A variable coefficient nonlinear Schr"{o}dinger equation with a four-dimensional symmetry group and blow-up of its solutions

A v ariable co efficien t nonlinear Sc hr¨ odinger equation with a four-dimens ional symmetry group and blo w-up of its soluti o ns F. G ¨ ung¨ or ∗ , M. Hasa no v ∗ , C. ¨ Ozemir † Octob er 23, 201 8 Abstract A canonical v ariable co efficient n onlinear Sc hr¨ odinger equation with a four dimensional symmetry group con taining SL(2 , R ) group as a subgrou p is considered. This t yp ical inv ariance is then used to transform b y a symmetry transformation a kno wn solution th at can b e derived b y truncating its P ainlev ´ e expansion and s tudy blo w-ups of these solutions in the L p -norm for p > 2, L ∞ -norm and in the sense of distributions. Keywor ds: SL(2 , R ) in v ariance, v ariable co efficien t nonlinear Schr¨ odinger equation, exact solutions, blo w-up AMS subje ct Classific ations: Primary 3 5Q55, 35B44, 3 5B06; Secondary 35A25 1 In tr o duc tion It is w ell kno wn that the linear heat a nd Sc hr¨ odinger equations f o r u ∈ R , ψ ∈ C u t = u xx , iψ t + ψ xx = 0 , x ∈ R , t > 0 ha v e isomorphic Lie symmetry groups. The symmetry gro up with the infinite-dimensional ideal refle cting linearit y factored out c an b e written as a semidirect pro duct of the three-dimensional Heisen b erg group H and SL(2 , R ) G = H ⋉ SL(2 , R ) . ∗ Department o f Ma thematics, F aculty of Arts and Sc ie nc e s, Do˘ gu¸ s Universit y , 3 4722 Ista nb ul, T urk ey , e - mail: fgungor @dogus.edu.tr and e-ma il: mhasansoy@dogus .e du.tr † Department of Mathema tics , F a culty o f Science and Letters, Ista n bul T ec hnical Univ ersity , 34469 Istanbul, T urkey , e-mail: oz e mir @itu.edu.tr 1 On the other hand, among the mo dular class of nonlinear Sc hr¨ odinger equations in one space dimension iψ t + ψ xx = F ( | ψ | ) ψ , (1.1) the o nly one preserving the symmetry group of the linear Sc hr¨ odinger equation except with infinite-dimensional symmetry of its linear coun terpart is the quin tic Schr¨ odinger equation iψ t + ψ xx = λ | ψ | 4 ψ . (1.2) In this a rticle w e lo ok a t a v ariable co efficien t extension of t he one-dimensional cubic NLSE (nonlinear Sc hr¨ odinger equation) iψ t + ψ xx + g ( x, t ) | ψ | 2 ψ + h ( x, t ) ψ = 0 , g = g 1 + ig 2 , h = h 1 + ih 2 , g j , h j ∈ R , j = 1 , 2 , g 1 6 = 0 . (1.3) V ariable co efficien t extensions of nonlinear ev olution t yp e equations tend to arise in cases when less idealized conditions such as inhomogeneities and v ariable top ographies are assumed in their deriv ation. The reader is referred to [8] and the r eferences therein for sev eral phys ically motiv ated applications. Symmetry classification of (1.4) w as giv en in [4]. W e men tion that the dimension of the maximal symmetry group is dim G = 5 and is a c hieve d only when the co efficien ts are constan t and additionally h = 0 whic h is reduced t o nothing more than the usual NLS equation. The symmetry group is isomorphic to the group of o ne- dimensional extended Galilei similitude algebra. W