Nonlinear Evolution Equations and Inverse Scattering. Abd-Alrahman Mahmoud Shehada Jabr

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1 Nonlinear Evolution Equations and Inverse Scattering By Abd-Alrahman Mahmoud Shehada Jabr Supervisor Prof.Dr. Gharib Mousa Gharib This Thesis was Submitted in Partial Fulfillment of the Requirements for the Master s Degree of Science in Mathematics Faculty of Graduate Studies Zarqa University Zarqa - Jordan November, 2016

2 ii جامعة الزرقاء نموذج التفويض أنا عبدالرحمن محمود شحادة جبر أفوض جامعة الزرقاء بتزويد نسخ من رسالتي للمكتبات أو المؤسسات أو الهيئات أو األشخاص عند طلبهم حسب التعليمات النافذة في الجامعة. التوقيع: التاريخ: Zarqa University Authorization Form I, Abd-Alrahman Mahmoud Shehadah Jabr, authorize Zarqa University to supply copies of my Thesis to libraries or establishments or individuals on request, according to the University regulations. Signature: Date:

3 iii COMMITTEE DECISION This Thesis (Nonlinear Evolution Equations and Inverse Scattering) was successfully defended and approved on Examination Committee Signature Dr. Gharib Mousa Gharib (Supervisor) Prof. of Mathematics Dr. Khaled Khalil Jaber (Member) Assoc. Prof. of Mathematics Dr. Naser Hassan Al-Zomot (Member) Assoc. Prof. of Mathematics Dr. Ali Mahmud Ateiwi (Member) Prof. of Mathematics

4 iv ACKNOWLEDGEMENT I am very grateful to many people for the completion of this project, the first of them and above all the Almighty Allah for the countless blessing, after that I especially thank my supervisor, Head of the Department of Mathematics, Prof.Dr. Gharib Musa Gharib for his skilled guidance and unwavering support throughout this project. I am most grateful to the Department of Mathematics at Zarqa University and all faculty members. I also extend my thanks to my family in general and specially my mother and father for their Doa a without which, this thesis could not have been accomplished. Finally I would like to thank my wife for her support, cooperation and patience.

5 v Authorization Form Committee Decision Acknowledgement Table of Contents Abstract Table of Contents ii iii iv vi vii Introduction 1 Chapter One: Background 1.1 Partial differential equation... (3) 1.2 Discover of IST. (4) 1.3 Soliton solution......(6) 1.4 Nonlinear evolution equation (7) Chapter Two: Inverse Scattering Transform 2.1 The components of the IST method...(9) 2.2 Linear Example of the Inverse Scattering Transform.... (10) 2.3 The IST for the KdV equation... (14) 2.4 One soliton solution for thkdv equation (23) 2.5 two soliton solution for thkdv equation (27)

6 vi Chapter Three: The Family of Equations 3.1 The AKNS system and IST for the family of equation (34) 3.2 Example1 The Sine-Gordon equation..... (40) 3.3 Example1 The Sinh-Gordon equation.....(42) 3.4. Conclusions (44) 3.5. References.....(45) Abstract(in Arabic) (48)

7 vii ABSTRACT The main original contribution of this thesis is the development of the inverse scattering transform (IST) method for nonlinear evolution equations. The equation we solve a general form of KdV equation, which is known as fully integrable model. In chapter one, we introduce historical background about PDE and some kind of solutions for PDE, also we talk about a nonlinear evolution equation which is very important in so many phenomena of waves, solitary waves, and soliton solution. In chapter 2, we introduce IST method for solving these equations and we write classifications for the integrable models of equations which are solvable by our main method IST and solve a linear example and summarize this method for solving KdV equation. In chapter 3, we talk about the family of equations and introduce the AKNS system for this family and give some examples for the family of equations like Sine-Gordon equation, Sinh-Gordon equation and Liovell s equation.

