Integrability And Symmetries Of Some Special Evolutionary Type Equations In 1+1- And 2+1-dimensions

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2012Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2012Evrim tipi denklemler, ısı yayılımı ve dalga hareketi gibi temel fiziksel olayların modelleri olarak ortaya çıkmaktadır. Isı denklemleri, nonlineer Schrödinger (NLS) tipi dalga denklemleri, Davey-Stewartson (DS) ve genelleştirilmiş Davey-Stewartson (GDS) sistemi, Korteweg-de Vries (KdV), Burgers ve Kadomtsev-Petviashvili (KP) denklemleri bu sınıf için en sık karşılaşılan denklemler olarak anılabilir. Bahsedilen denklemler, uygulamalı matematik ve matematiksel fizik alanındaki literatürün oldukça büyük bir kısmına konu olmaktadır. Sabit katsayılı denklemlerin değişken katsayılı genelleştirmeleri, türetildikleri modellerde homojen olmayan, konum ve/veya zamana göre değişim gösteren koşullar gözönüne alındığında elde edilir. Çoğunlukla bu genelleştirme sonucunda orijinal denklemin simetri cebiri ve integre edilebilirliği gibi özellikleri aynı kalmaz. Ancak değişken katsayılar belli koşulları sağladığında genelleştirilmiş denklem de Lax çifti, Painlevé özelliği gibi integre edilebilirlik özelliklerine sahip olabilir, simetri cebirinin tümünü veya alt cebirlerini taşıyabilir. Bu koşulları elde etmedeki seçeneklerden biri, değişken katsayılı denklemi sabit katsayılı denkleme dönüştüren nokta dönüşümlerinin bulunmasıdır. Painlevé özelliği, bir denklemin tüm çözümlerinin hareketli tekil noktalar civarında tek değerli olması, yani tüm çözümlerde en fazla kutup türünden tekillik bulunmasıdır. Painlevé özelliği integre edilebilirlik için gerek veya yeter koşul değildir. Ancak literatürde karşılaşılan integre edilebilir denklemlerin büyük bir kısmı aynı zamanda bu özelliğe de sahiptir. Bu özelliği kimi yazarlar, P-integre edilebilirlik olarak da adlandırmaktadır. Painlevé özelliğinin araştırılması bazı durumlarda bilgisayar yazılımları ile dahi yapılabilmektedir. Bu kolaylık nedeniyle Painlevé analizi, denklemlerin integre edilebilirlik ve çözüm analizinde iyi bir başlangıç noktası olmaktadır. İntegre edilebilir olmayan denklemler için Painlevé seri temsillerinin sonlu terimde kesilmesinin de tam çözüm elde etmede faydalı yöntemlerden biri olduğunu not etmek gerekir. Diferansiyel denklemin çözüm uzayını değişmez bırakan dönüşüm gruplarının elde edilmesi, analitik çözüm yöntemleri oldukça kısıtlı olan doğrusal olmayan denklemlerin analizinde en etkin sistematik araçlardan biridir. Lie grupları adını alan bu dönüşüm grupları ve ilişkili simetri cebirlerinin kullanılmasıyla bir kısmi diferansiyel denklemin değişken sayısının azaltılması ve yeterince zengin bir simetri cebiri varsa adi diferansiyel denklemlere indirgenerek tam çözümlerin bulunması mümkündür. Uygulama alanı fark denklemlerine değin uzanmaktadır. Görünüşte farklı olan iki denklemin simetri cebirleri, bir dönüşümle birbirine denk ise bu denklemler de aslında birbirine dönüştürülebilir. Bu açıdan Lie teorisi, diferansiyel denklemlerin sınıflandırılmasında bir araç olarak ortaya çıkar. Diferansiyel denklemlerin simetri grupları dikkate alınarak sınıflandırılması günümüze değin aktif olarak çalışılan bir konu olmuştur. Matematiksel açıdan, genel bir denklem sınıfına ait, belirli simetri cebirlerine sahip denklem ailelerinin belirlenerek ayırt edilmesi ilginç bir problemdir. Elde edilen cebirler, bu ailelerin temsilci denklemlerinin grup-değişmez çözümleri için de yol göstermektedir. Sonsuz boyutlu simetri cebirleri söz konusu olduğunda, elde edilen denklem ailelerinin integre edilebilirliği için de ışık tutabilmektedir. Fizikçiler için, bu faydalara ek olarak, uygulamada karşılaşılan denklemlerin genel sınıflarının cebirsel özellikleri dikkate alınarak yapılan sınıflandırma çalışmalarında ulaşılan, çok sayıda durumda problemin fiziksel doğasını yansıtan sonuçlar ilginç olmaktadır. Elde edilen denklem aileleri, belirli simetri özelliklerine sahip fiziksel olayları modellemede aday olmaktadır. Bu tez çalışmasında yukarıda belirtilen çerçevedeki analiz yöntemleriyle dört adet problem ele alınmıştır. İncelenen ilk problem, değişken katsayılı kübik nonlineer Schrödinger denklemi nin (NLSD) integre edilebilirliği üzerinedir. Değişken katsayılı NLSD ve türevli terimleri içeren genelleştirmeleri için Painlevé testine ilişkin sonuçlar elde edilmiştir. Değişken