e are particularly in terested in the case when at least the SL(2 , R ) in v ariance is con tained in the full symmetry group o f ( 1 .3). This is usually referred to as the pseudo-conformal inv ariance in the con text of qualitativ e analysis of PD Es. W e emphasize tha t this in v ariance manifests itself as a subgroup of t he full symmetry groups o f the NLSE, Da v ey-Stew artson (DS) equations and their p ossible generalizations [5] and has b een successfully applied to in v estigate blowup formation in these nonlinear ev olutio n mo dels [2, 7, 1, 3]. A study of self-similar solutio ns of the pseudo-conformally in v arian t nonlinear Schr¨ odinger equation can b e found in [6]. The p oin t is that when the v a riable co efficien ts are allo w ed in these equations, this t ypical symmetry is mostly destroy ed. Our in ten tio n here is to detect the sub cases in whic h suc h a symmetry remains intact in v ariable co efficien ts v ariants of the NLS equations. This will mak e it p ossible to generate new nontrivial solutio ns from known ones. W e quote the following result from [4] and note that throughout the pap er, an y solution will b e understo o d a s a p oin twise solution in the classical sense. Prop osition 1. Any e quation of the form (1.3) c ontaining SL(2 , R ) symmetry gr oup as a sub gr oup c an b e tr ansfo rme d by p o int tr ansform ations to the c anonic al form iψ t + ψ xx + ( ǫ + iγ ) 1 x | ψ | 2 ψ + ( h 1 + ih 2 ) 1 x 2 ψ = 0 , x ∈ R \ { 0 } , (1.4) wher e ǫ = ± 1 , γ , h 1 and h 2 ar e a rb i tr a ry r e al c onstants. 2 Note that like all equations in the mo dular class (1.1), equation (1.3) and also (1.4) are alw a ys inv arian t under the constant change of phase of ψ (gauge-in v ariance) while lea ving the ( x, t ) co ordinates unc hang ed. W e represen t the phase and mo dulus of ψ b y ρ , ω writing ψ = ρ ( x, t ) exp ( iω ( x, t )). Prop osition 2. Th e symmetry alge br a L of (1.4) is four-dimensiona l and sp ann e d by the ve ctor fie l d s T = ∂ t , D = 2 t∂ t + x∂ x − 1 2 ρ∂ ρ , C = t 2 ∂ t + xt∂ x − 1 2 tρ∂ ρ + x 2 4 ∂ ω , W = ∂ ω . The c ommutators among T , D , C satisfy [ T , D ] = 2 T , [ T , C ] = D , [ D , C ] = 2 C and W is the c enter eleme nt, n amely c ommutes with al l other elem e nts. The Lie algebr a L has the dir e ct sum structur e L = sl(2 , R ) ⊕ R . The elemen ts T , D , C , W generate time translations, scaling, (pseudo)-conforma l and gauge transformations, resp ectiv ely . A significan t consequence of these transformations is the group action on the solutions giv en by the follow ing. Prop osition 3. If ψ 0 ( x, t ) is a solution of (1.4 ) , then so is ψ ( x, t ) = ( a + bt ) − 1 / 2 e i bx 2 4( a + bt ) ψ 0  x a + bt , c + dt a + bt  for ad − bc = 1 . Pr o of. By exp onen tiating the infinitesimal generators T , D , C (i.e. solving Cauc hy problems for these v ector fields) and then comp osing the corresponding gro up transformations w e find the ab o v e SL(2 , R ) action on the solutions. Note that the action corresp onding to t generates M¨ obius transformations of t . Based o n this result, our main purp ose is to transform one known solution of the original equation to a more complicated one b y the transformations of t he SL(2 , R ) symmetry and then c ho ose group parameters a, b, c appropriately and pass t o limit of the w a v e function ψ as t → T − for some finite time T in the L p -norm and distributional sense as w ell. W e now use a truncation approac h to o btain a sp ecial exact solution of (1.4 ) . W e are going to find this sp ecial explicit solution b y t r uncating it s P ainlev ´ e expansion at the first term. F or con v enience w e write (1.4) together with its complex conjugate as the system iu t + u xx + ( ǫ + iγ ) 1 x u 2 v + ( h 1 + ih 2 ) 1 x 2 u = 0 , − iv t + v xx + ( ǫ − iγ ) 1 x uv 2 + ( h 1 − ih 2 ) 1 x 2 v = 0 . (1.5) 3 Here u w as emplo y ed instead of ψ and v denotes its complex conjugate, but in this setting they are view ed as indep enden t functions. W e first show that (1 .5) do es not pass the P ainlev ´ e test for PDEs and then pro ceed to obtain a n exact solution afterw ards. A partial differen tial equation is said to ha v e the Painlev ´ e prop ert y if all its solutions are single v alued aro und any non-c haracteristic mov able singularit y manifold. If this singularity manifold is denoted by Φ( x, t ) = 0 (actually a curv e in this case), w e shall lo ok fo r solutions of the system (1.5) in the form of a La ur ent expansion and w e expand u ( x, t ) = ∞ X j =0 u j ( x, t )Φ α + j ( x, t ) , v ( x, t ) = ∞ X j =0 v j ( x, t )Φ β + j ( x, t ) , (1.6) where u 0 , v 0 6 = 0 and u j , v j , Φ( x, t ) are analytic functions. α and β are negativ e in tegers to b e determined fro m the leading order analysis, so as to ensure absence of essen tial singularities a nd branc h p oints in a ll solutions. F or the determination of leading orders α and β , we substitute u ∼ u 0 Φ α and v ∼ v 0 Φ β in (1.5) and see that b y bala ncing the t erms of smallest order α + β = − 2 (1.7) and u 0 v 0 = − α ( α − 1) x ǫ + iγ Φ 2 x = − β ( β − 1) x ǫ − iγ Φ 2 x (1.8) m ust ho ld. (1.7) allows the negative in tegers α = − 1 and β = − 1 and with these leading orders, (1.8) forces γ = 0. After determination of t he leading o rders, w e substitute (1 .6) into ( 1 .5). Equating to zero the co efficien t of Φ − 3+ j , j ≥ 1 , j ∈ N , w e arriv e a t a linear system Q ( j )  u j v j  =  F j G j  (1.9) from whic h the co efficien ts u j , v j can b e f o und. Those v alues o f indices j f o r whic h det Q ( j ) = 0 are called r esonanc e s . In order that the expansion (1.6 ) includes correct n um b er of arbitra r y functions as required by the Cauc h y- K o v alewski theorem (where Φ( x, t ) should b e one of the arbitrary functions), some consistency conditions at resonances m ust b e satisfied. In the general case (1.3) , these constrain ts force the co efficien ts to b e prop erly related (See [8 ] for details). F or (1.4), this is not t he case, namely it cannot pass the Painlev ´ e test. It thus fails to satisfy the necessary condition for the equation t o ha v e Painlev ´ e prop erty . Not withstanding this fact, application of the P a inlev´ e expansion to noninte grable PDEs lik e ( 1 .4) (or more prop erly partial in tegrable) can allow pa rticular explicit solutions to b e obtained b y truncating the expansion. This approa c h imp o ses constrain ts on the arbitr a ry functions and the function Φ a s a result of compatibility of a n o v erdetermined PDE system. 