The sine-Gordon equation is a parti

The sine-Gordon equation is a partial differential equations which is known to have soliton solutions, hence it is also called one of the soliton equations. The discretized version of the equation could be done in various ways. Here, we will follow the method by describing its Lax-pair. By restriction to traveling wave solution; thus imposing periodic boundary condition; we derive an ordinary difference equation which is integrable as is the original equation.

In the literature, attention has been devoted to the integrability of the equation, the geometry it generates, symmetry in the system or the classification of integrable system. In 2010, the late J.J. Duistermaat wrote a seminal book called Discrete Integrable Systems, QRT Maps, and Elliptic Surfaces which provide us with novel way of looking
at integrable system. This book also originated from a discussion on a generalized discrete sine Gordon equation between one of the author of this paper and J.J. Duistermaat as is indicated in the preface of that book.

Our interest in studying the sine-Gordon discrete dynamical systems is on the dynamics and the bifurcations there in. To do this we need to have free parameters in the system. For this reason, we introduce a generalization to the sine-Gordon equation (originally this generalization was done by Tuwankotta and Quispel). Since integrability is a property to be preserved, we choose to generalize the Lax pair. By requiring the compatibility of the horizontal and vertical switch, we derive a mapping which we call: generalized sine-Gordon equation. To our knowledge, the study of a well-known integrable system from the point of view of dynamics and bifurcations has not been extensively performed.

We begin with formulating a generalized sine-Gordon equation, by introducing eight parameters into the Lax-pair matrices. By analyzing the so-called compatibility condition (or commutativity of the multiplication of the matrices), we derive a system of two algebraic homogeneous equations. We have two possibilities: the space of solutions of the system of homogeneous equations is one dimensional or two dimensional. In this paper, we restrict our self to consider only the latter. By doing this, we can reduce the number of parameter in the system to three.

Using the so-called staircase method, we derive an ordinary discrete integrable dynamical system, with three parameters. Further reduction to the number of parameters in the system, can be done by analyzing the integrals of the discrete system. For the case studies where the dimension of the phase space of the discrete system is two or three, we derive seven functions which contain the dynamics for all value of parameter. By analyzing the level sets of these functions, we derive some conclusion on the dynamics and bifurcations in the system.

We have observed an interesting local bifurcation of critical point in the system, namely: the period doubling bifurcation, where two period-2 points are created from a critical point. We have observed also a nonlocal bifurcation involving collision of homoclinic and heteroclinic connection between saddle type critical points. Furthermore, we have observed a change of stability of a critical point from a saddle type into an elliptic type of which we have not seen before in the literature.

Hasil (Inggris) 1:

The sine-Gordon equation is a partial differential equations which is known to have soliton solutions, hence it is also called one of the soliton equations. The discretized version of the equation could be done in various ways. Here, we will follow the method by self-describing its Lax-pair. By restriction to traveling wave solution; thus imposing periodic boundary condition; We derive an ordinary difference equation which is integrable as is the original equation.In the literature, attention has been devoted to the integrability of the equation, it generates the geometry, symmetry in the system or the classification of integrable systems. In 2010, the late J.J. Duistermaat wrote a seminal book called Discrete Integrable Systems, QRT Maps and Elliptic Surfaces which provide us with a novel way of lookingat integrable system. This book also originated from a discussion on a generalized discrete sine Gordon equation between one of the author of this paper and J.J. Duistermaat as is indicated in the preface of that book. Our interest in the Wadi sine-Gordon discrete dynamical systems is on the dynamics and bifurcations there in. To do this we need to have the free parameters in the system. For this reason, we introduce a generalization to the sine-Gordon equation (originally this generalization was done by Tuwankotta and Quispel). Since integrability is a property to be preserved, we choose to generalize the Lax pair. By requiring the compatibility of the horizontal and vertical switch, we derive a mapping which we call: generalized sine-Gordon equation. To our knowledge, the study of a well-known integrable system from the point of view of dynamics and bifurcations has not been extensively performed.We begin with formulating a generalized sine-Gordon equation, by introducing eight parameters into the Lax-pair matrices. By analyzing the so-called compatibility condition (or commutativity of the multiplication of the matrices), we derive a system of two homogeneous algebraic equations. We have two possibilities: the space of solutions of the homogeneous system of equations is one dimensional or two dimensional. In this paper, we restrict our self to consider only the latter. By doing this, we can reduce the number of parameters in the system to three.Using the so-called staircase method, we derive an ordinary discrete integrable dynamical system, with three parameters. Further reduction to the number of parameters in the system, can be done by analyzing the integrals of the discrete system. For the case studies where the dimension of the phase space of the discrete system is two or three, we derive seven functions which contain the dynamics for all value of the parameter. By analyzing the level sets of these functions, we derive some conclusion on the dynamics and bifurcations in the system. We have observed an interesting local bifurcation of critical points in the system, namely: the period doubling bifurcation, where two period-2 points are created from a critical point. We have also observed a nonlocal bifurcation involving collision of homoclinic and heteroclinic connection between saddle type critical points. Furthermore, we have observed a change of stability of a critical point from a saddle type into an elliptic type of which we have not seen before in the literature.