8 1 INTRODUCTION Differential equations are important because they explain relationships containing rate of change. Such relationships form the basis for studying phenomena in the sciences, engineering, economics, medicine and most human knowledge. Since the study of differential equations began, almost, problem had been solved only for linear equations those have at most one power of the unknown variables or its derivatives appeared in each term of the equation. The main reason for this is that linear equations can be solved by the superposition principle. That is, since differentiation is a linear operation, any linear combination of solutions of a linear equation is again a solution of the equation. Hence, the methods of Fourier series and Fourier transforms were developed to find the general solutions in terms of sums or integrals of certain basic solutions. However, in the last 40 years, the mathematicians focus in the study of nonlinear equations and in methods for their exact solution. A nonlinear partial differential equation relates to study a number of different physical systems like water waves, an harmonic lattices, plasma physics and elastic rods. It describes the long time evolution of amplitude dispersive waves. The field of nonlinear waves and integrable systems has a long and difficult history, see [1,2,3,5]. It began in the nineteenth century with the pioneering work of Stokes, Boussinesq and Korteweg and de Vries [1], all of whom studied the dynamics of fluids. Many of the models that were derived were nonlinear partial differential equations, and without computational assistance very little could be said at the time about their solutions. In the second half of the twentieth century, some of these models were then rediscovered by researchers such as Kruskal and Zabusky in 1965, they used a combination of mathematical analysis and computational power to explain the Fermi-Pasta-Ulam (FPU) paradox. There was an observation of recurring states of energy (rather than the expected dissipation) within a onedimensional string of connected masses with nonlinear spring interactions[1-5]. From studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed[1]. The recent literature contains many extensions for

9 2 nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are introduced. In this thesis, we will give a historical account of some of these equations, specifically concerning the method of inverse scattering transform (IST). In this thesis, we consider the one-dimensional problems solvable by the IST method. The IST is also connected to Backlound transformations, which relate solutions of partial differential equations to solutions of other equations. Also more recently were done for higher dimensional problems. The IST is one of the most important developments in mathematical physics, which is applied to solve many linear partial differential equations. The name inverse scattering method comes from the idea of recovering the time evolution of a potential from the time evolution of its scattering data [3,5,8]. Now before we study this method to find exact solution for nonlinear evolution equation we will introduce a classification of partial differential equation.

10 3 1.1: Partial differential equation Chapter 1 Background Recall that the words differential and equations clearly indicate solving some kind of equation involving derivatives. We have seen the classification of partial differential equations into linear, quasi linear, semi linear, homogeneous, non-homogeneous and nonlinear[1,4,7,8] Definition ( partial differential equation ). A partial differential equation (PDE) is an equation that has an unknown function depending on at least two variables, contains some partial derivatives of the unknown function [5] Definition An evolution equation is a partial differential equation for an unknown function of the form where is an expression involving only u and its derivatives with respect to. If this expression is nonlinear, equation (1.1) is called a nonlinear evolution equation (NLEE) [1] Definition. A solution to PDE is, generally speaking, any function (in the independent variables) that satisfies the PDE. However, from this family of functions one may be uniquely selected by imposing adequate initial and/or boundary conditions. A PDE with initial and boundary conditions constitutes the so-called initial-boundary-value problem (IBVP). Such problems are mathematical models of most physical phenomena [3,4] Definition. (A well-posed problem) An initial-boundary-value problem is well-posed if: it has a unique solution, and the solution varies continuously with the given inhomogeneous data, that is, small changes in the data should cause only small changes in the solution [11].

11 4 1.2: Discover of Inverse Scattering Transform : In 1965 Zabusky and Kruskal explained the FPU problem in terms of solitary wave solutions to the Korteweg-de Vries (KdV) equation. In their numerical analysis they observed solitarywave pulses, these pulses are called solitons because of their particle-like behavior, and noted that such pulses interact with each other nonlinearly but come out of interactions unaffected in size or shape except for some phase shifts [3]. At that time no one knew how to solve the IVP for the KdV equation, except numerically. In 1967 Gardner, Greene, Kruskal, and Miura presented a method, now known as the IST to solve that IVP, assuming that the initial profile decays to zero sufficiently rapidly as. They showed that the integrable nonlinear partial differential equation (NPDE), i.e. the KdV equation[1-6], is related to a linear ordinary differential equation (LODE), which is the one-dimensional Schraodinger equation, ( ) where is the eigenvalue associate the eigenfunction and that the solution to (1.2) can be recovered from the initial profile. They also explained that the soliton solutions to the KdV equation correspond to a zero reflection coefficient in the associated scattering data. Note that the variables x and t in (1.2) are spatial and time variable respectively and they throughout denote the partial derivatives with respect to those variables [1,2,3]. In 1972 Zakharov and Shabat showed [20], that the IST method is applicable also to the IVP for the nonlinear Schraodinger (NLS) equation,

12 5 where i denotes the imaginary number equation is the first-order linear system,. They proved that the associated linear differential { where λ is the spectral parameter and denotes complex conjugation for system (1.5) is now known as the Zakharov-Shabat system [2].. The After that, again in 1972 Wadati showed that the IVP for the modified Korteweg-de Vries (mkdv) equation see [1,3,4], can be solved by the inverse scattering problem for the linear system, { Next, in 1973 Ablowitz, Kaup, Newell, and Segur showed that [1,2,4,5] the IVP for the Sine- Gordon equation, can be solved in the same way by the inverse scattering problem associated with the linear system, { Since then, many other NPDEs have been discovered to be solvable by the IST method.