4 Before truncating the series (1.6) at some index j = N , w e first we ak en the condition that α and β assumes only integer v a lues. When w e solv e (1.7) and (1.8) together, w e find that f o r γ 6 = 0 α = − 1 − iδ , β = − 1 + iδ ; δ = − 3 ǫ ± p 8 γ 2 + 9 2 γ (1.10) and (1.8) is equiv alen t to u 0 v 0 = − 3 δ γ x Φ 2 x . (1.1 1) T runcating the P a inlev ´ e expansion at the first term ( j = 0), w e a ssume a solution of the form u ( x, t ) = u 0 ( x, t )Φ( x, t ) − 1 − iδ , v ( x, t ) = v 0 ( x, t )Φ( x, t ) − 1+ iδ . (1.12) W e substitute these in (1.5) and set the co efficien ts of the terms Φ − 3 ± iδ , Φ − 2 ± iδ , Φ − 1 ± iδ equal to zero. The conditio n at t he o r der Φ − 3 ± iδ is equiv alent to (1.11) and terms of order Φ − 2 ± iδ , Φ − 1 ± iδ disapp ear if Φ, u 0 , v 0 are c hosen to b e Φ( x, t ) =  x k 4 t + k 1  2 / 3 + k 2 (1.13) u 0 ( x, t ) = A 1 x 1 / 6 ( k 4 t + k 1 ) 2 / 3 exp  i  k 4 x 2 4( k 4 t + k 1 ) + k 3   (1.14) v 0 ( x, t ) = A 2 x 1 / 6 ( k 4 t + k 1 ) 2 / 3 exp  − i  k 4 x 2 4( k 4 t + k 1 ) + k 3   (1.15) for arbitrary real constan ts A 1 , A 2 and k 1 , ..., k 4 with the constrain ts h 1 = 5 / 36 and h 2 = 0. W e require t o ha v e v = u ∗ , whic h implies A 1 = A 2 . Let us rename this constan t A . When w e c heck the condition (1.11), w e find that A 2 = − 4 δ 3 γ . Since this square m ust b e p ositiv e, it is necessary that δ γ = − 3 ǫ ± p 8 γ 2 + 9 2 γ 2 < 0 . (1.16) F or b oth ǫ = ± 1, it is seen that the minus sign m ust b e pic ke d in the formula for δ of (1 .10). Ha ving fo und a consisten t truncation, w e can write the solution t o (1 .4) as u ( x, t ) = Ax 1 / 6 x 2 / 3 + k 2 ( k 4 t + k 1 ) 2 / 3 exp  i  k 4 x 2 4( k 4 t + k 1 ) − δ ln  x 2 / 3 ( k 4 t + k 1 ) 2 / 3 + k 2  + k 3  . (1.17) Cho osing k 4 = 0, we obtain the stationary solution u ( x ) = Ax 1 / 6 x 2 / 3 + k 2 / 3 1 k 2 exp h i  − δ ln( x 2 / 3 + k 2 / 3 1 k 2 ) + k 3 i , (1.18) 5 where h 1 = 5 36 , h 2 = 0 , A = ( − 4 δ 3 γ ) 1 / 2 , δ = − 3 ǫ − p 8 γ 2 + 9 2 γ , (1.19) and k 1 , k 2 , k 3 ( k 3 is r elab elled) are a r bitrary real constants. W e summarize: Prop osition 4. The fol lowi n g solves e quation (1.4) for arbitr a ry c onstants k 1 , k 2 , k 3 and fo r the p ar am eters h 1 = 5 / 36 , h 2 = 0 u ( x, t ) = A x 1 / 6 x 2 / 3 + k 2 / 3 1 k 2 exp h i  − δ ln( x 2 / 3 + k 2 / 3 1 k 2 ) + k 3 i , (1.20) wher e A = ( − 4 δ 3 γ ) 1 / 2 and δ = − 3 ǫ − √ 8 γ 2 +9 2 γ . 