On integrability of (2+1)-dimensional quasilinear systems

Log-Sobolev Ineqaulities:Different Roles ofRic and Hess ∗Feng-Yu WangDepartment of Mathematics,University of Wales Swansea,Singleton Park,SA28PP,UKEmail:F.Y.Wang@October 24,2007AbstractLet P t be the diffusion semigroup generated by L :=∆+∇V on a completeconnected Riemannian manifold with Ric ≥−(σ2ρ2o +c )for some constants σ,c >0and ρo the Riemannian distance to a fixed point.It is shown that P t is hypercontrac-tive,or the log-Sobolev inequality holds for the associated Dirichlet form,provided −Hess V ≥δholds outside of a compact set for some constant δ>(1+√2)σ√d −1.This indicates,at least in finite dimensions,that Ric and −Hess V play quite dif-ferent roles for the log-Sobolev inequality to hold.The supercontractivity and theultracontractivity are also studied.AMS subject Classification:60J60,58G32.Keywords:Log-Sobolev inequality,Ricci curvature,Riemannian manifold,diffusion semi-group.1IntroductionLet M be a d -dimensional completed connected non-compact Riemannian manifold and V ∈C 2(M )such that Z := M e V (x )d x <∞,where d x is the volume measure on M .Let µ(d x )=Z −1e V (x )d x.Then P t ,the semigroup of the diffusion process generated by L :=∆+∇V ,is symmetric in L 2(µ).It is well-known by the Bakry-Emery criterion that (see [4])(1.1)Ric −Hess V ≥K ∗Supported in part by NNSFC(10121101),the 973-Project in China and RFDP(20040027009).for some constant K>0implies the Gross log-Sobolev inequality[15](1.2)µ(f2log f2):=Mf2log f2dµ≤Cµ(|∇f|2),µ(f2)=1,f∈C1(M)for C=2/K.This result was extended by Chen and the author[10]to the situation that Ric−Hess V is uniformly positive outside a compact set.In the case that Ric−Hess V is bounded below,sufficient concentration conditions ofµfor(1.2)to hold are presented in[20,1,21].Obviously,in a condition on Ric−Hess V the Ricci curvature and−Hess V play the same role.What can we do when Ric−Hess V is unbounded below?It seems very hard to confirm the log-Sobolev inequality with an unbounded below condition of Ric−Hess V.So,in this paper we try to clarify the roles of Ric and−Hess V in the study of the log-Sobolev inequality.Let usfirst recall the gradient estimate of P t,which is a key point in the above references to prove the log-Sobolev inequality.Let x t be the L-diffusion process starting at x,and let v∈T x M.Due to Bismut[6] and Elworthy-Li[12],under a reasonable lower bound condition of Ric−Hess V,one has∇P t f,v =E ∇f(x t),v t ,t>0,f∈C1b(M),where v t∈T xtM solves the equationD t v t:=//−1t→0dd t//t→0v t=−(Ric−Hess V)#(v t)for//t→0:T xtM→T x M the associated stochastic parallel displacement,and(Ric−Hess V)#(v t)∈T xtM with(Ric−Hess V)#(v t),X :=(Ric−Hess V)(v t,X),X∈T xtM.Thus,for the gradient of P t,which is a short distance behavior of the diffusion process,a condition on Ric−Hess V appears naturally.On the other hand,however,Ric and−Hess V play very different roles for long distance behaviors.For instance,Letρo be the Riemannian distance function to afixed point o∈M.If Ric≥−k and−Hess V≥−δfor some k≥0,δ∈R,the Laplacian comparison theorem impliesLρo≤k(d−1)cothk/(d−1)ρo+δρo.So,for largeρo the Ric lower bound leads to a bounded term while that of−Hess V provides a linear term.The same phenomena appears to the formula on distance of coupling by parallel displacement(cf.[3,(2.3),(2.4)]),which implies the above Bismut-Elworthy-Li formula by letting the initial distance tend to zero(cf.[16]).Here,k≥0is essential for our framework since the manifold has to be compact if Ric is bounded below by a positive constant.Since the log-Sobolev inequality is always available on bounded regular domains,it is more likely a long distance behavior of the diffusion process.Thus,we would conclude that Ric and −Hess V take different roles to ensure the validity of the log-Sobolev inequality.Indeed,it has been observed by the author [21]that (1.2)holds for some C >0provided Ric is bounded below and −Hess V is uniformly positive outside a compact set.So,for the validity of the log-Sobolev inequality,the positivity of −Hess V is a dominative condition,which allows the Ricci curvature has arbitrary negative constant lower bound,and hence allows Ric −Hess V to be globally negative.The first aim of this paper is to search for the weakest possibility of curvature lower bound for the log-Sobolev inequality to hold under the condition(1.3)−Hess V ≥δoutside a compact setfor some constant δ>0.This condition is reasonable as the log-Sobolev inequality implies µ(e λρ2o )<∞for some λ>0,see e.g.[2,18].According to the following Theorem 1.1and Example 1.1,we conclude that under (1.3)the optimal curvature lower bound condition for (1.2)to hold is(1.4)Ric ≥−(c +σ2ρ2o )for some constants c,σ>0.More precisely,let θ0>0be the smallest positive constant such that for any connected complete non-compact Riemannian manifold M and V ∈C 2(M ),the conditions (1.3)and (1.4)with δ>σθ0√d −1implies Z <∞and (1.2)for some C >0.Due to Theorem 1.1and Example 1.1below,we conclude thatθ0∈[1,1+√2].The exact value of θ0is however unknown.Recall that P t is called supercontractive if P t 2→4<∞for all t >0while ultracon-tractive if P t 2→∞<∞for all t >0(see [11]).In the present framework these two properties are stronger than the hypercontractivity: P t 2→4≤1for some t >0,which is equivalent to (1.2)due to Gross [15].Theorem 1.1.Assume that (1.3)and (1.4)hold for some constants c,δ,σ>0with δ>(1+√2)σ√d −1.Then (1.2)holds for some C >0.Further more,P t is supercontractive if and only if µ(exp[λρ2o ])<∞for all λ>0,while it is ultracontractive if and only if P t exp[λρ2o ] ∞<∞for all t,λ>0.Example 1.1.Let M =R 2be equipped with the rotationally symmetric metricd s 2=d r 2+ re kr 2 2d θ2under the polar coordinates (r,θ)∈[0,∞)×S 1at 0,where k >0is a constant.Then (see e.g.