New analytic solutions of nonlinear-partial differential equations of mathematical physics via isovector, equivalence transformations and direct methods.

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CERTIFICATE

This is to certify that the thesis entitled :

`NEW ANALYTIC SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS

VIA ISOVECTOR, EQUIVALENCE TRANSFORMATIONS AND DIRECT METHODS'

which is being submitted by Ms. Lipika Bhattacharya, Research Scholar, Mathematics Department to the Indian Institute of Technology, Delhi, for the award of the DEGREE OF DOCTOR OF PHILOSOPHY in MATHEMATICS, is a record of bonafide research work carried out by her under my guidance and supervision and has fulfilled all the requirements for the submission of this thesis.

The results contained in this thesis have not been submitted in part or full, to any other University or Institute for the award of any degree or diploma.

O.P. BHUTANI Professor,

Department of Mathematics, Indian Institute of Technology, Delhi.

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ACKNOWLEDGMENTS

It gives me an immense pleasure in expressing my sincere regards and gratitude to Professor O.P. Bhutani, F,N.A., Department of Mathematics, Indian Institute of Technology, Delhi for his helpful guidance and constant encouragement throughout the preparation of this thesis. But for his keen interest and skilful attitude this work could not have been accomplished.

My thanks are also due to the authorities of the Indian Institute of Technology, Delhi and Head of the Department of Mathematics for providing me necessary facilities for carrying out tnis work. Financial assistance in the form of fellowship for the full period of research, from the Council of Scientific and Industrial Research, India is gratefully acknowledged.

I also take this opportunity to thank Dr. K. Vijayakumar for many stimulating discussions and Ms. Soma Gupta and Ms. Kavitha Sharma for their generous help during the prepa-ation of this thesis.

I shall be failing in my duty if I do not express my gratitude to my mother Mrs. Bhawani Bhattacharjee and husband Mr. A. Roy Chowdhury for the sacrifices they made in seeing the work through.

(Lipika Bhattacharya)

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ABSTRACT

The study being presented as a thesis entitled New Analytical Solutions Of Nonlinear Partial Differential Equations of Mathematical Physics via Isovector, Equivalence Transformations and Direct Methods" highlights the usefulness of the analytical methods of solution mentioned in the title by way of application to a large number of problems from the realm of theoretical physics. The isovector and the equivalence transformation approaches have direct links with the Lie's theory of continuous groups of transformations. Precisely speaking, the isovector fields and the equivalence generators, in the isovector approach and the equivalence transformations method respectively, play the same role as that of the infinitesimal operator in Lie's theory. Using the similarity transformations obtained from these one-parameter groups and which essentially reduce the number of independent variables by one, the nonlinear partial differential equations (nlpdes) in two independent variables are reduced to nonlinear ordinary differential equations (nlpdes), the solutions of which eventually yield solutions of the nlpdes. Beside these two group theoretic methods, some other methods, which dtmot evoke group invariance, are also used. These direct methods are WTC technique or analysis of the weak Painleve property for nlpdes, Clarkson and Kruskal's direct method and travelling wave solutions technique by Jeffrey and Xu. In the next few paragraphs we briefly mention the material presented in the seven chapters of the thesis.

Chapter 1 gives a brief description of the methods used in handling equations of the remaining chapters and the survey of some related works. In Chapter II, some

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Klein-Gordon and Liouville type equations in n-Euclidean dimensions are taken up for similarity reduction. The resulting nlodes are solved for some special forms of the arbitrary functions involved and the results are tabulated.

A Poisson type equation in n-Euclidean dimensions in which the inhomogeneous part is an arbitrary function of the dependent and independent variables as well as the first order partial derivatives of the dependent variable, is studied in Chapter HI by the equivalence transformation and isovegctor approaches and the results are compared. Isogroups are calculated for the Harry Dym equation in Chapter IV and ten group theoretic reductions are tabulated, some of which are solved to get some implicit and some explicit solutions.

In Part I of Chapter V, an integro-differential equation derived by Hirota and Satsuma by Backlund transformation of the Boussinesq equation is taken up and some exact solutions are reported. In the second part, the third order Boltzmann equation is studied via the isovector and WTC techniques, to get exact solutions. It is observed .that some of the similarity transformations work for the (p+ 1) th order equation as well, from which results for the Krook

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Wu model (p =1) are deduced.

In Chai er VI the small disturbance potential flow equation of nonequilibrium transonic gasdynamics is considered through the isovector approach and the direct method due to Clarkson and Kruskal, which again transforms the nlpde to nlode but does not require invariance under a group of transformation.

Beside giving exact solutions of the equation under consideration, solvability and nilpotency of the Lie algebra of the transformation groups yielded by the isovectors are examined.

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Unlike Chapters II - VI wherein we utilize group theoretic methods for determining the exact solutions of nlpdes, in this chapter, the technique given by Jeffrey and Xu to find travelling wave solutions, is applied to a class of conservative nonlinear evolution equations that embeds nonlinearity and dispersion. Results for some physically interesting particular cases such as the generalized Kuramoto-Sivashinski equation, the Korsunsky equation in a special case, a generalized Boussinesq equation, a modified Boussinesq equation, a nonlinear atomic chain model equation etc. are deduced from the general solution.

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2.1 Introduction 29

2.2 Transformation of Eqs (2.1.1) - (2.1.2) to Symmetric Variables 31 2.3 Generalized Klein-Gordon Equation and the Isovector Approach 32 2.4 Similarity Transformations for Eq (2.3.10) 40 2.5 Similarity Solutions of Eq (2.4.26) • 46 2.6 Liouville Type Equation in n-Euclidean Dimensions 51 2.7 Similarity Transformations and Similarity Reductions of Eq(2.6.3) 53

2.8 Concluding Remarks 55

CHAPTER III ON THE EQUIVALENCE TRANSFORMATIONS AND ISOVECTOR APPROACH APPLIED TO

cp + f(xl , x2,..., x n , = 0 56-86

3.1 Introduction 57

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3.2 The Equivalence Generators 58

3.3 Applications of Theorem 3.1 64

3.4 Examples 71

3.5 The Isovector for Eq (3.2.1) 82

CHAPTER IV ISOGROUPS AND EXACT SOLUTIONS OF THE

HARRY-DYM EQUATION 87-106

4.1 Introduction 88

4.2 The Isovector for the HD Equation 89

4.3 The Orbital Equations and Similarity Transformations 93 4.4 Reduction to odes and Exact Solutions 98

CHAPTER V ISOGROUPS AND EXACT SOLUTIONS FOR THE HIROTA-SATSUMA INTEGRO-DIFFERENTIAL EQUATION AND HIGHER ORDER BOLTZMANN

EQUATION AND ITS ITEGRABILITY 107-145

PART I THE HIROTA-SATSUMA INTEGRO-DIFFERENTIAL EQUATION

5.1 Introduction 108

5.2 The Isovector Field and Similarity Reductions 109 5.3 Similarity Solutions of the Hirota-Satsuma Equation 116

PART 11 THE HIGHER ORDER BOLTZMANN EQUATION

5.4 Introduction 127

5.5 Isovector Approach and Similarity Reductions 130 5.6 Similarity Solutions of the Boltzmann Equation 138

5.7 Painleve Analysis of Eq (5.5.1) 139

New analytic solutions of nonlinear-partial differential equations of mathematical physics via isovector, equivalence transformations and direct methods.

CERTIFICATE

ACKNOWLEDGMENTS

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CONTENTS

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