13 6 1.3: Soliton solution The first observation of a soliton was made in (1834) by the Scottish engineer John Scott Russell at the Union Canal between Edinburgh and Glasgow [3,4,5]. Russell reported his observation to the British Association of the Advancement of Science in September (1844), but he did not succeed in convincing the scientific community. The Dutch mathematician Korteweg and de Vries published [14] a paper in (1895) based on de Vries Ph.D. dissertation, in which surface waves in shallow water, narrow canals were modeled by what is now known as the KdV equation. The importance of this paper was not understood until (1965), even though it contained as a special solution what is now known as the one-soliton solution Definition. A Soliton is the part of a solution to an integrable nonlinear partial differential equation due to a pole of the transmission coefficient ins the upper half complex plane. The soliton solution that satisfies nonlinear equations usually has the following properties [3]: 1) solitons are waves dying out at infinity and they have profiles, which are unaltered after colliding with other solitons. 2) they evolve with time and therefore they satisfy certain evolution equations. 3) they are stable solutions and they do not disperse apart when they collide with other solitons. 4) in collision with other solitons, there is nonlinear interaction. However, they retain their original shape shortly afterwards, only slightly displased. 5) soliton with larger amplitude pulse moves faster and is narrower in width than the smaller soliton.

14 7 1.4: Nonlinear evolution equation. A nonlinear evolution equation focuses on the study of a number of different physical systems like water waves, plasma physics, harmonic lattices, and elastic rods KdV equation The equation that Kruskal and Zabusky[1], found a model as where δ is a parameter, which is a NLEE in two independent variables and is known as the KdV equation. This is in fact the equation found by Korteweg and de Vries when they studyied shallow water waves. These waves are localised waves, unlike linear waves, interact elastically with neighbouring waves, and have a relationship between amplitude and speed,this particlelike nature led is called solitons. An example of a two-soliton solution of the KdV equation. [ ] / whose graph as a function of and is shown in Figure 1[1,13]. FIGURE 1a. 3-D Two-soliton of the KdV equation

15 8 FIGURE 1b. Two-soliton the KdV equation Nonlinear Schrödinger equation and sine-gordon equation After the KdV system was found there was a great number of advancements as researchers found ways of applying this new method of solution to a number of important systems. One of the important applications was in (1971) from Zakharov and Shabat [5], who used ideas of Lax to solve the initial-value problem for the nonlinear Schrödinger equation for solutions with decaying boundary conditions. Like the case of the KdV equation, the researchers found a soliton solutions and an infinite number of conservation laws. In (1972) Wadati solved the mkdv (2.4), and in (1973) Ablowitz, Kaup, Newell and Segur (AKNS) [1-5] applied this method to solve the sine-gordon equation, for which they found soliton solutions, breather solutions and an infinite number of conservation laws. The wide applicability of this method then led AKNS [4] [5] to show that equations (1.10), (1.11), (1.12) and (1.13) are in fact all related to a single matrix eigen value problem, from which many physically important systems are obtainable. Noting the similarity between this method of solving partial differential equations and the method of Fourier transform, they also labelled it the IST.

16 9 Chapter 2 Inverse Scattering Transform The Inverse Scattering Transform (IST) is a method of finding solutions to linear and integrable nonlinear partial differential equations. In this chapter, we look at some definitions and the mathematical structure of the IST and its application to solve the heat equation and nonlinear KdV equation. 2.1: The components of the IST method : Direct scattering problem : The problem of determining the scattering data corresponding to a given potential in a differential equation Inverse scattering problem : The problem of determining the potential that corresponds to a given set of scattering data in a differential equation Lax method : A method introduced by Lax in 1968 that determines the integrable nonlinear partial differential equation associated with a given linear ordinary differential equation so that the initial value problem for that nonlinear partial differential equation can be solved with the help of an inverse scattering transform Scattering data : The scattering data associated with a linear ordinary differential equation usually consists of a reflection coefficient which is a function of the spectral parameter, a finite number of constants that correspond to the poles of the transmission coefficient in the upper half of the complex plane, and the bound-state norming constants, whose number for each bound-state pole is the same as the order of that pole. It is desirable that the potential in the linear ordinary differential equation is uniquely determined by the corresponding scattering data and vice versa.