[14])Ric =−d 2d r2(r e kr 2)r e kr =−4k −4k 2r 2.Thus,(1.4)holds for σ=2k.Next,take V =−δ2ρ2o .By the Hessian comparison theorem and the negativity of the sectional curvature,we obtain (1.3).Since d =2ande V (x )d x =r e kr 2−δr 2/2d r d θ,one has Z <∞if and only if δ>2k =σ√d −1.Therefore,θ0≥1.Following the line of [20,21],the key point in the proof of Theorem 1.1will be a proper Harnack inequality of type(P t f (x ))α≤C α(t,x,y )P t f α(y ),t >0,x,y ∈M for any nonnegative f ∈C b (M ),where α>1is a constant and C α∈C ((0,∞),M 2)is a positive function.Such an inequality was established in [20]for Ric −Hess V bounded below and extended in [3]to a more general situation with Ric satisfying (1.4).The Harnack inequality presented in [3]contains a leading term exp[ρ(x,y )4],which is however too large to be integrability w.r.t.µ×µunder our conditions.So,to prove Theorem 1.1,we shall present a sharper Harnack inequality in Section 3by refining the coupling method introduced in [3](see Proposition 3.1below).This inequality,together with the concentration of µensured by (1.3)and (1.4),will imply the hypercontractivity of P t .To establish this new Harnack inequality,some necessary preparations are presented in Section 2.Finally,Theorem 1.1is extended in Section 4to the case that −Hess V ≥Φ◦ρo outside a compact set for an increasing positive function Φwith Φ(r )↑∞as r ↑∞.2PreparationsWe first study the concentration of µby using (1.3)and (1.4),for which we need to estimate Lρo from above according to [5]and references within.Lemma 2.1.If (1.3)and (1.4)hold then there exists a constant C 1>0such that(2.1)Lρ2o≤C 1(1+ρo )−2 δ−σ√d −1 ρ2oholds outside cut(o ),the cut-locus of o .If moreover δ>σ√d −1then Z <∞and µ(e λρ2o )<∞for all λ<12(δ−σ√d −1).Proof.By (1.4)and the Laplacian comparison theorem,∆ρo ≤ (c +σ2ρ2o )(d −1)coth(c +σ2ρ2o )/(d −1)ρo holds outside cut(o ).Thus,outside cut(o )one has∆ρ2o ≤2ρo (c +σ2ρ2o )(d −1)coth (c +σ2ρ2o )/(d −1)ρo +2≤2d +2ρo (c +σ2ρ2o )(d −1),(2.2)where the second inequality follows from the fact thatr cosh r ≤(1+r )sinh r,r ≥0.On the other hand,for x /∈cut(o )and U the unit tangent vector along the unique minimal geodesic form o to x ,by (1.3)there exists a constant c 1>0independent of x such that∇V,∇ρo (x )= ∇V,U (o )+ρo (x )0Hess V (U,U )( s )d s ≤c 1−δρo (x ).Combining this with (2.2)we prove (2.1).Finally,let δ>σ√d −1and 0<λ<12(δ−σ√d −1).By (2.1)we haveL e λρ2o ≤λe λρ2o C 1(1+ρo )−2 δ−σ√d −1 ρ2o +4λρ2o≤c 2−c 3ρ2o eλρ2o for some constants c 2,c 3>0.By [5,Proposition 3.2],this implies Z <∞andMρ2o e λρ2o d µ≤c 2c 3<∞.Lemma 2.2.Let x t be the L -diffusion process with x 0=x ∈M.If (1.3)and (1.4)hold with δ>σ√d −1,then for any δ0∈(σ√d −1,δ)there exists a constant C 2>0such thatE exp (δ0−σ√d −1)24 T 0ρo (x t )2d t ≤exp C 2T +14(δ0−σ√d −1)ρo (x )2 ,T >0,x ∈M.Proof.By Lemma 2.1,we haveLρ2o ≤C −2(δ0−σ√d −1)ρ2o outside cut(o )for some constant C >0.Then the Itˆo formula for ρo (x t )due to Kendall[17]implies that(2.3)d ρ2o (x t )≤2√2ρo (x t )d b t + C −2(δ0−σ√ρ2o (x t ) d t holds for some Brownian motion b t on R.This implies that the L-diffusion process is non-explosive so thatT n:=inf{t≥0:ρo(x t)≥n}→∞as n→∞.For anyλ>0and n≥1,it follows from(2.3)thatE exp2λδ0−σ√d−1T∧T nρ2o(x t)d t≤eλρ2o(x)+CλT E exp2√2λT∧T nρo(x t)d b t≤eλρ2o(x)+CλTE exp16λ2T∧T nρ2o(x t)d t1/2.Thus,takingλ=18(δ0−σ√d−1)we obtainE exp14δ0−σ√d−12 T∧T nρ2o(x t)d t≤exp14δ0−σ√d−1ρ2o(x)+C2Tfor some C2>0.Then the proof is completed by letting n→∞.Finally,we recall the coupling argument introduced in[3]for establishing the Harnack inequality of P t.Let T>0and x=y∈M befixed.Then the L-diffusion process starting from x can be constructed by solving the following Itˆo stochastic differential equationd I x t=√2Φt d B t+∇V(x t)d t,x0=x,where d I is the Itˆo differential on manifolds introduced in[13](see also[3]),B t is the d-dimensional Brownian motion,andΦt is the horizontal lift of x t onto the orthonormal frame bundle O(M).To construct another diffusion process y t starting from y such that x T=y T,as in[3] we add an additional drift term to the equation(as explained in[3,Section3],we may and do assume that the cut-locus of M is empty):d I y t=√2P xt,y tΦt d B t+∇V(y t)d t+ξt U(x t,y t)1{t<τ}d t,y0=y,where P xt,y t is the parallel transformation along the unique minimal geodesic from x t toy t,U(x t,y t)is the unit tangent vector of at y t,ξt≥0is a smooth function of X t to be determined,andτ:=inf {t ≥0:x t =y t }is the coupling time.Since all terms involved in the equation are regular enough,there exists a unique solution y t .Furthermore,since the additional term containing 1{t<τ}vanishes from the coupling time on,one has x t =y t for t ≥τdue to the uniqueness of solutions.Lemma 2.3.Assume that (1.3)and (1.4)hold with δ≥2σ√d −1.Then there existsa constant C 3>0independent of x,y and T such that x T =y T holds for ξt :=C 3+2σ√d −1ρo (x t )+ρ(x,y )T.Proof.According to Section 2in [3],we have(2.4)d ρ(x t ,y t )= I (x t ,y t )+ ∇V,∇ρ(·,y t ) (x t )+ ∇V,∇ρ(x t ,·) (y t )−ξt d t,t <τ,whereI Z (x t ,y t )=d −1i =1 ρ(x t ,y t )0 |∇U J i |2− R (U,J i )U,J i ( s )d s for R the Riemann curvature tensor,U the unit tangent vector of the minimal geodesic:[0,ρ(x t ,y t )]→M from x t to y t ,and {J i }d −1i =1the Jacobi fields along which,togetherwith U ,consist of an orthonormal basis of the tangent space at x t and y t and satisfyJ i (y t )=P x t ,y t J i (x t ),i =1,...,d −1.LettingK (x t ,y t )=sup {c +σ2ρ2o },we obtain from (1.4)and [22,Theorem 2.14](see also [9,8])that(2.5)I (x t ,y t )≤2 K (x t ,y t )(d −1)tanh ρ(x t ,y t )2 K (x t ,y t )/(d −1) .Moreover,(1.3)implies∇V,∇ρ(·,y t ) (x t )+ ∇V,∇ρ(x t ,·) (y t )= ρ(x t ,y t )Hess V (U,U )( s )d s ≤c 1−δρ(x t ,y t )for some constant c 1>bining this with (2.4),(2.5)and ξt =C 3+2σ√d −1ρo (x t )+ρ(x,y )T,we arrive at d ρ(x t ,y t )≤ 2 K (x t ,y t )(d −1)+c 1−δρ(x t ,y t )−C 3−2σ√d −1ρo (x t )−ρ(x,y )Td t for t <τ.Noting that K (x t ,y t )≤ c +σ2ρo (x t )+ρ(x t ,y t ) 2 1/2≤√c +σ ρo (x t )+ρ(x t ,y t ) and δ≥2σ√d −1,one has2 K (x t ,y t )(d −1)−δρ(x t ,y t )−2σ√d −1ρo (x t )≤2 c (d −1).Thus,when C 3≥c 1+2 c (d −1)we haved ρ(x t ,y t )≤−ρ(x t ,y t )T,t <τ.This implies that τ≤T and hence,x T =y T .3Harnack inequality and proof of Theorem 1.1We first prove the following Harnack inequality using results in Section 2.Proposition 3.1.Assume that (1.3)and (1.4)hold with δ>(1+√2)σ√d −1.Thenthere exist C >0and α>1such that(3.1)(P T f (y ))α≤(P T f α(x ))exp CT ρ(x,y )2+C (T +ρo (x )2) holds for all x,y ∈M,T >0and nonnegative f ∈C b (M ).Proof.According to Lemma 2.3,we takeξt =C 3+2σ√d −1ρo (x t )+ρ(x,y )Tsuch that τ≤T and x T =y T .LetR =exp 1√2 τ0 P x t ,y t Φt d B t ,ξt U (x t ,y t ) −14 τ0ξ2t d t .By the Girsanov theorem,the H¨o lder inequality,and noting that x T =y T ,we have P T f (y )=E [f (y T )R ]=E [f (x T )R ]≤(P T f α(x ))1/α(E R α/(α−1))(α−1)/α.That is,(3.2)(P T f (y ))α≤(P T f α(x ))(E R α/(α−1))α−1.Since for any exponential integrable martingale M t and any β>1one has(3.3)E e βM t −β2 M t =E e βM t −β2p 2 M t ·e β(βp −1)2 M t ≤ e βp (βp −1)2(p −1)M t (p −1)/p ,p >1,by taking β=α/(α−1)we obtain(3.4)(E R α/(α−1))α−1≤ E exp pα(pα−α+1)8(p −1)(α−1)2 T 0ξ2t d t (α−1)(p −1)/p ,p >1.Since δ>(1+√2)σ√d −1,we may take δ0∈((1+√2)σ√d −1,δ),small ε >0andlarge C 4>0,independent of T,x and y ,such thatξ2t = C 3+2σ√o (x t )+ρ(x,y )T 2≤(1−ε ) C 4+C 4ρ(x,y )2T 2+2(δ0−σ√d −1)2ρo (x t )2 holds.Moreover,since(3.5)lim p ↓1lim α↑∞pα(pα−α+1)8(p −1)(α−1)2=18,there exist p,α>1such thatpα(pα−α+1)8(p −1)(α−1)2 T 0ξ2t d t ≤C 4T +C 4ρ(x,y )2T +(δ0−σ√d −1)24 T 0ρo (x t )2d bining this with (3.4)and Lemma 2.2,we obtain (E R α/(α−1))α−1≤exp C 5T +C 5ρ(x,y )T+C 5ρo (x )2 ,T >0,x ∈M for some constant C 5>0.This completes the proof by taking large enough T >0such that C 5≤T ε.Proof of Theorem 1.1.By Proposition 3.1,let α>1and C >0such that (3.1)holds.Since δ>σ√d −1,we may take T >0such that C T ≤ε:=18δ−σ√d −1 .Then for any nonnegative f ∈C b (M )with µ(f α)=1,it follows from (3.1)that1= M P T f α(x )µ(d x )≥(P T f (y ))α M e −ερ(x,y )2−C (1+ρo (x )2)µ(d x )≥(P T f (y ))α {ρo ≤1}e −ε(1+ρo (y ))2−2C µ(d x )≥ε (P T f (y ))αexp[−2ερo (y )2],y ∈Mfor some constant ε >0.Thus,M(P T f (y ))2αµ(d y )≤1ε M e 4ερo (y )2µ(d y )<∞according to Lemma 2.1.This implies thatP T L α(µ)→L 2α(µ)<∞.Therefore,the log-Sobolev inequality (1.2)holds for some constant C >0due to the uniformly positively improving property of P t (see the proof of [21,Theorem 1.1]and[1]).Finally,due to the Harnack inequality (3.1),the assertions on supercontactivity and ultracontractivity follow from the proof of [19,Theorem 2.3]4Extension to general negative curvature lower bounds In this section we work with a stronger lower bound condition on −Hess V :(4.1)−Hess V ≥Φ◦ρo holds outside a compact subset of M,for a positive increasing function Φwith Φ(r )↑∞as r ↑∞.We aim to search for reasonable conditions on positive increasing function Ψsuch that(4.2)Ric ≥−Ψ◦ρoimplies the supercontractivity and/or ultracontractivity.Theorem 4.1.If (4.2)and (4.1)hold for some increasing positive functions Φand Ψsuch that(4.3)lim r →∞Φ(r )=lim r →∞( r 0Φ(s )d s )2Φ(r )=∞,(4.4) Ψ(r +t )(d −1)≤θ rΦ(s )d s +12 t/20Φ(s )d s +C,r,t ≥0for some constants θ∈(0,1/(1+√2))and C >0.Then P t is supercontractive.If furthermore (4.5)∞1d s √s √r0Φ(u )d u<∞,then P t is ultracontractive.More precisely,forΓ1(r ):=1√r√r 0Φ(s )d s,Γ2(r ):=∞rd s√s√sΦ(u )d u,r >0,(4.5)implies(4.6) P t 2→∞≤expc +c t1+Γ−11(c/t )+Γ−12(t/c ) <∞,t >0for some constant c >0andΓ−11(s ):=inf {t ≥0:Γ1(t )≥s },s ≥0.Proof.(a)Replacing c +ρ2o by Ψ◦ρo and noting that Hess V ≤−Φ◦ρo for large ρo ,theproof of Lemma 2.1implies (4.7)Lρ2o ≤c 1(1+ρo )−2ρoρoΦ(s )d s −Ψ◦ρo (d −1)for some constant c 1>bining this with (4.4)and noting that 1ρoρoΦ(s )d s →∞as ρo →∞,we conclude that for any λ>0L e λρ2o ≤C −2λρo √21+√2e λρ2o ρo0Φ(s )d s +4λ2ρ2oe λρ2o ≤C +C (λ)−λρo e λρ2o ρoΦ(s )d s,(4.8)where C >0is a universal constant andC (λ):=sup r>0r e λr 24λ2r −λ(1+√2)2 rΦ(s )d s=sup r 2≤Γ−114(1+√2)2λ r e λr2 4λ2r −λ(1+√2)2 r0Φ(s )d s ≤4λ2Γ−11 4(1+√2)2λ exp λΓ−11 4(1+√2)2λ≤exp 4λ+2λΓ−11 4(1+√2)2λ <∞.(4.9)Therefore,Z <∞and (4.10)µ(e λρ2o )<∞,λ>0.(b)By (4.4),(4.7)and Kendall’s Itˆo formula [17]as in the proof of Lemma 2.2,we haved ρ2o (x t )≤2√o (x t )d b t + C 1−2√2ρo (x t)(1+ε)1+√2 ρo (x t )Φ(s )d s d t for some constants ε,C 1>0,where x t and b t are in the proof of Lemma 2.2.Let(4.11)ϕ(r )=r0d s√s√sΦ(u )d u,r ≥0.We arrive atd ϕ◦ρ2o (x t )≤2√2ρo (x t )ϕ ◦ρ2o (x t )d b t +4ρ2o (x t )ϕ ◦ρ2o (x t )d t+ϕ ◦ρ2o (x t ) C 1−2√2ρo (x t)(1+ε)1+√2 ρo (x t )0Φ(s )d s d t.From (4.3)we see thatρo ϕ ◦ρ2o ϕ ◦ρ2o ρo 0Φ(s )d s≤Φ◦ρo 2( ρo 0Φ(s )d s )2which goes to zero as ρo →∞.Then there exists a constant C 2>C 1such thatd ϕ◦ρ2o (x t )≤2√2ρo (x t )Φ(s )d sd b t +C 2d t −2√21+√2 ρo (x t )0Φ(s )d s 2d t.This implies that for any λ>0E exp2√2λ1+√2 T 0 ρo (x t )Φ(s )d s 2d t ≤e C 2λT +λϕ◦ρ2o (x )E exp 2√2λ T 0ρo (x t )0Φ(s )d s d b t ≤eC 2λT +λϕ◦ρ2o (x )E exp 16λ2Tρo (x t )Φ(s )d s 2d t 1/2.Takingλ=√28(1+√2),we arrive at(4.12)E exp12(1+√2)2Tρo(x t)Φ(s)d s2d t≤e2C2T+ϕ◦ρ2o(x)√2/8(1+√2).(c)Letγ:[0,ρ(x t,y t)]→M be the minimal geodesic from x t to y t,and U its tangent unit vector.By(4.2)we have∇V,∇ρ(·,y t) (x t)+ ∇V,∇ρ(x t,·) (y t)=ρ(x t,y t)0Hess V(U s,U s)d s≤−ρ(x t,y t)/2Φ(s)d s.