The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations
The evolution process of four class soliton solutions is investigated by basic calculus theory. For any given , we describe the special curvature evolution following time for the curve of soliton solution and also study the fluctuation of solution curve.
Engineer J. Scott Russel is always mentioned when one talks about solitary phenomenon, because he is the first person who discovered the solitary phenomenon and reported it. Korteweg and de Vries constructed partial differential equation, which named KdV equation, to describe such solitary phenomenon. Kruskal and Zabusky proved that the solitary solutions of KdV equation possessed the particle property, and they firstly created the terminology “soliton” in 1965. Their works aroused an upsurge of study on solitary. In 1993, Camassa and Holm proposed a new model on shallow water wave motion, named as Camassa-Holm equation (CH equation). The significant property of CH equation is that the equation has a special weak solution with solitary and a peak at point , where is the wave propagation speed (see ). Constantin and Strauss proved the stability on peak solitary solution of the CH equation (see ). These different solitaries such as the smooth solitary of KdV equation and peakon of CH equation attracted attention which comes from different research areas, such as mathematics, physics, communication technology, image process, and artificial intelligence. In recent twenty years, there are tremendous amount of works about solitary solutions, and more types of solitary are established such as compact soliton, kink soliton, and bright and dark soliton. Soliton theory and application have formed interesting great mass fervor of nonlinear partial differential equation (see [1, 3–5]).
On the other hand, the profiles of plane curves or intersurface attracted more and more attention. There have a mount of works about curvature motion and curvature motion flow (see [6–11]). Particularly, the curve evolution problem played important role in applied sciences. The study of immersed closed curves, which are introduced from their curvature functions, can apply to crystal growth (see [12, 13]), flame propagation (see [14, 15]), and curve shortening (see [16, 17]). And also the bound for the evolution of length and curvature was investigated by . Inspirited by above works, here we are interested in the evolution about some geometrical characteristics for the solutions of evolution partial differential equations with initial data; especially, we focus on the development of geometric properties of initial curve. This paper shall review some solitary solutions in view of evolution which arise from the KdV and modified KdV equation, CH equation, and Sine-Gordon equation.
The reminder of this paper is organized as follows. In Section 2, we introduce some basic notations and concepts which will be used in the following section. In Section 3, we investigate the evolution process of four class solitary solutions.
Suppose be a solution of partial differential equation with initial data , where is a spatial variable and is a time variable; we consider plane curve , then the curvature of the plane curve is defined asand the sign of the curvature indicates the orientation of curve bending.
Introducing a cumulative curvature of the curve over interval as by computation, we have And denote the length of the curve by , that is, . Thus, we obtain
We will investigate these evolutional properties of solitary solutions as follows:(1) KdV equation as in , it is easy to check that it has solitary solution where is a constant.(2) Modified KdV equation as in , it has kink solution where is a constant.(3) Sine-Gordon equation as in , it has bright an dark soliton solution where is a constant. (4) Camassa-Holm equation as in , it has peakon solution where is a constant.
Considering as a family of single parametric curve with parameter , we shall investigate the evolution procedure of some properties of parametric curve , such as monotonicity, extremum, and curvature.
3. Evolution Procedure of Solitary Solution
3.1. KdV Equation
In order to compute and , firstly we need to calculate . And it follows from expression (6) thatTake a derivative (13) with respect to , Take a derivative (13) with respect to , Through above formulas, we obtainBy differential (16) with respect to , And it is follows from the definition of thatwhere .
From calculus theory, we know that the extremum point from at develops to at and the inflection point of from develops to . It is well known that the inflection point can change the sign of curvature . For any fixed , the extremum will arrive at time ; it means that the curve over will not reach extreme value and the curve over must be an extremum at some time. The concave-convex of curve for given will change twice at and , respectively, where the interval of two times is invariant for every . If , the change does not happen.
Lemma 1. Suppose and , then there exists a unique , such that .
Proof. It is clear that and , so we get which satisfies . We know that arrived maximum at and ; it indicates that and is unique. Thus the proof of Lemma 1 is completed.