17 Time evolution of the scattering data : The evolvement of the scattering data from its initial value at to its value at a later time. In this section we well introduced an important PDE and applied the IST to find exact solution of them. The Heat equation, and the KdV equation, The heat equation and the KdV equation are both partial differential equations in one spatial dimension which is and one temporal dimension which is, however one fundamental difference between these two equations is that (2.1) is linear in, but (2.2) is not. Despite this difference, they satisfy the compatibility condition for an associated linear system, which is the basis for the IST. The IST for the heat equation is simple, because of the fact that the equation itself is linear. 2.2: Solution of the heat equation by Inverse Scattering Transform Consider the heat equation (2.1) with an initial condition satisfying this can be solved by separation of variables or Fourier transform to give the general solution as,

18 11 where is the Fourier transform of the initial condition. Alternatively one can generate [5] the Lax pair, which is a pair of matrices or operators satisfy Lax equation whose consistency gives. Equation (2.5a) arising the forward scattering problem, we solve it for along the initial time. The second Lax equation (2.5b) is then used to evaluate from which may then be constructed from (2.5a). To solve (2.5a) for we start with the boundary condition, but satisfies Faddeev condition then, then we have then the solution of is for some constant. We then introduce the unique Jost solutions boundary conditions, and φ of (2.5a) with the φ in terms of the initial condition we have, φ by (2.3) these integrals will exist in the half-planes and

19 12 respectively. Furthermore since these are both solutions of (2.5a), it follows that φ for some function B which is independent of, and we can write B as, That is the direct scattering procedure. These expressions are not enough to determine the time dependent Jost solutions φ and since this would involve the knowledge of, which is the same thing needed to find. Now consider the time dependence of the function, defined by, φ Substitute this equation into equation (2.5b) and taking the limit we have since that and as for all, then we have φ

20 13 and so provided that is bounded and that it follows that φ ( ) ( ) So we have φ ( ) Thus the solution of the heat equation becomes ( ) This is clearly the result where we recognize condition as the Fourier transform of the initial 2.3: Inverse Scattering Transform for the KdV equation The steps for solving the KdV equation by the IST are more complicated than the heat equation. This method was first discovered by Gardner, Greene, Kruskal and Miura [1-5] in the

21 s. Here we give an important mathematical features of these steps, and for a detailed analysis on the forward scattering problem considered here see [3], while the inverse problem is treated in [6]. All mathematicians agree with that the Lax pair for the KdV equation is given by where is constant depends on the normalization of and is the spectrum parameter. The direct problem at time, given. The spectrum of these equations consists of a finite number of discrete eigenvalues,,, for and a continuum,, for < 0 [1,2,3]. Suppose that, then the Lax pair for KdV equation becomes where, provided that since and are independent of time then,, since the consistency of the system of to satisfy the KdV equation, then first equation (2.22a) defines the forward scattering problem, while the second (2.22b) defines the time evolution of the scattering problem. Assume that there exists some real-valued initial condition which satisfies

22 15 which is associate to integrability condition, this initial condition is called Faddeev condition [3]. The forward scattering problem is used to determine to This equation is a second-order equation which is linear for, and it is called a Sturm- Louiville equation. Now to solve this equation we first consider the limit, in which equation (2.22a) becomes and the solution for becomes for some constant A and B. Then we introduce two unique Jost solutions by the boundary conditions 2 φ φ 2 since equation (2.27) is invariant under the transformation, it follows that φ φ and. Furthermore since the solution of equation (2.27) is two-dimensional space we may write φ φ

23 16 where the two functions, and are independent of, which satisfy where, are the complex conjugate of respectively[1,3]. More of that the functions and satisfy the condition and this satisfies Now consider the Wronskian equation φ φ φφ φ φ on. Note that this implies that A does not vanish on. We now state several properties about the Jost solutions and the functions A and B. Proposition The Jost solutions and the spectral functions A and B have the following analyticity properties: φ and exist and they are continuous at the eigenvalue in the closed half-plane Im( ) 0, and are analytic in in the open half-plane. φ and exist and they are continuous at the eigenvalue in the closed half-plane, and are analytic in in the open half-plane. exists and is continuous in in the closed half-plane, and is analytic in in the open half-plane. B(0; ) exists and is continuous in on.