katsayılı NLSD için sabit katsayılı denkleme dönüşüm formülleri elde edilmiş, elde edilen sonuçların simetri cebirleriyle ilişkisi bir örnek üzerinde ele alınmıştır. Sunulan sonuçlar arasında bazı tam çözümler de bulunmaktadır. Literatürdeki sonuçlara göre, değişken katsayılı NLS denkleminin Lie simetri cebirlerinin maksimal boyutu beştir ve maksimal cebir, denklem sabit katsayılı denkleme dönüştürülebildiğinde gerçeklenmektedir. Bunun yanında, dört boyutlu simetri cebirlerine sahip NLS denklemleri, birbirine denk olmayan beş farklı sınıfa ait olabilir. İlk problemde elde edilen sonuçlara göre, değişken katsayılı NLSD, Painlevé testini geçtiği koşulda standart NLS denklemine dönüştürülebilir ve beş boyutlu bir simetri cebirine sahiptir. Bu, denklemin integre edilebilir durumu olarak adlandırılır. Eğer bir değişken katsayılı NLS denkleminin simetri cebirinin boyutu dört ise, sabit katsayılı denkleme dönüştürülemez. Dört boyutlu simetri cebirlerinin kanonik denklemleri için, bir boyutlu cebirlerin optimal sistemi kullanılarak adi diferansiyel denklemlere indirgeme yapılabilir. Bu şekilde mümkün tüm indirgemelerin elde edilmesi ve çözümlerinin analizi, ele alınan ikinci problemdir. Bunun yanında, kanonik kısmi diferansiyel denklemlerin Painlevé serilerinin ilk terimde kesilmesi yoluyla oldukça ilginç tam çözümler elde edilmiştir. Değişken katsayılı kübik-kuintik Schrödinger denklemi (KKSD), özellikle fiber optik uygulamalarında model olarak kullanılan bir denklemdir. Simetri cebirinin maksimal boyutunun belirlenmesi ve denklem ailesinin sahip olabileceği sonlu boyutlu simetri cebirlerinin kanonik sınıflarının bulunması literatürde mevcut kübik durum ile paralellik göstermektedir. Analiz sonuçlarına göre kübik-kuintik denklem için maksimal simetri cebiri dört boyutlu, kübik durumda beş boyutlu, kuintik durumda ise altı boyutludur. Burada elde edilen sonuçlar, ikinci problemdeki analize benzer şekilde, dört boyutlu simetri cebirlerine sahip kübik-kuintik denklemler için grup-değişmez çözümlerin araştırılması imkanını verir. Son olarak ele alınan problem, değişken katsayılı KP-Burgers denkleminin sonsuz boyutlu Lie simetri cebirlerine sahip sınıflarının belirlenmesidir. Literatürde integre edilebilirliği bilinen $2+1$-boyutlu denklemler için Kac-Moody-Virasoro tipinde simetri cebirine sahip olmak tipik bir özelliktir. Ele alınan denklem ailesi için, Virasoro ve Kac-Moody tipinde simetri cebirlerinin denklemin değişmezlik cebiri olarak gerçeklenebildiği gösterilmiş, bulunan kanonik denklem aileleri için Painlevé özelliği, tam çözüm ve indirgenmiş denklemler üzerinde durulmuştur. İntegre edilebilirlik ve simetri araçlarını kullanarak, değişken katsayılı evrim tipi denklemlerden iki farklı sınıf dalga yayılımı denklemi üzerine literatürde mevcut sonuçlara katkıda bulunduğumuzu düşünmekteyiz.Evolutionary type equations tend to arise as models of basic physical phenomena such as heat conduction and wave propagation. Heat equations, wave equations of nonlinear Schrödinger (NLS) type, Davey-Stewartson and the generalized Davey-Stewartson equations, Korteweg-de Vries, Burgers and Kadomtsev-Petviashvili equations can be mentioned as the equations encountered very frequently belonging to this class. It can be said that quite a big portion of the literature on applied mathematics and mathematical physics is devoted to the analysis of these equations. Variable coefficient extensions of nonlinear evolution type equations mostly arise in cases when less idealized conditions such as inhomogeneities and variable topographies are assumed in their derivation. For example, variable coefficient Korteweg-de Vries and Kadomtsev-Petviashvili equations describe the propagation of waves in a fluid under the more realistic assumptions including non-uniformness of the depth and width, the compressibility of the fluid, the presence of vorticity and others. While these conditions lead to variable coefficient equations, all or some of the integrability properties of their standard counterparts, namely when the coefficients are set equal to constants, are in general destroyed. However, when the coefficients are appropriately related or have some specific form, the generalized equation can still be integrable as evidenced by the presence of Painlevé property, Lax pairs, symmetries and other attributes of integrability. One of