(4.13)To understand the last inequality,we assume,for instance,thatρo(x t)≥ρo(y t)so that by the triangle inequality,ρo(γs)≥ρo(x t)−s≥ρ(x t,y t)/2−s,s∈[0,ρ(x t,y t)/2].For the coupling constructed in Section3,one concludes from(4.13)and the proof of Lemma2.3that(4.14)dρ(x t,y t)≤2K(x t,y t)(d−1)+C3−ρ(x t,y t)/2Φ(s)d s−ξtd t,t<τholds for some constant C4>0,whereK(x t,y t):=supΨ◦ρo≤Ψ(ρo(x t)+ρ(x t,y t))and is the minimal geodesic from x t to y bining this(4.4)and(4.14),we obtaindρ(x t,y t)≤C4+2θρo(x t)Φ(s)d s−ξtd t,t<τ.So,takingξt=C4+2θρo(x t)0Φ(s)d s+ρ(x,y)T,we arrive atdρ(x t,y t)≤−ρ(x,y)Td t,t<τ.This impliesτ≤T and hence x T=y T a.s.Combining(4.4)with(3.4)and(3.5)we conclude that for the present choice ofξt there existα,p,C5>1such thatE Rα/(α−1) p/(p−1)≤E exp12(1+√2)2Tρ(x t)Φ(s)d s2d t+C5T+C5Tρ(x,y)2.Combining this with(4.12)and(3.2)we obtain(4.15)(P T f(y))α≤(P T fα(x))expCT+CTρ(x,y)2+Cϕ◦ρ2(x)holds for someα,C>1,any positive f∈C b(M)and all x,y∈M,T>0.(d)For any positive f∈C b(M)withµ(fα)=1,(4.15)implies that(P T f(y))αB(o,1)exp−CT−CTρ(x,y)2−Cϕ2(x)µ(d x)≤1.Therefore,there exists a constant C >0such that(4.16)(P T f(y))α≤expC (1+T)+CTρ(y)2,y∈M,T>0.Combining this with(4.10)we obtainP T α→pα<∞,T>0,p>1.This is equivalent to the supercontactivity by the Riesz-Thorin interpolation theorem and P t 1→1=1.Thus,thefirst assertion holds.(e)To prove(4.6),it suffices to consider t∈(0,1]since P t 2→∞is decreasing in t>0. So,below we assume that T≤1.By(4.16)and the fact that(P2T f)α≤P T(P T f)α,we have(4.17) P2T α→∞≤ P T e2C ρ2o/T ∞e C (1+T),T>0.So,by the Riesz-Thorin interpolation theorem and P t 1→1=1,for the ultracontractivity it suffices to show that(4.18) P T eλρ2o ∞<∞,λ,T>0.SinceΦis increasing,it is easy to check thatη(r):=√r√rΦ(s)d s,r≥0is convex,and so is s→sη(log sλ)forλ>0.Thus,it follows from(4.8)and the Jenseninequality thathλ,x(t):=E eλρ2o(x t)<∞,x0=x∈M,λ,t>0andd+d thλ,x(t)≤C+C(λ)−λhλ,x(t)η(λ−1log hλ,x(t)),t>0.This implies(4.18)provided(4.5)holds.This can be done by considering the following two situations.(1)Since hλ,x(t)is decreasing providedλhλ,x(t)η(λ−1log hλ,x(t))>C+C(λ),ifλhλ,x(0)η(λ−1log hλ,x(0))≤2C+2C(λ)thenhλ,x(t)≤sup{r≥1:λrη(λ−1log r)≤2C+2C(λ)}≤1λ(2C+2C(λ))+Cfor some constant C >0.(2)Ifλhλ,x(0)η(λ−1log hλ,x(0))>2C+2C(λ),then hλ,x(t)is decreasing in t up totλ:=inf{t≥0:λhλ,x(t)η(λ−1log hλ,x(t))≤2C+2C(λ)}.Indeed,d+ d t hλ,x(t)≤−λ2hλ,x(t)η(λ−1log hλ,x(t)),t≤tλ.Thus,∞hλ,x(T∧tλ)d rrη(λ−1log r)≥λ2(T∧tλ).This is equivalent toΓ2(λ−1log hλ,x(T∧tλ))≥12(T∧tλ).Hence,hλ,x(T∧tλ)≤expλΓ−1212(T∧tλ).Since it is reduced to case(1)if T>tλby regarding tλas the initial time,in conclusion we havesup x∈M hλ,x(T)≤maxexpλΓ−12T/2,C +1λ(2C+2C(λ)).Therefore,(4.6)follows from(4.17),(4.9)withλ=2C /T,and the Riesz interpolation theorem.Finally,we note that a simple example for conditions in Theorem4.1to hold isΦ(s)=sα−1,Ψ(s)=εs2αforα>1and small enoughε>0.In this case P t is ultracontractive withP t 2→∞≤exp[c(1+t−(α+1)/(α−1))],t>0for some c>0.References[1]S.Aida,Uniformly positivity improving property,Sobolev inequalities and spectralgap,J.Funct.Anal.158(1998),152–185.[2]S.Aida,T.Masuda and I.Shigekawa,Logarithmic Sobolev inequalities and exponen-tial integrability,J.Funct.Anal.126(1994),83–101.[3]M.Arnaudon,A.Thalmaier,F.-Y.Wang,Harnack inequality and heat kernel esti-mates on manifolds with curvature unbounded below,Bull.Sci.Math.130(2006),no.3,223–233.[4]D.Bakry,D.and M.Emery,Hypercontractivit´e de semi-groupes de diffusion,C.R.Acad.Sci.Paris.S´e r.I Math.299(1984),775–778.[5]V.I.Bogachev,M.R¨o ckner and Feng-Yu Wang,Elliptic equations for invariant mea-sures onfinite and infinite dimensional manifolds,J.Math.Pures Appl.80:2(2001), 177–221.[6]J.M.Bismut,Large Deviations and the Malliavin Calculus,Boston:Birkh¨a user,MA,1984.[7]J.Cheeger,M.Gromov and M.Taylor,Finite propagation speed,Laplace operator,and geometry of complete Riemannian manifolds,J.Diff.Geom.17(1982),15–53.[8]Mu-Fa Chen and F.-Y.Wang,Application of coupling method to thefirst eigenvalueon manifold,Sci.Sin.(A),1994,37(1):1–14.[9]M.Cranston,Gradient estimates on manifolds using coupling,J.Funct.Anal.99(1991),110–124.[10]M.-F.Chen and F.-Y.Wang,Estimates of logarithmic Sobolev constant:an improve-ment of Bakry-Emery criterion,J.Funct.Anal.144(1997),287–300.[11]E.B.Davies and B.Simon,Ultracontractivity and heat kernel for Schr¨o dinger oper-ators and Dirichlet Laplacians,J.Funct.Anal.59(1984),335–395. [12]K.D.Elworthy and X.-M.Li,Formulae for the derivatives of heat semigroups,J.Funct.Anal.125(1994),252–286.[13]M.Emery,Stochastic calculus in manifolds,Springer-Verlag,Berlin,1989.With anappendix by P.-A.Meyer.[14]R.E.Greene and Hong-Xi Wu,Function Theory on Manifolds Which Possess a Pole,Lecture Notes in Math.699,Springer-Verlag,1979.[15]L.Gross,Logarithmic Sobolev inequalities,Amer.J.Math.97(1976),1061–1083.[16]E.P.Hsu,Logarithmic Sobolev inequalities on path spaces over Riemannian mani-folds,Comm.Math.Phys.189(1997),9–16.[17]W.S.Kendall,The radial part of Brownian motion on a manifold:a semimartingaleproperty,Ann.Probab.15(1987),no.4,1491–1500.[18]M.Ledoux,Concentration of measure and logarithmic Sobolev inequalities,S´e minairede Probabilit´e s,XXXIII,120–216,Lecture Notes Math.1709,Springer,Berlin,1999.[19]M.R¨o ckner and F.-Y.Wang,Harnack and functional inequalities for generalizedMehler semigroups,J.Funct.Anal.203(2003),237–261.[20]F.-Y.Wang,Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,Probability Theory Relat.Fields109(1997),417–424.[21]F.-Y.Wang,Logarithmic Sobolev inequalities:conditions and counterexamples,J.Operator Theory46(2001),183–197.[22]F.-Y.Wang,Functional Inequalities,Markov Properties,and Spectral Theory,SciencePress,Beijing/New York,2005.。

Abstract The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for twopoint resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a M¨ obius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.