24 17 To prove that see [1,3,4,5]. Proposition The Jost solutions and the spectral functions A and B have the following asymptotic properties: φ ( ) { ( ) φ ( ) { ( ) ( ) ( ) Theorem The function has a finite number of zeroes in the open halfplane, and does not vanish on. Moreover all of these zeroes are simple and lie on the imaginary axis. At each we have φ = for some constant to prove see [1]. Now, we study the evolution of the normalization constant for eigenfunction φ By definition, the normalization constant is defined as : { φ }

25 18 In order to study the evolution of the normalization constant, we differentiate both sides with respect to the time and use the second Lax equation φ φ φ φφ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ [ φ φ φ φ ] the solution for the above equation gives us the equation of evolution for the normalizing constant After that the transmission and reflection coefficients since λ gives rise to unbound state, then defines the following φ

26 19 0 φ φ φ φ 1 ( φ ) 0 ( ) 1 whence, in order to eliminate these exponential functions, we must equate the coefficients of and. Simply, we may first rewrite the above expression as since and are linearly independent then in order to vanish the second term, the coefficient must be equal to zero similarly φ

27 20 [ φ φ φ ] 0 1 and the relationship between the reflection and transmission coefficient satisfies, Finally, we can state the following which summarizes our analysis for the scattering data. If are given as above then { where and are obtained from the initial data for the KdV equation The n zeroes of A form a set of discrete eigenvalues, which we will show are in fact associated with the N solitons which exist within the solution of the KdV equation. The inverse transform involves using A and B to reconstruct the time dependent solution As for the heat equation this is done from the setting of a Riemann-Hilbert problem. Importantly, since the boundary conditions for u are independent of time, the analyticity and asymptotic results of Propositions and and Theorem continue to hold for all. By these results the relation

28 21 φ. / defines a condition between the two functions and along the real -axis, with known boundary condition. For Im( ) > 0, the solution of this is given by the singular integral. /. / This is a closed-form singular integral equation for known from., where all time dependence is One can also isolate the dependence of on the spectral parameter by taking the form now to find K, inserting expression (2.53) into (2.35), thereby obtaining a Goursat problem for K. It can be shown that the solution of this Goursat problem exists and is unique [7]. The motivation for this option comes from the fact that one of the boundary conditions for K gives a simple relation between it and the solution to the KdV equation: [ ] In order to obtain an equation for we substitute equation (2.53) into the singular integral equation for. Since K is related to the Fourier transform of, by taking the inverse Fourier transform we obtain the following Gel fand-levitan equation, valid for :

29 22 where the quantity L is given by. /. / furthermore by considering the Wronskian one can also show that ( ) which follows from the fact that the Jost solutions are real whenever the knowledge of. Thus from { { }} one can construct the full time-dependent solution through the linear Volterra-type integral equation (2.52). The quantity is known as the transmission coefficient and is known as the reflection coefficient. Initial conditions for which the reflection coefficient is identically zero on are known as reflectionless potentials. Now, we study two examples of soliton solution.

30 23 2.4: One soliton solution Consider the initial value problem With the initial condition, then we have, where n is the number of soliton, let where corresponds to the vertical displacement of the water from the equilibrium at the location at time. Replacing by amounts to replacing by in (2.58). Also, by scaling and, i.e. by multiplying them with some positive constants, it is possible to change the constants in front of each of the three terms on the left-hand side of (2.58). Solution First we find the eigenvalues and eigenfunction ( ) where Let this transformation map from for to [-1,1] for s. So we have Therefore, the Sturm-Louiville problem becomes

31 24 [ ] [ ] Comparing this equation with the associated Legender equation [ ] 0 1 we get It is clear that this is the only one eigenvalue and the corresponding eigenfunction can be found from the Lgender polynomials, where n=0,1,2,,n, and let N=1 then we have,

32 25 So the eigenvalue and the eigenfunction Second we Normalize the eigenfunction Therefore, the normalized eigenfunction is Third determination of and by using the definition Therefore, the evolution equation for the normalization constant is given by After that, we determine integration kernel ( )