the approaches to determine these relations is to transform the variable-coefficient equation to its constant-coefficient counterpart. A differential equation is said to have the Painlevé property if all of its solutions are single-valued around any movable singular point; that is, they have singularities of at most pole-type. Having the Painlevé property is not a necessary nor a sufficient condition for the integrability of an equation. However, most of the equations known to be integrable in the literature also have this property. Some authors name this property as P-integrability . Applying the Painlevé test to a differential equation can be made using a computer package in some cases, therefore it is a good starting point for an analysis of integrability and solutions of an equation. If a partial differential equation (PDE) passes this test, it can roughly be said that the equation has a good chance of being integrable. For integrable PDEs it is usually possible to construct auto-Backlund transformations which relate equations to themselves via differential substitutions and also Lax pairs, which then ensures the sufficient condition of integrability. Application of the Painlevé series expansion to nonintegrable PDEs can allow particular explicit solutions to be obtained by truncating the expansion at a finite term. This usually requires compatibility of an overdetermined PDE system. Let us mention that the method of truncated expansion has been successfully applied to many nonintegrable PDEs in constructing exact solutions. Transformation groups which leave the solution space of an equation invariant is one of the most efficient systematic tools for analysis of nonlinear equations, for which methods of finding analytical solutions are very restricted. By the use of these symmetry groups, which are called Lie groups, and the associated symmetry algebras it is possible to lower the number of independent variables of a PDE, and even reduce it to an ordinary differential equation (ODE) if the symmetry algebra is rich enough. Application of the theory is also possible to difference equations. If two equations have symmetry algebras which are equivalent by a point transformation, then the equations are also equivalent to each other by this transformation. From this point of view the theory of Lie appears as a tool for classifying differential equations. Classifying evolutionary type PDEs has long been an active research area both for mathematicians and physicists. There are various approaches for the classification of PDEs with respect to point or higher symmetries under the action of diffeomorphisms. So far classifications of considerably large classes of equations in 1+1-dimensions under the low-dimensional abstract finite-dimensional Lie algebras have been done. On the other hand, it is well known that a number of physically significant integrable partial differential equations in 2+1-dimensions typically have infinite-dimensional symmetry algebras with a specific Kac-Moody-Virasoro (KMV) structure. Among them, KP equation, modified KP, cylindrical KP equation, all equations of the KP hierarchy, Davey-Stewartson system and three-wave resonant interaction equations are the most prominent ones. For the classification of symmetry algebras of an equation having arbitrary function(s) one first needs to find the equivalence transformations of the equation, which means finding the transformations those leave the equation form-invariant (keeping the differential structure the same but changing the arbitrary elements). This can be obtained by trying a direct transformation or by the infinitesimal method. The symmetry generator is found by the algorithm due to Lie, which is a calculation based on solutions of a overdetermined system of linear PDEs. The infinitesimal is determined up to some unknown functions, and these unknowns satisfy a set of PDEs with the arbitrary functions appearing in the equation. Then equivalence transformations are used to obtain non-equivalent forms of the symmetry generator, possibly in a simpler configuration. This is called the linearization of the vector field and these non-equivalent forms are generators for canonical one-dimensional Lie algebras which can be admitted by the equation. The remaining task is to check which of the two and higher dimensional Lie algebras can