N
Ii = 0.
i=1
(1)
The Kirchhoff law states
N j =1 ′
cij (Vi − Vj ) = Ii ,
i = 1, 2 , · · · , N ,
(2)
where the prime denotes the omission of the term j = i. Explicitly, (2) reads LV =I where c1 −c12 L = . . . −c1N

工程最优化问题及实用算法部分专业词汇简撷Aaccurateadj.正确的, 精确的 active constraint 主动约束 adjoint1.n.[数]伴随(矩阵)2.adj.伴随的, 共轭的, 相结合的 adjoint matrix 伴随(矩)阵 aestheticsn.美学, 美术理论, 审美学, 美的哲学 algorithmn.[数]运算法则, 算法 analytic(al)adj.分析的, 解析的 annealingn.(低温)退火, 焖火, 煅烧, 韧化, 热处理 annealing effect 退火效应artificial variables 人工变量 aspectn.方面, 方位, 朝向, 视点, 角度 axialadj.轴的, 轴向的, 轴心的 axialityn.轴对称性 axial vibration 轴向振动Bbar1.n.条, 棒, 杆尺, 规2.n.障碍物, 横杠, 梁, 拉杆, 挡杆, 隔板,挡杆3.n.巴(单位)4.n.棒材, 筋(条), 型钢 basic feasible solution 基(本)可行解 basic variable 基(本)变量 binaryadj.二进位的, 二元的 bisectionn.两断, 对分 bisection method 平分法, 等分法 boundaryn.边界, 分界线 bracketn.托架, 支架, 括号 brake1.n.闸, 刹车, 加载装置2.v.刹车 brake cam 制动凸轮 brittle failure脆断, 脆性破坏 brittle fracture脆性破裂, 脆性断裂Ccalculate 1.n.微积分2.n.演算, 计算(法)3.v.计算, 考虑, 计划, 打算 calculusn.微积分学 Cauchy ’s method 柯西法 categoryn.种类, 别, [逻]范畴 centroid n.质心characteristic1.adj.特有的, 表示特性的, 典型的，特性曲线的2.n.特性, 特征3.n.指数, 指标4.n.规格coefficient1.n.[数]系数, 率2.n.折算3.n.程度coil1.v.盘绕, 卷2.n.绕线圈, 螺线管3.n.线圈, 绕组4.n.盘条, 卷材columnn.圆柱, 柱壮物, 纵列combinationn.结合, 联合, 合并, 化合, 化合物complex search method联合搜索法component1.n.成分, 组件2.n.元件, 构件, 部件3.n.分力, 分量4.n.支量, 支命题5.adj.组成的, 构成的compressiven.有压力的, 压缩的compressive rigidity抗压刚度compressive stress压应力computationn.计算, 估计, 计算方法, 计算结果concave1.adj.凹的, 凹入的2.n.凹, 凹面conjugate1.v.共轭, 相配2.v.成对立物3.adj.对合的, 配合的4.adj.共轭的5.n.轭合物(用复数)conjugate direction共轭方向conjugate gradient method共轭梯度法consequencen.结果, [逻]推理, 推论, 因果关系, 重要的地位constitutevt.构成, 组成, 等同于, 相等constrainedadj.约束的constraint qualification约束条件contourn.轮廓, 周线, 等高线contraction1.n.收缩, 缩写式, 紧缩2.n.收敛段3.n.浓集, 密封convergencen.集中, 收敛convergence criteria收敛准则conversionn.变换, 转化, 换算convertvt.使转变, 转换, 换算convexadj.表面弯曲如球的外侧, 凸起的convex function凸函数criterian.标准, 准则criticaladj.评论的, 鉴定的, 批评的, 危急的, 临界的crossovern.交叉cross-section1.n.横截面, 剖面2.n.典型, 样品cubicadj.立方体的, 立方的current1.adj.当前的, 通用的, 流通的, 现在的, 最近的2.n.涌流, 趋势, 电流, 水流, 气流cutting plane method割平面法Ddamper1.n.阻尼器, 制动器, 阻尼线圈, 减震器2.n.风档,调节风门3.n.挡板damper coefficient阻尼系数decision variable决策变量decompose1.v.分解, (使)腐烂2.v.衰变, 蜕变decouplev.减弱(核爆炸)震波, 去耦, 分离deductionn.减除, 扣除, 减除额, 推论, 演绎deflectionn.挠度, 挠曲, 偏斜, 偏转, 偏差derivation1.n.分支, 分路2.n.偏转, 偏差3.n.导出, 求导数, 公式推导derivative1.adj.引出的, 系出的2.n.派生的事物, 派生词3.n.导数, 微商descent direction下降方向descent method下降法devise1.vt.设计, 发明, 图谋, 作出(计划), 想出(办法), 配置2.n.装置, 器械, 仪表, 元件3.n.方法, 手段4.n.图样, 草案DFP(Davidon-Fletcher-Powell) method DFP算法(拟牛顿法的一种)diametern.直径differential1.adj.微分的2.n.微分direct search method直接搜索法discard1.vt.丢弃, 抛弃2.v.放弃discrepancyn.相差, 差异, 矛盾discreteadj.不连续的, 离散的, 分开的dogleg method折线方法dualityn.二元性, 二重性, 二象性, 对偶性duality principle[数]对偶(性)原理, 二象性原理dualadj.双的, 二重的, 双重dual phase method两阶段法durabilityn.耐久性, 耐用性, 稳定性Eengineering optimization工程最优化enumerativeadj.列举的, 计数上的even1.adj.平的, 平滑的, 偶数的, 一致的, 恰好的, 平均的, 连贯的2.adv.恰好, 正当3.vt.使平坦, 使相等4.vi.变平, 相等5.n.偶数, 偶校验exhaustive search method穷举搜索法expansionn.扩充, 开展exploratoryadj.试探的, 探测的exploratory development应用研究, 探索研究exponential1.n.指数, 例子, [数]指数2.adj.指数的, 幂数的exponential series指数级数[系列]extendv.扩充, 延伸, 伸展, 扩大Ffeasible direction可行方向, 容许方向feasible direction method可行方向法feasible point可行点Fibonacci search method斐波拉契搜索法finite element mothod有限元法formulate1.vt.用公式表示, 明确地表达, 作简洁陈述2.v.阐明fractionn.小部分, 片断, 分数frequencyn.频率, 周率, 发生次数fundamental1.adj.基础的, 基本的2.n.基本原则, 基本原理Ggarbage1.n.作废信息, 无用数据, 废纸2.n.习惯上约定的字符Gauss-elimination method高斯消元法Gaussian distribution高斯分布(曲线), 正态分布geneticadj.遗传的, 起源的genetic algorithms(GAs)遗传算法geometric(al)adj.几何的, 几何学的global optimization全局最优化golden section search method黄金分割法gradientn.梯度gradient-based method既约梯度法gradient projection method投影梯度法gradient vector梯度向量graphicaladj.绘成图画似的, 绘画的, 图示的, 图解的graphical interpretation图释, 示意图gridn.格子, 栅格, 坐标网, 方格网Hheadwayn.进展, 净空高度helicaladj.螺旋状的Hessian matrixHessian矩阵hierarchicaladj.分等级的Hooke-Jeeves pattern search methodH-J模式搜索法hypercuben.超立方体Iimplicitadj.含蓄的, 隐式的inconsistentadj.不一致的, 不协调的, 矛盾的incrementn.增量, 增加indicatevt.指出, 简要地说明inductiveadj.诱导的, 感应的, 归纳的infinite1.n.无限的东西(如空间、时间), [数]无穷大2.adj.无穷的, 无限的, 无数的, 极大的infinite barrier penalty无穷大障碍惩罚inflectionn.变形inflexionn.[数]拐点ingredientn.成分, 因素initialadj.最初的, 初始的initializevt.初始化integern.整数integer program(m)ing整数规划(法)integer type整型integrability可积分(性)interactiveadj.交互式的intermediate1.adj.中间的2.n.媒介interpolatevt.插值, 内插interpolationn.插值法, 内插法interpretationn.解释, 阐明intervaln.间隔, 区间, 距离n.时间间隔interval halving method区间分半法, 等分区间法interval method区间法interval of convergence收敛区间inverse1.adj.倒转的, 反转的2.n.反面, 倒数, 逆量inverse penalty逆惩罚，反向惩罚iterationn.反复, 迭代iteration method迭代法iterative1.adj.重复的, 反复的, [数]迭代的2.n.反复体Jjournaln.轴颈, 枢轴Kkernel1.n.核心, 要点, 精髓, 内核, 中心, 型心2.n.影响函数kinematic(al)adj.[物] 运动学的, 运动学上的Kuhn-Tucker conditions龙格—库塔条件LLagrange multiplier拉格朗日乘子linear1.adj.(直)线性的, (直)线的, 沿线的, 沿轴作用的2.adj.平直linear programming线性规划local optimum局部最优loop1.n.环, 线(绳)圈, 弯曲部分, 回路, 回线2.n.循环Mmagnituden.大小, 数量, 巨大, 广大, 量级manipulate1.vt.操作, 使用2.vt.(巧妙地)处理(数据等)matrixn.矩阵mechanismn.机械装置, 机构, 机制method of multiplier乘子法minimalityn.最小(性)modal1.adj.形式的, 样式的, 形态上的, 模态的2.adj.[计](对话框等)模式的,modulusn.系数, 模数molaradj.质量的multipliern.乘子, 乘数, 增效器, 乘法器multivariableadj.[统][数]多变量的,多元的multivariable optimization多变量最优化mutationn.