# The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

Authors: Wenxia Chen; Danping Ding; Xiaoyan Deng; Gang Xu
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

The evolution process of four class soliton solutions is investigated by basic calculus theory. For any givenx, we describe the special curvature evolution following time t for the curve of soliton solution and also study the fluctuation of solution curve.


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## Body

## 1. Introduction

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 pointx=ct, where c is the wave propagation speed (see [1]). Constantin and Strauss proved the stability on peak solitary solution of the CH equation (see [2]). 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 [18]. 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 Section2, 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.


## 2. Notations

Supposeu(x,t) be a solution of partial differential equation with initial data u(x,0)=u0(x), where x∈R is a spatial variable and t>0 is a time variable; we consider plane curve Γt={(x,u(x,t))}, then the curvature of the plane curve is defined as(1)Kt,x=uxx1+ux23/2,and the sign of the curvature indicates the orientation of curve bending.Introducing a cumulative curvature of the curveΓt over interval (-∞,x) as (2)It,x=∫-∞xKt,xdx,by computation, we have (3)It,x=ux1+ux2.And denote the length of the curve by g(t,x), that is, g(t,x)=(1+ux2)1/2. Thus, we obtain (4)dKt,xdt=1g3t,xuxxt-3Kt,xg2t,xdgt,xdt,dIt,xdt=uxtg3t,x.We will investigate these evolutional properties of solitary solutions as follows:(1)
KdV equation(5)ut+uux+μuxxx=0;as in [5], it is easy to check that it has solitary solution(6)ut,x=ux-Dt=3Dsech2D2x-Dt,whereD is a constant.(2)
Modified KdV equation(7)ut-6u2ux+uxxx=0;as in [5], it has kink solution(8)ut,x=ux+Dt=±D2tanh⁡D2x+Dt,whereD is a constant.(3)
Sine-Gordon equation(9)utt-uxx=sin⁡u;as in [19], it has bright an dark soliton solution(10)ut,x=ux-Dt=arctan⁡1-D2x-Dt,whereD is a constant.(4)
Camassa-Holm equation(11)ut-uxxt+3uux=2uxuxx+uuxxx;as in [1], it has peakon solution(12)ut,x=ce-x-ct,wherec is a constant.ConsideringΓt={(x,u(x,t))} as a family of single parametric curve with parameter t, we shall investigate the evolution procedure of some properties of parametric curve Γt, such as monotonicity, extremum, and curvature.