2 T ransforming soluti o ns b y SL(2 , R ) gro up Blo wup in the L p , L ∞ norms and in the distributional sense No w w e w ould like to illustrate how the SL(2 , R ) group action can b e useful in establishing blo w-up profiles of initia l v alue problems for v ariable co efficien t NLS equations just as they w ere used for their constan t co efficien t coun terparts. Let u ( x ) b e the stationary solution to (1.4), defined b y (1.18)-(1.1 9). W e set ψ 0 ( x ) := u ( x ) a nd use Prop osition 3 . By this pro p osition for arbitrary a, b ∈ R ψ ( x, t ) = ( a + bt ) − 1 / 2 exp  ibx 2 4( a + bt )  ψ 0  x a + bt  (2.1) is a lso a solution to equation (1.4). W e can assume a > 0 , b < 0 and denote ε := a + bt = b ( t − − a b ) = b ( t − T ) , where T = − a b > 0. Hence, t → T − ⇔ ε → 0 + . By using this nota t io n w e can write solution (2.1) in the form ψ ε ( x ) = ε − 1 / 2 exp  ibx 2 4 ε  ψ 0  x ε  . W e are go ing to sho w t ha t these solutions will blowup in the L p -norm when p > 2, L ∞ -norm and in the sense o f generalized functions, resp ectiv ely . Note: In the following theorems, w e do not imp ose an initial condition but rather w e limit to t he o ne a s dictated by the solution (2.1) in the form ψ ( x, 0) = 1 √ a exp  ibx 2 4 a  ψ 0  x a  . Also, for a blow-up a t some finite t ime we ha v e to fix b figuring in (2.1). 6 L p -blo w-up solutions. Theorem 1. F or any T > 0 ther e is a solution ψ ( x, t ) to e quation (1.4) such that lim t → T − k ψ ( x, t ) k p = + ∞ for al l p > 2 , (2.2) wher e k ψ ( x, t ) k p =  R + ∞ −∞ | ψ | p dx  1 /p . Pr o of. Let T > 0 b e a finite time. W e can a lwa ys arrang e tw o n um b ers a > 0 and b < 0 suc h that T = − a b . Setting these n umbers in (2 .1) w e get the function ψ ( x, t ) whic h will b e instrumen tal in the pro of. W e rewrite this function, a s g iv en ab o v e, in the form ψ ε ( x ) = ε − 1 / 2 exp  ibx 2 4 ε  ψ 0  x ε  . Then w e hav e lim ε → 0 + k ψ ε ( x ) k p = lim t → T − k ψ ( x, t ) k p . (2.3) By t he definition of ψ 0 ( x ) w e hav e ψ 0  x ε  = Aε − 1 6 x 1 6 ε − 2 3 x 2 3 + C exp  iϕ ( x, ε )  , where ϕ ( x, ε ) = − δ ln   x ε  2 3 + k 2 3 1 k 2 + k 3  , C = k 2 3 1 k 2 and we c ho ose C > 0. Thus , ψ ε ( x ) = Ax 1 6 x 2 3 + ε 2 3 C exp  i  bx 2 4 ε + ϕ ( x, ε )   , and Z ∞ −∞ | ψ ε ( x ) | p dx = 2 Z ∞ 0 A p x p/ 6  x 2 / 3 + ε 2 / 3 C  p dx. By ( 2 .3) lim t → T − k ψ ( x, t ) k p = + ∞ if and only if: i) R ∞ 0 | ψ ε ( x ) | p is finite fo r all p > 2 and ε > 0, ii) lim ε → 0 + R ∞ 0 | ψ ε ( x ) | p dx = ∞ . Let us substitute x = εy . Then Z ∞ −∞ | ψ ε ( x ) | p dx = 2 A p Z ∞ 0 ε p 6 y p 6 ε 2 p 3  y 2 3 + C  p εdy = 2 A p ε ( p − 2) / 2 Z ∞ 0 y p 6  y 2 3 + C  p dy = = 2 A p ε ( p − 2) / 2 h Z 1 0 y p 6  y 2 3 + C  p dy + Z + ∞ 1 y p 6  y 2 3 + C  p dy i , (2.4) 7 where R 1 0 y p 6  y 2 3 + C  p dy is con v ergen t by con tin uit y of the function. On t he o t her hand, y p 6  y 2 3 + C  p ≤ y p 6 y 2 p 3 = 1 y p 2 . Consequen tly , R ∞ 1 y p 6  y 2 3 + C  p dy ≤ R ∞ 1 1 y p 2 dy implies that the integral R ∞ 1 y p 6  y 2 3 + C  p dy is con v ergent fo r all p > 2. Hence, R ∞ 0 | ψ ε ( x ) | p dx is con v ergen t for p > 2. Finally , by taking the limit w e find lim ε → 0 + Z ∞ −∞ | ψ ε ( x ) | p dx = lim ε → 0 + 2 A p ε ( p − 2) / 2 Z ∞ 0 y p 6  y 2 3 + C  p dy = ∞ . Blo w-up solutions in L ∞ - norms. In t he following theorem w e pro v e tha t the ab ov e defined solutions ψ ε ( x ) will blo wup in L ∞ - no r m to o. Theorem 2. F or any T > 0 ther e is a solution ψ ( x, t ) to e quation (1.4) such that lim t → T − k ψ ( x, t ) k ∞ = ∞ , (2.5) wher e k ψ ( x, t ) k ∞ = ess sup x ∈ [0 , ∞ ) | ψ ( x, t | , t < T . Pr o of. Let aga in ψ ( x, t ) b e the function defined b y (2 .1) and ψ ε ( x ) b e the ab o v e defined function. By the construction o f ψ ε ( x ) lim ε → 0 + k ψ ε ( x ) k ∞ = lim t → T − k ψ ( x, t ) k ∞ and therefore we are done if w e can sho w that lim ε → 0 + k ψ ε ( x ) k ∞ = ∞ . W e hav e | ψ ε ( x ) | = A | x | 1 6 x 2 3 + ε 2 3 C . Let A = C = 1. | ψ ε ( x ) | is an ev en function so that we can r estrict ourselv es to the in terv al [0 , ∞ ). Since | ψ ε ( x ) | is contin uous on [0 , ∞ ), | ψ ε (0) | = 0 and lim x →∞ | ψ ε ( x ) | = 0, then there exists x 0 ∈ (0 , ∞ ) suc h that k ψ ε ( x ) k ∞ = ess sup x ∈ [0 , ∞ ) | ψ ε ( x | = max x ∈ [0 , ∞ ) | ψ ε ( x ) | = x 1 6 0 x 2 3 0 + ε 2 3 C . A simple computation yields x 0 = ε √ 27 . Therefore, k ψ ε ( x ) k ∞ = C ε 1 6 ε 2 3 = C √ ε . Consequen tly , lim ε → 0 + k ψ ε ( x ) k ∞ = ∞ . 8 δ -Blowup solutions in the sense of generalized functions. Let D = C ∞ 0 (0 , ∞ ) b e the space of infinitely differen tiable functions with compact supp ort in (0 , ∞ ). The dual of D is called the space of generalized functions and is denoted by D ′ . Definition 1. L et f , f n ∈ D ′ . We say that the se quenc e f n c onver ges to f if and only if h f n , ϕ i → h f , ϕ i for al l ϕ ∈ D , wher e h f , ϕ i denotes the value of the functional f at ϕ . No w we presen t δ -blow up solutions in the sense of generalized functions. Theorem 3. F or al l p > 2 ε ( p − 2) / 2 | ψ ε ( x ) | p → K δ ( x ) as ε → 0 + , wher e K = A p R ∞ −∞ | y | p 6  y 2 3 + C  p dy , C > 0 and δ ( x ) denotes the D ir ac distribution at the origin. Pr o of. By Definition 1 w e ha v e to sho w that lim ε → 0 + Z ∞ −∞ ε ( p − 2) / 2 | ψ ε ( x ) | p ϕ ( x ) dx = K ϕ (0) for all ϕ ∈ D . Eviden tly , ε ( p − 2) / 2 Z ∞ −∞ | ψ ε ( x ) | p ϕ ( x ) dx = ε ( p − 2) / 2 Z ∞ −∞ A p | x | p/ 6  x 2 / 3 + ε 2 / 3 C  p ϕ ( x ) dx. By setting the substitution x = εy we obta in ε ( p − 2) / 2 Z ∞ −∞ | ψ ε ( x ) | p ϕ ( x ) dx = ε ( p − 2) / 2 Z ∞ −∞ ε p 6 A p | y | p/ 6 ε 2 p 3  y 2 / 3 + C  p ϕ ( εy ) εdy = ε ( p − 2) / 2 Z ∞ −∞ A p | y | p/ 6 ε ( p − 2) / 2  y 2 / 3 + C  p ϕ ( εy ) dy = Z ∞ −∞ A p | y | p/ 6  y 2 / 3 + C  p ϕ ( εy ) dy . (2.6) Let f ε ( y ) := A p | y | p/ 6  y 2 / 3 + C  p ϕ ( εy ). The sequence f ε ( y ) satisfies the follow ing tw o conditions: i) | f ε ( y ) | ≤ C ε A p | y | p/ 6  y 2 / 3 + C  p ∈ L 1 ( −∞ , ∞ ) if p > 2, ii) lim ε → 0 + f ε ( y ) = A p | y | p/ 6  y 2 / 3 + C  p ϕ (0). Then b y L eb esgue’s dominated con v ergence theorem w e obtain lim ε → 0 + Z ∞ −∞ f ε ( y ) dy = ϕ (0) Z ∞ −∞ A p | y | p/ 6  y 2 / 3 + C  p dy = K ϕ (0) . By D efinition 1, this implies that ε ( p − 2) / 2 | ψ ε ( x ) | p → K δ ( x ) , as ε → 0 + . 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