变化, 转变Nnegative-definite matrix负定矩阵Newton’s method牛顿法nonbasic variable非基变量non-constrainedadj.无约束的nonlinearadj.非线性的nonlinear programming problom 非线性规划问题nontraditional optimization method 非传统优化方法Ooddadj.奇数的, 单数的one-dimensional search一维搜索optimaladj.最佳的, 最理想的, 最优的optimality criteria最优性标准, 最优性原则optimizationn.最佳化, 最优化optimumadj.最佳的, 最有利的orthogonaladj.直角的, 直交的, 正交的Pparallel1.adj.平行的, 相同的, 类似的, 并联的2.n.平行线, 平行面, 类似, 相似物3.v.相应, 平行parametern.参数, 参量, 补助变数partial derivative偏导数, 偏微商pattern move模式移动pattern search method模式搜索法penaltyn.处罚, 惩罚penalty function method惩罚函数法phase1.n.阶段, 状态, 相, 相位2.v.定相polynomial1.adj.[数]多项式的2.n.多项式, 由两字以上组成的学名positive-definite matrix正定矩阵Powell’s conjugate direction method Powell共轭方向法preudo-objective function伪目标函数probability of crossover相交的概率, 交叉的可能性proceduren.程序procedure analysisn.工序分析, 过程分解propertyn.性质, 特性property value特征值, 本征值, 固有值prunev.剪除Qquadrate1.adj.方形的, 长方形的2.n.方形3.vt.使适合, 使一致quadratic1.adj.二次的, 平方的2.n.二次方程式Rrandom search method随机搜索法rear and front axle stand前后车轴架rectanglen.长方形, 矩形rectangular1.adj.矩形的, 长方形的2.adj.(成)直角的, 正交的, 方格式的reflectionn.映射region-elimination method区域消去法represent dynamically动力模拟reversal1.n.颠倒, 反转, 反向, 逆转, 撤销2.n.符号变换, 反号, 变号reversal condition倒逆条件, 可逆条件rigorous method严格方法，定点法Runge-Kutta method龙格－库塔方法Ssaddle point鞍点, 驻点secant1.adj.切的, 割的, 交叉的2.n.割线, 正切sectionaladj.剖面的, 局部的, 可组合的sensitivity analysis灵敏度分析sequencen.次序, 顺序, 序列shearv.剪切shearing stress(抗)剪应力, 切应力slack variable松驰变量, 松弛变数slope1.n.斜坡, 斜面, 倾斜2.v.(使)顺斜simplex1.adj.单纯的, 单一的2.n.单纯形simplex search method单纯形搜索方法simplifyn.简化simulation1.n.仿真, 假装, 模拟2.n.模拟实验single-variable optimization单变量最优化sinusoidaladj.正弦曲线(的), 窦状小管, 窦状隙stationaryn.静止的, 稳定的stationary pointn.驻点, 稳定点steepadj.陡峭的, 急剧升降的, 不合理的steepest descent direction最速下降方向steepest descent method最速下降法stiffnessn.坚硬, 硬度stochasticadj.随机的, 偶然的, 推测的subroutinen.[计]子程序subsequence1.n.后继, 随后, 顺序2.n.子序列subspacen.子空间substantiven.(独立存在的)实体substitute1.n.代用品, 代替者, 替代品2.v.代替, 替换, 替代successiveadj.继承的, 连续的suspensionn.悬挂, 悬架suspension arm吊杆, 弹簧吊架sustainvt.支撑, 撑住, 维持, 持续symmetryn.对称, 匀称Ttangent1.adj.接触的, 切线的, 相切的2.n.切线, [数]正切terminatev.停止, 结束, 终止terminationn.终止thresholdn.开始, 开端, 极限topology1.n.拓扑, 布局2.n.拓扑学transform1.vt.转换, 改变, 改造, 使...变形2.vi.改变, 转化, 变换3.n.[数]变换(式) transmissibilityn.可传送性, 可遗传性, 变速性truss1.vt.捆绑2.n.构架, 桁架tyre1.n.轮胎2.vt.装轮胎于tyre capacity车胎承载能力Uunidirectionaladj.单向的, 单向性的unimodaladj.[统](频率曲线或分布)单峰的, (现象或性质)用单峰分布描述的uniqueadj.唯一的, 独特的Vvalidatevt.确认, 证实, 验证vanishvi.[数]成为零variable1.n.[数]变数, 可变物, 变量2.adj.[数]变量的variable elimination method变量消去法, 消元法variationn.变更, 变化variation range变化范围, 变化区间vector1.n.[数]向量, 矢量2.vt.无线电导引vector solution矢量解法vertical1.adj.垂直的, 直立的, 顶点的2.n.垂直线, 垂直面, 竖向(in the) vicinity (of sth)在周围地区, 在附近vibrationn.振动, 颤动, 摇动, 摆动volumen.体积, 量, 大量Wworm gearn.蜗轮wrench1.n.扳手2.n.扭转XYZzig-zag1.n.Z字形, 锯齿形, 蜿蜒曲折2.adj.曲折的, 锯齿形的, Z字形的3.adv.作Z字形地, 弯弯曲曲地4.v.成Z字形, 作Z字形行进, 曲折前进Zoutendijk methodZoutendijk(可行方向)法REFERENCESENGLISH-CHINESE DICTIONARY OF MACHINE TECHNOLOGY, BEIJING, SCIENCEPRESS, 1983OXFORD ADV ANCED LEARNER’S ENGLISH-CHINESE DICTIONARY(Fourth Edition), THE COMMERCIAL PRESS&OXFORDUNIVERSITY PRESS,1997。

(2+1)维高阶Broer Kaup方程的多种局域相干结构黄文华;卢颖【摘要】应用WTC截断法获得了(2+1)维高阶Broer Kaup方程的一个含有三个任意变量函数的一般解析解.从一般解出发,选取适当的任意函数可以分别构造出多种特殊的局域相干结构,如多dromion解、双孤子解、环孤子解等,并通过解析和图示方法揭示了其奇异性质.%With the Weiss-Tabor-Carnevale truncation method,the (2 + 1)-dimensional higher-order Broer-Kaup equation is solved,and it obtained a general analytical solution including three arbitrary functions.Staring from the general solution,some special types of localized coherent structures such as multi-dromion solutions,two-soliton solutions and ring soliton solution are built by selecting proper arbitrary functions respectively.The interesting novel features and evolution properties are revealed analytically and graphically.【期刊名称】《湖州师范学院学报》【年(卷),期】2018(040)010【总页数】7页(P1-6,29)【关键词】高阶Broer Kaup方程;WTC截断法;局域相干结构【作者】黄文华;卢颖【作者单位】湖州师范学院理学院,浙江湖州313000;湖州师范学院理学院,浙江湖州313000【正文语种】中文【中图分类】TN20 IntroductionIn recent decades, there has been noticeable progress in the research of the modern nonlinear physics. In the study of nonlinear physical models, to find physically significant solutions especially soliton solutions and localized coherent structures is very important. These exact solutionsand and coherent structures can help one to well understand the mechanism of the complicated physical phenomena and dynamical processes. A vast variety of methods has been developed in recent years to seek exact solutions and construct localized coherent structures such as multilinear variable separation approach[1,2], the generalized homogeneous balance method[3], the extended tanh-function[4],the generalized projective Riccati equation method[5],etc. In high dimensions, especially for the(2+1)-dimensional nonlinear physical systems, many types of single-valued and multi-valued localized excitations have been found and some interesting novel features and interaction properties are revealed[1-13].In this paper, we will consider the (2+1)-dimensional higher-order Broer-Kaup (HBK) equationuyt+4(uxx+u3-3uux+3uw)xy+12(uv)xx=0,vt+4(vxx+u2v+3uvx+3vw)x=0,(1)wy-vx=0,which was obtained from the inner parameter dependent symmetry constraints of the Kadomtsev-Petviashivili(KP) equation [14] and is applied to the statistical physics, the plasma physics and the nonlinear optics. When take y=x,Eq.