## 3. Evolution Procedure of Solitary Solution

### 3.1. KdV Equation

In order to computeK(t,x) and I(t,x), firstly we need to calculate ux(x,t). And it follows from expression (6) that(13)uxt,x=-3D3/2sech2ξtanh⁡ξ,whereξ=D2x-Dt.Take a derivative (13) with respect to t, (14)uxt=3D32sech2ξ3sech2ξ-2.Take a derivative (13) with respect to x, (15)uxx=3D22sech2ξ2-3sech2ξ.Through above formulas, we obtain(16)It,x=-3D3/2gsech2ξtanh⁡ξ.By differential (16) with respect to t, (17)dIdt=32Dg3sech2ξ3sech2⁡ξ-2.And it is follows from the definition of K(t,x) that(18)Kt,x=32DDg3sech2ξ2-3sech2⁡ξ,(19)dKt,xdt=3D74g5Asech2⁡ξtanh⁡ξ,where A=A(ξ)=4-12sech2⁡ξ-72D3sech4ξ+216D3sech6ξ-135D3sech8ξ.From calculus theory, we know that the extremum point fromΓ0 at x=0 develops to Γt at x=Dt and the inflection point of Γ0 from (ln⁡(6±2)/2,2D) develops to (Dt+ln⁡((6±2)/2),2D). It is well known that the inflection point can change the sign of curvature K(t,x). For any fixed x∈R, the extremum will arrive at time t1(x)=x/D; it means that the curve over (-∞,0) will not reach extreme value and the curve over (0,+∞) must be an extremum at some time. The concave-convex of curve for given x will change twice at t2(x)=x/D+(1/D)ln⁡(6-2)/2 and t3(x)=x/D+(1/D)ln⁡((6+2)/2), respectively, where the interval of two times is invariant for every x≥ln⁡((6-2)/2). If x<ln⁡(6-2)/2, the change does not happen.Lemma 1.
Suppose0<D3<3/8 and A(x)=4-12x-72D3x2+216D3x3-135D3x4, then there exists a unique ζ∈(0,1), such that A(ζ)=0.Proof.
It is clear thatA(0)=4 and A(1)=-8+9D3<0, so we get ζ∈(0,1) which satisfies A(ζ)=0. (20)A′x=-12-144D3x+648D3x2-540D3x3,A′′x=4D3-36+324x-405x2.We know that A′(x) arrived maximum at x0=(135+8914)/405 and A′(x0)<32D3-12<0; it indicates that A′(x0)<0 and ζ is unique. Thus the proof of Lemma 1 is completed.Without loss of generality, we can assume thatζ=q/p, where p>q>0 and they are relatively prime. It means sech2ξ=ζ=q/p, then ξ1=ln⁡(p-p-q)/q and ξ2=ln⁡(p+p-q)/q solve the equation. By direct computation, we obtain that ξ1<0 and ξ2>0. The evolution procedure of the curvature at given x can be described as follows.Theorem 2.
Suppose0<D3<3/8, K(t,x) is given by (18), and then(a)
if0<x<(2/D)ξ1, the curvature K(t,x) decrease with respect to t;(b)
if(2/D)ξ1<x<0, the curvature K(t,x) increase when 0<t<(1/D)(x-(2/D)ξ1) and decrease if t>(1/D)(x-(2/D)ξ1) with respect to t;(c)
if0<x<(2/D)ξ2, the curvature K(t,x) decrease when 0<t<x/D; increase when x/D<t<(1/D)(x-(2/D)ξ1); and decrease again when t>(1/D)(x-(2/D)ξ1);(d)
ifx>(2/D)ξ2, the curvature K(t,x) increase when 0<t<(1/D)(x-(2/D)ξ2); decrease when (1/D)(x-(2/D)ξ2)<t<x/D and increase again if x/D<t<(1/D)(x-(2/D)ξ1); decrease again when t>(1/D)(x-(2/D)ξ1).For more perceptual intuition, we choose some different value forx to show above properties via graph; one can see Figure 1.Figure 1
The curvatureK(t,x) for different given x.
(a)
x
=
-
0.5
(b)
x
=
0.5On the other hand, we can obtain the description for evolution procedure of cumulative curvature at givenx. If x∈(-∞,2/Dln⁡(6-2)/2), the cumulative curvature I(t,x) is always decreased with respect to time t; if x∈(2/Dln⁡6-2/2,2/Dln⁡6+2/2), I(t,x) increase when t<1/D(x-2/Dln⁡6-2/2) and decrease when t>(1/D)(x-2/Dln⁡(6-2)/2). If x∈(2/Dln⁡(6+2)/2,+∞), I(t,x) decrease when t<1/D(x-2/Dln⁡(6+2)/2) and increase when 1/D(x-2/Dln⁡6+2/2)<t<1/D(x-2/Dln⁡(6-2)/2); meanwhile I(t,x) decrease again when t>(1/D)(x-2/Dln⁡(6-2)/2). For I(t,x), the changes of monotone indicate the orientation of curve bending of some segment of Γt changing at time t; it coincides with the existence of inflection point.Thus, for any givenx>0, the curve Γt(x) always has extreme value at t∗=x/D, but it is not true for any given x<0.