be realized as the invariance algebra of the equation. This is faciliated by the use of well-known structural results on the classification of low-dimensional Lie algebras. The procedure is carried on till the arbitrary function appearing in the equation is completely determined. The result is a list of representative equations with canonical invariance algebras, classified up to equivalence transformations. Results concerning symmetry classification of PDEs are drawing interests of both mathematicians and physicists. From the mathematical point of view, classification itself is interesting for the reason that we can distinguish between subclasses of a family of equations regarding their symmetry properties. It serves as an initial guide for the solutions of the equations which arise as representatives of the subclasses. In the case of infinite-dimensional symmetries, classification with respect to Lie symmetries can turn out to be a way of detecting integrable systems. On the other hand, it is known that fundamental equations of physics have rich symmetry structures. A classification of their generalizations exploiting the possible different symmetry properties allow physicists to choose equations for modeling real phenomena with certain physical properties. In this thesis we deal with four problems using methods mentioned above. We first analyze the integrability of a variable coefficient nonlinear Schrödinger (VCNLS) equation. We obtain the results concerning the Painlevé tests of the VCNLS equation and its generalizations including some derivative terms. We present the explicit transformation formulae for VCNLS to the standard NLS equation and consider the relations with the NLS symmetry algebra on a specific equation. We complete this analysis by giving some exact solutions. According to the results already available in the literature, maximal dimension of Lie algebra of a VCNLS equation is five and it is achieved when the equation is transformable to the standard NLS. On the other hand, there are five possible four-dimensional non-equivalent Lie algebra that can be admitted. According to the results obtained in the first problem, if a VCNLS equation passes the Painlevé test it can be transformed to the standard NLS, therefore it has a five-dimensional Lie algebra. This is the integrable case. Transformation to the standard NLS is not possible in the four-dimensional case. The canonical variable coefficient PDEs having four-dimensional Lie algebras can be reduced to ODEs. For a systematic reduction we need an optimal system of one-dimensional subalgebras. Therefore we determine all such reductions in this non-integrable case and work on their solutions. Furthermore, we obtained interesting exact solutions of the canonical PDEs by truncating their Painlevé series at the first term. Variable coefficient cubic-quintic nonlinear Schrödinger equation is a typical model equation of fiber optics. Determination of the maximal dimension of the symmetry algebra and the canonical classes of the finite-dimensional symmetry algebras together with the corresponding equations follows by similar lines with the cubic case existing in the literature. According to our results, maximal dimension of the symmetry algebra is four in the genuine cubic-quintic case, five in the cubic case, and it is six in the quintic case. Results obtained in this part can be used to obtain the reduced equations for cubic-quintic equations having four-dimensional symmetry algebras and hence to find the group-invariant solutions, as in the cubic case of the second problem. The last problem we worked on is the classification of infinite-dimensional Lie symmetry algebras of variable coefficient KP-Burgers equations. As we mentioned, integrable equations in 2+1-dimensions typically have symmetry algebras of Kac-Moody-Virasoro type. It is shown that Virasoro and Kac-Moody type Lie algebras can be realized as invariance algebras of specific subclasses of equations and some results on Painlevé property, exact solution and reductions of the canonical equations are presented. We believe that we have made some contributions to the results existing in the literature on two classes of variable coefficient evolutionary wave propagation equations using the tools of integrability and symmetries. Evolutionary type equations tend to arise as models of