-338

［21］Degasperis A，Procesi M.Asymptotic integrability//Degasperis A，Gaeta G.Symmetry and Perturbation Theory.World Scientific，1999:23-37

［22］Degasberis A，Holm D D，Hone A N W.A new integrable evolution equation with peakon solution.Theor Math Phys，2002，133:1461-1472

［23］Liu Y，Yin Z Y.Global existence and blow-up phenomena for the Degasperis-Procesi equation.Comm Math Phys，2006，267:801-820

［24］Yuen M.Self-similar blow up solutions to the Degasperis-Procesi shallow water system.Comm Non Sci Numer Simul，2011，16:3463-3469

［25］Yuen M.Perturbed blow up solutions to the two-component Camassa-Holm equations.J Math Anal Appl，2012，390:596-602

［26］Galaktionov V A，Svirshchevski S.Exact solutions and Invariant subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics.London:Chapman and Hall，2007

［27］Galaktionov V A.Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities.Proc R Soc Edinburgh，1995，125:225-246

［28］Svirshchevski S.Invariant linear spaces and exact solutions of nonlinear evolution equations.J Non Math Phys，1996，3:164-169

［29］Ji L N，Qu C Z.Conditional Lie-B¨acklund symmetries and invariant subspaces to nonlinear diffusion equations with convection and source.Stud Appl Math，2013，131:266-301

［30］Zhu C R，Qu C Z.Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators.J Math Phys，2011，52:043507

［31］Zhdanov R Z.Conditional Lie-B¨acklund symmetries and reductions of evolution equations.J Phys A:Math Gen，1995，128:3841-3850

［32］Fokas A S，Liu Q M.Nonlinear interaction of traveling waves of nonintegrable equations.Phys Rev Lett，1994，72:3293-3296

［33］Qu C Z，Ji L N，Wang L Z.Conditional Lie-B¨acklund symmetries and sign-invarints to quasi-linear diffusion equations.Stud Appl Math，2007，119:355-391

［34］Qu C Z.Group classification and generalized conditional symmetry reduction of the nonlinear diffusionconvection equation with a nonlinear source.Stud Appl Math，1997，99:107-136

［35］Qu，C Z.Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method.IMA J Appl Math，1999，62:283-302

［36］Ji L N，Qu C Z，Ye Y J.Solutions and symmetry reductions of the n-dimensional nonlinear convectiondiffusion equations.IMA J Appl Math，2010，75:17-55

［37］Basarab-Horwath P，Zhdanov R Z.Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries.J Math Phys，2000，42:376-389

［38］Zhdanov R Z，Andreitsev A Y.Non-classical reductions of initial-value problems for a class of nonlinear evolution equations.J Phys A:Math Gen，2000，33:5763-5781

∗March 31，2015;revised November 24，2015.This work is supported by NSFC（11471260）and the Foundation of Shannxi Education Committee（12JK0850）.

†Jing Kang