33 26 Then we write Gel fand and Levitan equation We solve the above equation by the separation of variables by assuming Substituting (2.72) into (2.71), we get Comparing coefficients of in equation (2.73) gives So we have Multiply by, so we get so the kernel becomes

34 27 and the potential can be solved as. / ( ) Thus the solution becomes FIGURE 1. One-soliton solutions of the KdV equation. 2.5: Two soliton solution Consider the initial value problem

35 28 with the initial condition let Solution First we find the eigenvalues and eigenfunctions ( ) where Let this transformation map from for x to [-1,1] for s. So we have Therefore, the Sturm-Louiville problem becomes [ ] [ ] Comparing this equation with the generalized Legender equation [ ] 0 1 we get These are two eigenvalues for the Sturm-Louiville problem and in order to find the corresponding eigenfunctions, we use the associated Legender polynomials with,

36 29. / and ( ) So the eigenvalues and the eigenfunctions After that, normalization of the eigenfunction Therefore, the normalized eigenfunctions are Then, we determine and

37 30 by using the definition Therefore, the evolution equation is given by After that, determination of integration kernel ( )

38 31 Then we write Gel fand and Levitan equation [ ] We solve the above equation by separation of variables by assuming Then by substitution in the above we have [ ][ ] Comparing coefficients of and equation (2.93) gives [ ] The second equation is [ ] [ ] [ ][ ]

39 32 [ ] [ ] From ( 2.95a ) and ( 2.96b ), we have the following system: [ ] [ ] { [ ] [ ] The above system of algebraic equations can be solved by using Cramer s rule where Substituting the above results into equation (2.92), we obtain [ ] [ ] Thus [ ] To derive the solution

40 33. [ ] / ( ) where Thus the solution becomes FIGURE 3. Two-soliton solutions of the KdV equation.

41 34 Chapter 3 The Family of Equations In the previous chapter, we introduced a historical and steps of the IST, as we saw the IST is a very difficult method and has a lot of scattering data, so in this chapter we will introduce this method to solve a general formula of important -dimention of equations which describes a pseudospherical surfaces (pss) and gives some examples and compare the solution of these equations with the solutions given by other methods. First we define that equation which describes pss, that is an equation describes a surface with Gaussian curvature. Beals, Rabelo and Tenenblat (BRT) [18] introduced the family of equations as follows: where [ ] where is a differentiable function of with are real constants, such that This family includes the sine-gordon, sinh-gordon and Liouville equations. 3.1 The AKNS system and IST for the family of equation : Let be two-dimensional differentiable function space with coordinates A DE for a real function describes a pss. If it s a necessary and sufficient condition for the existence of differentiable functions [3] Dependent of and its derivatives such that the form

42 35 satisfies the definition of equations that describes pss i.e The above definition is equivalent to that DE for problem so we can write is the integrability condition for the ( φ ) where d denotes exterior differentiation, Ƥ is 2Χ2 matrix such that ( ) Take ( ) where which is a parameter independent of and while and are functions of and, now we have [3] If we assume that the above equations are compatible, that satisfies that Q must satisfy, then P and [ ]

43 36 and consider the 2 x 2 scattering problem { φ φ φ and the linear time dependence is given by { φ φ φ where A, B, C and D are scalar functions of and λ, independent φ. Now, we just specify that ( ) ( ) Now, when, then (3.8) reduced to the Schrodinger scattering problem It is interesting to note that the most interesting nonlinear evolution equations phenomena when or (or if q is real). This procedure provides a simple technique which allows us to find a nonlinear evolution equations expressible in the form (3.7). The compatibility condition of equations (3.8-9), that is requiring that φ φ, and assuming that the parameter is time-independent, that is, allows a set of conditions which A, B, C and D must satisfy. Therefore φ φ

44 37 φ φ φ φ φ φ φ φ Respectively, do the same thing to then we have the following results So without loss of generality we assume that then we have Since the solution is related to take and. and if we choose A, B and C as. / ( ) ( ) Then A,B and C satisfy equation (3.16), so the linear time dependence is φ. / φ ( ) { ( ) φ. /

45 38 Since we have where ( ) The solution of equation (3.19) is where is constant column vector i.e. / ( ) Now we choose ( ) then we have { φ [ ( ) ] [ ( ) ] Konno and Wadati introduced the function φ This function first appeared and explained the geometric context of pss equations, now

46 39 Now we write the kernel integral as writing Gel fand and Levitan equation 2 0. / / 1/ Thus the solution of the family of equation 2 0. / 13