(1) is reduced to the usual (1+1)-dimensional HBK equation. Lin et.al [15] used the Painlev′e analysis to derive fractal dromion and multi-peakon structures, and a variety of multi-kink solutions and multi-soliton solutions are also obtained based on Bäcklund transformation very recently[16].As is known, the Painlev′e analysis developed by Weiss, Tabor and Carnevale [17] is one of the most powerful methods to provethe integrability of a model, and also can be extended to find some exact solutions[16,18]. Directly staring from the Weiss-Tabor-Carnevale (WTC) truncation expansion, we further study the (2+1)-dimensional HBK equation. A general solution including three arbitrary functions is obtained, and by introducing arbitrary functions appropriately special types of dromions, solitoffs and ring solitary wave solutions are constructed.1 WTC truncation method for the (2+1)-dimensional HBK systemTo solve the (2+1)-dimensional HBK system, according to the idea of WTC, we truncate the painlev′e expansion of Eq.(1) at the constant termu=φ-1u0+u1,v=φ-2v0+φ-1v1+v2,w=φ-2w0+φ-1w1+w2,(2)where φ is the singular manifold variable, and u1,v2, w2 are arbitrary known solutions of Eq.(1). For simplicity, we take the trivial seed solutionsu1=0,u2=0,w2=w2(x,y,t).(3)Substituting (2) with (3) into Eq.(1) and equating the coefficients of like powers of φ, we obtain(4)and φ,w2 satisfy the system of equations4φxyφxxx-4φxφxxxy+φxyφt-φxφyt=0,(5)4φxxx+φt+12w2φx=0.From (5), one can find a class of solutions(6)where f≡f(x,t), g≡g(y), τ≡τ(t)a re arbitrary functions of indicated variables and “a” is an arbitrary constant. Obviously, here w2 is a functionof{x,t}.The substitution of (4) and (6) into (2) yields a general solution to Eq.(1)(7)(8)Thanks to the arbitrariness of the function s f,g and τ, the general solutions Eqs.(7)～(9) for the (2+1) dimensional HBK equation reveal a diversity of exact solutions and quite rich coherent structures when the functionsf,g and τ are selected appropriately and avoid the singularities. In the next section, we focus our attention on some special possible localized coherent structures.2 Localized coherent structuresNow, we study the field quantity v determined by Eq.(8). Considering the arbitrary function τ in Eq.(8) affect structures of the field v mainly on its amplitude, for simplicity, we fix τ(t)=b=constant, then the field quantity v is rewritten as(10)and a bunch of localized coherent structures can be derived.2.1 Dromion solutionsIn (2+1) dimensions, one special type of localized coherent soliton structures called dromion solution,decaying exponential in all directions, exist in many nonlinear models. A dromion localize at a cross point of underlying(ghost) straight or curved line solitons subject to suitable dispersion relations[1,6]. Multi-dromion solutions and dromion lattice solutions are also derived in many nonlinear models[8]. Multi-dromion solutions are usually built by multiple straight-line solitons subject to appropriate conditions. The dromions are localized around the cross pointsof the corresponding straight lines. For the field v expressed by Eq.(10), we restrict the arbitrary functions f and g as followf=exp(x2+t), g=exp(y2),(11)then we obtain a special type of multi-dromion solution with two peaksand two holes, which is depicted in Fig.1(a)-(b).If some a periodicity in x and y is included into functions f and g, one can obtain dromion solutions with oscillating tails in some a spatial directions. For instance, takingf=exp(1.05x+sin(2x)/2+t2), g=exp(y),(12)we drive a dromion featuring oscillations along x and the localizationalong y, as show in Fig.2 (a). If we setf=exp(1.2x+t2), g=exp(1.5sin(y)),(13)another type of oscillating dromion solution is built, which is plotted in Fig.2 (b).From Fig.2 (b) and the corresponding contour map Fig.2(e), one can seethe quantity v oscillate in y direction with peaks and holes assembling periodically and localized in x direction. Further more, if periodical functions are both introduced in the functions f and g, a “dromion lattice” solution can be constructed. Fig.2 (c) displays such a “dromion lattice” solution by fixingf=exp(1.2sin(x)+t), g=exp(1.2sin(y)).(14)With a=b=1 at t=0. Note that the individual dromions building the lattice are located at the cross points of the “ghost” (underlying) straight-line solitons.2.2 Solitoff solutionsThe solitoff is another special type of coherent soliton solution in (2+1)-dimensional nonlinear models, where the wave field decays exponentially in all directions except for a preferred direction[19,20]. Solitoffs constitute an intermediate state between dromions and plane solitons. In Ref.[20],a solitoff solution and two-solitoff solution are displayed, and for the two-solitoff solution,it is shown 2-straightline solitons become only one half straight line soliton due to resonance effect. Here, for the field v of the HBK system, we will build new two special types of two-solitoff solutions. When setting the functions f and g in Eq.