basic physical phenomena such as heat conduction and wave propagation. Heat equations, wave equations of nonlinear Schrödinger (NLS) type, Davey-Stewartson and the generalized Davey-Stewartson equations, Korteweg-de Vries, Burgers and Kadomtsev-Petviashvili equations can be mentioned as the equations encountered very frequently belonging to this class. It can be said that quite a big portion of the literature on applied mathematics and mathematical physics is devoted to the analysis of these equations. Variable coefficient extensions of nonlinear evolution type equations mostly arise in cases when less idealized conditions such as inhomogeneities and variable topographies are assumed in their derivation. For example, variable coefficient Korteweg-de Vries and Kadomtsev-Petviashvili equations describe the propagation of waves in a fluid under the more realistic assumptions including non-uniformness of the depth and width, the compressibility of the fluid, the presence of vorticity and others. While these conditions lead to variable coefficient equations, all or some of the integrability properties of their standard counterparts, namely when the coefficients are set equal to constants, are in general destroyed. However, when the coefficients are appropriately related or have some specific form, the generalized equation can still be integrable as evidenced by the presence of Painlev\ e property, Lax pairs, symmetries and other attributes of integrability. One of the approaches to determine these relations is to transform the variable-coefficient equation to its constant-coefficient counterpart. A differential equation is said to have the Painlev\ e property if all of its solutions are single-valued around any movable singular point; that is, they have singularities of at most pole-type. Having the Painlev\ e property is not a necessary nor a sufficient condition for the integrability of an equation. However, most of the equations known to be integrable in the literature also have this property. Some authors name this property as P-integrability . Applying the Painlev\ e test to a differential equation can be made using a computer package in some cases, therefore it is a good starting point for an analysis of integrability and solutions of an equation. If a partial differential equation (PDE) passes this test, it can roughly be said that the equation has a good chance of being integrable. For integrable PDEs it is usually possible to construct auto-B\ acklund transformations which relate equations to themselves via differential substitutions and also Lax pairs, which then ensures the sufficient condition of integrability. Application of the Painlev\ e series expansion to nonintegrable PDEs can allow particular explicit solutions to be obtained by truncating the expansion at a finite term. This usually requires compatibility of an overdetermined PDE system. Let us mention that the method of truncated expansion has been successfully applied to many nonintegrable PDEs in constructing exact solutions. Transformation groups which leave the solution space of an equation invariant is one of the most efficient systematic tools for analysis of nonlinear equations, for which methods of finding analytical solutions are very restricted. By the use of these symmetry groups, which are called Lie groups, and the associated symmetry algebras it is possible to lower the number of independent variables of a PDE, and even reduce it to an ordinary differential equation (ODE) if the symmetry algebra is rich enough. Application of the theory is also possible to difference equations. If two equations have symmetry algebras which are equivalent by a point transformation, then the equations are also equivalent to each other by this transformation. From this point of view the theory of Lie appears as a tool for classifying differential equations. Classifying evolutionary type PDEs has long been an active research area both for mathematicians and physicists. There are various approaches for the classification of PDEs with respect to point or higher symmetries under the action of diffeomorphisms. So far classifications of considerably large classes of equations in (1+1)-dimensions under the low-dimensional abstract finite-dimensional Lie algebras have been done. On the other hand, it is well known that a number of physically sig