Hasil (Inggris) 2:

The sine-Gordon equation is a partial differential equations the which is known to have soliton solutions, hence It is also called one of the soliton equations. The discretized version of the equation could be done in various ways. Here, we will follow the method by describing its Lax-pair. By restriction to traveling wave solution; Thus Spake imposing periodic boundary condition; we derive an ordinary difference equation is the which is integrable as is the original equation. In the literature, attention has been devoted to the integrability of the equation, it generates the geometry, symmetry in the system or the classification of integrable system. In 2010, the late JJ Duistermaat wrote a seminal book called Discrete Integrable Systems, QRT Maps, and Elliptic Surfaces roomates provide us with a novel way of looking at integrable system. This book Also Originated from a discussion on a generalized discrete sine Gordon equation between one of the authors of this paper and JJ Duistermaat as is indicated resources in the Preface of that book. Our interest in studying the sine-Gordon discrete dynamical systems is on the dynamics and the bifurcations there in. To do this we need to have free parameters in the system. For this reason, we introduce a generalization to the sine-Gordon equation (originally this generalization was done by Tuwankotta and Quispel). Since integrability is a property to be preserved, we choose to generalize the Lax pair. By requiring the compatibility of the horizontal and vertical switch, we derive a mapping of the which we call: generalized sine-Gordon equation. To our knowledge, the study of a well-known integrable system from the point of view of dynamics and bifurcations has not been extensively performed. We begin with formulating a generalized sine-Gordon equation, by introducing eight parameters into the Lax-pair matrices. By analyzing the so-called compatibility condition (or commutativity of the multiplication of the matrices), we derive a system of two homogeneous algebraic equations. We have two possibilities: the space of solutions of the system of homogeneous equations is one-dimensional or two dimensional. In this paper, we restrict our self to Consider only the latter. By doing this, we can reduce the number of parameters in the system to three. Using the so-called staircase method, we derive an ordinary integrable discrete dynamical system, with three parameters. Further reduction to the number of parameters in the system, can be done by analyzing the integrals of the discrete system. For the case studies where the dimension of the phase space of the discrete system is two or three, we derive seven functions roomates Contain the dynamics for all values ​​of the parameters. By analyzing the level sets of Reviews These functions, we derive some conclusion on the dynamics and bifurcations in the system. We have observed an interesting local bifurcation of critical points in the system items, namely: the period doubling bifurcation, where two period-2 points are created from a critical point. We have observed Also a collision involving nonlocal bifurcation of homoclinic and heteroclinic connection between saddle-type critical points. Furthermore, we have observed a change of stability of a critical point from a saddle-type into an elliptic type of the which we have not seen before in the literature.