(10) as(15)we derived a two-solitoff solution holding two single solitoff while the ridges emerging and prolonging away in opposite directions, which is plotted in Fig.3 (a)-(b). If the functions fand g is fix asf=exp(1.2x+t2),g=exp(-y2)，(16)another two-solitoff solution is shown in Fig.3 (c)-(d), which hold a positive single solitoff and a negative solitoff and the ridges emerging and prolonging away in the same direction.2.3 Ring soliton solutionsIn high dimensions, in addition to the point-like coherent excitation, there may be some types of ring soliton solutions which are not equal to zero identically apart from the closed curves and decay exponentially apart from the closed curves. A ring-like nonlinear wave can be viewed as a curved-line nonlinear wave closing on itself. Furthermore, it can feature different wave structures, such as plateau, basin, bowl, saddle, etc [2,8]. For the HBK system, the ring soliton solution can be easily obtained. We now construct a special type of saddle-like ring soliton solution for the field v by selecting f and g asf=exp(-x2+0.8t2),g=exp(-y2)，(17)a=b=1 at t=0,2.5,5. The shape of the respective ring-like nonlinear waves and the corresponding contour maps are displayed in Fig. 4. Interestingly, from Fig.4 (a)-(b), one can see the saddle ring soliton solution is originated from a multi-dromion solution with two peaks and two holes at t=0, and grow up with peaks and holes turning into ring bridges and the ring’s radius amplifying with the time t increase.3 ConclusionTo summarize, the WTC truncation method is applied to the (2+1)-dimensional HBK equation and a general analytic solution is derived. The general solution depends on the set of three arbitrary functions. By introducing proper functions and avoiding the singularities, the general solution reveals a diversity of exact solutions and quite rich localized coherent structures. We construct many new special types of localized soliton structures such as multi-dromion solutions, dromion lattice, two-solitoff solutions and ring soliton solution respectively.The interesting features and evolution behavior for these localized structures are investigated analytically and graphically. More about the HBK system and whether this phenomena of exact solutions and coherent structures exist in other higher-dimensional processes is worth further studying.References:【相关文献】[1] LOU S Y.On the coherent structures of the Nizhnik-Novikov-Veselov equation[J].Phys LettA,2000,277:94-100.[2] TANG X Y,LOU S Y,ZHANG Y.Localized excitations in(2+1)-dimensional systems[J].Phys RevE,2002,66(2):046 601.[3] ZHANG J F,LIU Y L.Localized coherent structures of the(2+1)-dimensional higher order Broer-Kaup equations[J].Applied Mathematics and Mechanics,2002,23:549-556.[4] DAI C Q,ZHANG J F.Novel variable separation solutions and exotic localized excitations via the Etion TM innonlinear soliton systems[J].J Math Phys,2006,47(4):043 501.[5] DAI C Q,CHEN R P.Solitons with fusion and fission properties in the(2+1)-dimensional modified dispersive water-wave system[J].Z Naturforsch,2006,61:307-315.[6] BOITI M,LEON J J P,MARTINA L,et al.Scattering of localized solitons in the plane[J].Phys Let A,1988,132:432-439.[7] RADHA R,LAKSHMAN M.Singularity analysis and localized coherent structuresin(2+1)dimensional generalized Korteweg de Vries equations [J].J Math Phys,1994,35:4 746-4 756.[8] LOU S Y,TANG X Y.Methods of Nonlinear Mathematical Physics [M].Beijing:Science Press,2006.[9] HUANG W H,LIU Y L,LU Z M.Doubly periodic wave and folded solitary wave solutions for(2+1)-dimensional higher-order Broer-Kaup equation[J].Chaos,Solitons and Fractals,2007,31:54-63.[10] HUANG W H.Periodic semifolded solitary waves for(2+1)-Dimensional variable coefficient broer kaup mun[J].Commun Theor Phys,2008,49:1 383-1 388. [11] HUANG W H.Periodic folded waves for a (2+1)-dimensional modified dispersive water wave equation[J].Chinese Physics B,2009,18:3 163-3 168.[12] DAI C Q,XUE C,WU S.Exotic localized structures based on a variable separation solution of the(2+1)-dimensional higher-order Broer-Kaup system[J].Nonlinear Analysis:Real world Applications,2009,10:259-265.[13] DAI C Q, LIU C Y.Solitary wave fission and fusion in th Nonlinear Analysis:Real world Applications,e(2+1)-dimensional generalized-Broer-Kaup system[J].Nonlinear Analysis:Modelling and Control,2012,17:271-279.[14] LOU S Y,HU X B.Infinitely many Lax Pairs and symmetry constraints of the KP equation[J].J Math Phys,1997,38:6 401-6 427.[15] LIN J,Li H M.Painlev′e integrability and abundant localized structures of(2+1)-dimensional higher order Broer-Kaup system[J].ZNaturforsch A,2002,57:929-936.[16] WAZWAZ A M.On the solutions of the(2+1)-dimensiona higher-order Broer-Kaup systems[J].Communications in Numerical Analysis,2014,14:00 193.[17] WEISS J,TABOR M,CARNEVALE G.The Painlev′e property for partial differential equations[J].J Math Phys,1983,24:522.[18] LOU S Y.Extended painleve expansion,nonstandard truncation and special reductions of nonlinear evolution equations[J].Z Naturforsch A,1998,53:251-258.[19] CHOW K W.“Solitoff”Solutions of Nonlinear E volution Equations[J].J Phys Soc Jpn,1996,65:1 971-1 976.[20] HUANG W H,ZHANG J F，SHENG Z M.Coherent soliton structures of the(2+1)-dimensional long-wave-short- wave resonance interaction equation[J].Chinese Physics,2001,11:1 101-1 106.。