47 40 3.2: Example 1: the sine-gordon equation : In the family of equation if we choose then we have and where is real valued and the subscripts denote the partial derivatives with respect to the spatial coordinate x and the temporal coordinate t. This equation is called the sine-gordon equation, The solution of the sine-gordon equation depends on the solution of the family of equation as follows. So we start with the scattering problem { φ φ φ If we choose and called the potential corresponding to a solution of the sine-gordon equation. So we have the following lax pair φ { φ φ We notice that, if φ are bounded and sufficiently rapidly as, then 2 φ And we have

48 41 ( ) ( ) ( ) So the linear time dependence is φ ( ) φ ( ) { ( ) φ ( ) Since the solution is related to through so that it follows that, as. Then we have φ { φ And the solution becomes, * ( ) +-

49 42 3.3: Example 2: the Sinh-Gordon equation : In the family of equation if we choose then we have and where is real valued and the subscripts denote the partial derivatives with respect to the spatial coordinate x and the temporal coordinate t. This equation is called the Sinh-Gordon equation. The solution of the Sinh-Gordon equation depends on the solution of the family of equation. So we start with the scattering problem { φ φ φ If we choose and called the potential corresponding to a solution of the sine-gordon equation. So we have the following lax pair φ { φ φ We notice that, if φ are bounded and sufficiently rapidly as, then 2 φ And we have:

50 43 ( ) ( ) ( ) So the linear time dependence is φ ( ) φ ( ) { ( ) φ ( ) Since the solution is related to through so it follows that, as. φ { φ And the solution becomes { [ ( ) ]}

51 44 3.4: Conclusion The application of the inverse scattering transform (IST) as a method of solving physically relevant nonlinear partial differential equations has much grown and used since its discovery in the late 1960s. It provides a way of linearizing these systems, that is a means of obtaining their solutions through the solving of linear equations. Some of the major successes of the IST in mathematical physics have been the solving of the Korteweg-de Vries equation and its variants, the nonlinear Schrodinger equation and the sine- Gordon equation, all of which have great importance in the field. The IST gives a wide class of solutions satisfying decaying boundary conditions, which are typically the most physically relevant scenarios. We found in chapter 2 that inverse scattering transform is similar to Fourier transform but Fourier transform cannot solve the nonlinear models because we have unsolvable integral equation when we do the inverse Fourier transform unlike inverse scattering transform. After that we concluded that the new inverse scattering method with using Tanh-transform to solve the linear system of Lax pair gave identical results to the results of the original way and we found that in one soliton and two soliton solution examples. And we can see that when we use this transform making the IST easier. After that in chapter 3 we solved the family of equation which included the Sine-Gordon and Sinh-Gordon equations, then we saw that the solution of these equations by the general solution of the family of equation is related to the solution of these equations in several method of solving that kind of equations.

52 45 3.5: References [1] Ablowitz, M.J. and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge Univ. Press, Cambridge. [2] M. Ablowitz, G. Biondini, and B. Prinari. Inverse Scattering Transform for the Integrable Discrete Nonlinear Schrodinger Equation with Nonvanishing Boundary Conditions. Inv. Prob., 23( ), [3] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, [4] M. Ablowitz, D. Kaup, A. Newell, and H. Segur. Method for Solving the Sine-Gordon Equation. Phys. Rev. Lett., 30: , [5] M. Ablowitz, D. Kaup, A. Newell, and H. Segur. The Inverse Scattering Transform - Fourier Analysis for Nonlinear Problems. Stud. Appl. Math., 53: , [6] M. Ablowitz and J. Ladik. On the Solutions of a Class of Nonlinear Partial Difference Equations. Stud. Appl. Math., 57:1 12, [7] B. Dubrovin, V. Matveev, and S. Novikov. Nonlinear Equations of Korteweg-de Vries Type, Finite Zoned Linear Operators, and Abelian Varieties. Russ. Math. Surv., 31:59 146, [8] B. Dubrovin and S. Novikov. Periodic and Conditionally Periodic Analogues of the Many-soliton Solutions of the Korteweg-de Vries Equation. Sov. Phys. JETP, 40: , 1974.