Without loss of generality, we can assume that , where and they are relatively prime. It means , then and solve the equation. By direct computation, we obtain that and . The evolution procedure of the curvature at given can be described as follows.
Theorem 2. Suppose , is given by (18), and then(a)if , the curvature decrease with respect to ;(b)if , the curvature increase when and decrease if with respect to ;(c)if , the curvature decrease when ; increase when ; and decrease again when ;(d)if , the curvature increase when ; decrease when and increase again if ; decrease again when .
For more perceptual intuition, we choose some different value for to show above properties via graph; one can see Figure 1.
On the other hand, we can obtain the description for evolution procedure of cumulative curvature at given . If , the cumulative curvature is always decreased with respect to time ; if , increase when and decrease when . If , decrease when and increase when ; meanwhile decrease again when . For , the changes of monotone indicate the orientation of curve bending of some segment of changing at time ; it coincides with the existence of inflection point.
Thus, for any given , the curve always has extreme value at , but it is not true for any given .
3.2. Modified KdV Equation
From (8), we obtain thatwhere , with being a constant.
It is clear that the curve is always monotone, increasing, or decreasing which is depending on choosing sign “+” or “−.” Without loose of generality, we just consider it in case of sign “+” in the remainder of this subsection. The inflection point of curve develops to from and there is no extremum.
Lemma 3. Suppose that , then there is a unique such that .
Proof. By solving equation straightly, we have which belongs to .
Thus, Lemma 3 is proved.
For convenience, we denote by , where are constants depending on . Then the evolution procedure of the curvature at given can be described as follows.
Theorem 4. The curvature is given by (25), then (a)if then always increase with respect to for any ;(b)if then decrease when and increase when ;(c)if then increase when and decrease when and increase again when .
Theorem 5. is given by (26), then (a)if , always decrease with respect to ;(b)if , increase when and decrease when .
There is no extremum for which is regarded either as a curve of for given or as a curve of for fixed .
3.3. Sine-Gordon Equation
Lemma 6. Suppose that and , then there exist , such that .
Proof. By differential with respect to , we can get Solving equation , we obtain a stationary point . Throughout rigmarole computing, we obtain , and Noticing that and hold true for large enough , according to calculus basic theory, we assert that there is which satisfies, respectively, . Then lemma is proved.
Theorem 7. Suppose , , and are given by Lemma 3, then (a)if , defined by (36) always increase with respect to ;(b)if , decrease when and increase when ;(c)if , increase when , decrease when , and increase again when ;(d)if , decrease when , increase when , decrease again when , and finally increase again when ;(e)if , increase when , decrease when , increase when , decrease again when , and finally increase again when .
Theorem 8. Assume that is given by (37), then (a)if , always increase with respect to time ;(b)if , decrease when and increase when .
There is no extremum for which is regarded either as a curve of for given or as a curve of for fixed .
3.4. Camassa-Holm Equation
It is clear that the extremum point of the curve which is defined by (12) develops to from , and there is no inflection point for the curve .
Theorem 9. Suppose that is given by (45), then (a)if , always decrease with respect to time for any ;(b)if , increase when and decrease when ;(c)if , decrease when , increase when , and finally decrease again when ;(d)if , increase when , decrease when , increase again when , and finally decrease again when .
For any given , the curve always has extreme value at , but it is not true for any given .
From the above discussion, the behavior of the curve group as a function of for any given is different with that as a function of for any given . The evolution procedure of curvature which is given by (6) and (12), respectively, is similar, but it is different to those curves given by (8) and (10). Even though, for self-similar curve group, different points possess different evolution procedures of curvature, all evolution results are the same. For some points, the curvature of uniformly monotonously tends to final state when goes to infinity, but for other points, the curvature of tends to final state after several fluctuations. Besides this, this paper investigates fluctuation for every given with respect to time . We find that, for some , will have its extreme value at some , but for another case, it never can happen. It is an interesting phenomenon.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was supported by the National Nature Science Foundation of China (Grants nos. 11371175, 11501253, 11571141, and 11571140), the Nature Science Foundation of Jiangsu Province (Grant no. BK20140525), and the Advanced Talent of Jiangsu University (Grants nos. 14JDG070 and 15JDG079).