53 46 [9] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1967), "Method for Solving the Korteweg-deVries Equation", Physical Review Letters 19: , [10] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1974), "Korteweg-deVries Equation and Generalization. VI. Methods for Exact Solution.", Comm. Pure Appl. Math. 27: , doi: /cpa , MR [11] H. Blohm. Solutions of a Discrete Inverse Scattering Problem and the Cauchy Problem of a Class of Discrete Evolution Equations. J. Math. Phys., 40: , [12] J. Shaw, Mathematical Principles of Optical Fiber Communications, SIAM, Philadelphia, [13] J. Atkinson, J. Hietarinta, and F. Nijhoff. Seed and Soliton Solutions of Adler s Lattice Equation. J. Phys. A, 40:F1 F8, [14] Liu, S., Z. Fu, S. Liu and Q. Zhao, Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289: [15] M. Boiti, F. Pempinelli, B. Prinari, and A. Spire. An Integrable Discretization of KdV at Large Times. Inv. Prob., 17: , 2001.

54 47 [16] N. Asano, Y. Kato, Algebraic and Spectral Methods for Nonlinear Wave Equations, Longman Scientific & Technical, Essex, England, [17] R.K. Bullough, P.J. Caudrey. "Solitons" Topics in Current Physics 17. Springer Verlag, Berlin-Heidelberg-New York, [18] Rogers, C. and W.F. Shadwick, Backlund Transformations. Aca. Press, New York. [19] V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, [20] V. Adler, A. Bobenko, and Y. Suris. Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases. Funct. Anal. Appl., 43(1):3 17, [21] Zhang, J., D. Zhang and D. Chen, Exact Solutions to a Mixed Toda Lattice Hierarchy through the Inverse Scattering Transform. Journal of Physics A: Mathematical and Theoritical, doi: / /44/11/

55 48 المعادالت غير الخطية المتطورة ومعكوس التشتت إعداد عبذانشح يح د شحادة جبش إشراف د. غريب موسى غريب الملخص ان ذف انشئ ض نهشصانت اصخعشاض غش مت يعك س انخشخج نحم ان عادالث غ ش انخط ت ان خط سة ي أيثهخ ا ظاو يعادالث ك د ف ح ث حطشل ا ف انجزء األ ل نهخعش فاث ان خعهمت بان عادالث انخفاظه ت حص ف ا ركش بعط انطشق نحم ز ان عادالث خصص ا بانزكش ان عادالث غ ش انخط ت ان خط سة يذ أ خ ا ف انعه و انف ز ائ ت انظ ا ش انطب ع ت بعذ رنك حعشف ا عه بعط أ اع انحه ل ن ز ان عادالث يثم انحم ان ج نه عادالث انخفاظه ت انحم انضهخ. ثى ف انجزء انثا دسص ا غش مت يعك س انخشخج حعشف ا عه أ خ ا ف إ جاد انحم نه عادالث عاللخ ا بطشق اخش ل ا بخحه م بعط انخط اث ف ز انطش مت ك ا غبم ا ز انطش مت عه يثال خط ي ان عادالث يعادنت حذفك انحشاسة ان ش سة غبم ا ا عا ز انطش مت عه حانت خاصت ي يعادالث ك د ف ثى أدخه ا حح م خخصش بعط انخط اث ف انحم ثى اصخعشظ ا بعط األيثهت عه أ اع انحم انضهخ ن عادنت ك د ف. ثى انجزء انثانث حطشل ا نه ذف انشئ ض حم ص سة عايت ي ان عادالث حض بعائهت ان عادالث ح ث حشخ م عه بعط ان عادالث ان ت يثم يعادالث كال -ج سد يعادالث ن فم. ع ذ حم عائهت ان عادالث كا ج أ ى خط ة إ جاد ان ظاو انخط ان شحبػ بانعائهت يا ض ب ظاو الكش ثى بعذ رنك أ جذ ا ان صف فت ان عشفت ن زا ان ظاو ف فعاء ان عادالث 2 2 ي ثى حابع ا خط اث انحم ف غش مت يعك س انخشخج اصخك ه ا ع اصش انخشخج األياي ل ا بع م انخط ش ن ز انع اصش ن حصم عه ان عك س ان خشخج ان شاد حه ع غش ك يعادنت حكايه ت حض ب جم فا ذ ي ثى إ جاد انحم انضهخ بعذ حم ان عادنت اصخعشظ ا حم نه عادالث ان شخمت ي ز ان عادنت يثم يعادالث كال -ج سد ب ماس ت زا انحم انضهخ جذ ا أ ي اثم نحه ل اخش بطشق اخش ن ز ان عادالث.