### 3.2. Modified KdV Equation

From (8), we obtain that(21)uxt,x=±D2sech2⁡ξ,where ξ=D/2(x+Dt), with D>0 being a constant.By differential (21) with respect to t, (22)uxtt,x=∓D2D2sech2⁡ξtanh⁡ξ.By differential (21) with respect to x, (23)uxxt,x=∓DD2sech2ξtanh⁡ξ.Denoting (24)g=1+D44sech4ξ1/2,direct computation yields(25)Kt,x=∓Dg3D2sech2ξtanh⁡ξ,(26)It,x=±D2gsech2⁡ξ.By differential (25) and (26) with respect to t, respectively, we obtain(27)dKt,xdt=∓D38g5Bξsech2⁡ξ,where B(ξ)=D2sech4⁡ξ+12sech2⁡ξ-8.(28)dIt,xdt=∓D2g3D2sech2⁡ξtanh⁡ξ.It is clear that the curveΓt 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 Γt develops to (-Dt,0) from (0,0) and there is no extremum.Lemma 3.
Suppose thatB(x)=D2x2+12x-8, then there is a unique x0∈(0,1) such that B(x0)=0.Proof.
By solving equation straightly,(29)Bx=D2x2+12x-8=0,we have (30)x0=36+8D2-6D2,which belongs to (0,1).
Thus, Lemma3 is proved.For convenience, we denotex0 by q/p, where p>q>0 are constants depending on D. Then the evolution procedure of the curvature at given x can be described as follows.Theorem 4.
The curvatureK(t,x) is given by (25), then (a)
if(31)x>2Dln⁡p+p-qq,thenK(t,x) always increase with respect to t for any t>0;(b)
if(32)2Dln⁡p-p-qq<x<2Dln⁡p+p-qq,thenK(t,x) decrease when t<1/D2/Dln⁡((p+p-q)/q)-x/D and increase when t>1/D2/Dln⁡((p+p-q)/q)-x/D;(c)
if(33)x<2Dln⁡p-p-qq,thenK(t,x) increase when t<1/D2/Dln⁡(p-p-q)/-q-x/D and decrease when 1/D2/Dln⁡(p-p-q)/-q-x/D<t<1/D2/Dln⁡((p+p-q)/q)-x/D and increase again when t>1/D2/Dln⁡((p+p-q)/q)-x/D.Theorem 5.
I
(
t
,
x
) is given by (26), then (a)
ifx>0, I(t,x) always decrease with respect to t;(b)
ifx<0, I(t,x) increase when t<-x/D and decrease when t>-x/D.There is no extremum forΓt(x) which is regarded either as a curve of t for given x or as a curve of x for fixed t.


### 3.3. Sine-Gordon Equation

By differential (10) with respect to x, we get(34)uxt,x=-1-D2x-Dt2+1-D2,and then by direct computation, we have (35)uxtt,x=-2D1-D2x-Dtx-Dt2+1-D22,uxxt,x=21-D2x-Dtx-Dt2+1-D22,uxxtt,x=2D1-D2x-Dt2+1-D2x-Dt2+1-D23.Let g=(1+(1-D2)/((x-Dt)2+1-D2)2)1/2>1; direct computation yields(36)Kt,x=21-D2x-Dtg3x-Dt2+1-D22,(37)It,x=-1-D2gx-Dt2+1-D2.By differential (36) and (37) with respect to t,(38)dKt,xdt=2D1-D2g5x-Dt2+1-D25Cx-Dt,where (39)Cx-Dt=x-Dt6+31-D2x-Dt4+31-D22x-Dt2-51-D2x-Dt2+1-D222-D2,dIt,xdt=-2D1-D2x-Dtg3x-Dt2+1-D22.Lemma 6.
Suppose thatD2≤3/4 and C(x)=(x)3+3(1-D2)(x)2+3(1-D2)2(x)-5(1-D2)(x)+(1-D2)2(2-D2), then there exist 0<x1<x2, such that C(x1)=C(x2)=0.Proof.
By differentialC(x) with respect to x, we can get (40)C′x=3x2+61-D2x-1-D23D2+2.Solving equation C′x=0, we obtain a stationary point x0=(151-D2-31-D2)/3>0. Throughout rigmarole computing, we obtain min⁡C(x)=C(x0), and (41)Cx0=-1591-D23/2+31-D22-51531-D25/2+31-D23<0.Noticing that C(0)>0 and C(x)>0 hold true for large enough x, according to calculus basic theory, we assert that there is 0<x1<x2 which satisfies, respectively, C(x1)=C(x2)=0. Then lemma is proved.From (38) and Lemma 3, we derive the following results.Theorem 7.
SupposeD2<3/4, x1, and x2 are given by Lemma 3, then (a)
ifx<-x2, K(t,x) defined by (36) always increase with respect to t;(b)
if-x2<x<-x1, K(t,x) decrease when t<(x+x2)/D and increase when t>(x+x2)/D;(c)
if-x1<x<x1, K(t,x) increase when t<(x+x1)/D, decrease when (x+x1)/D<t<(x+x2)/D, and increase again when t>(x+x2)/D;(d)
ifx1<x<x2, K(t,x) decrease when t<(x-x1)/D, increase when (x-x1)/D<t<(x+x1)/D, decrease again when (x+x1)/D<t<(x+x2)/D, and finally increase again when t>(x+x2)/D;(e)
ifx>x2, K(t,x) increase when t<(x-x2)/D, decrease when (x-x2)/D<t<(x-x1)/D, increase when (x-x1)/D<t<(x+x1)/D, decrease again when (x+x1)/D<t<(x+x2)/D, and finally increase again when t>(x+x2)/D.Theorem 8.
Assume thatI(t,x) is given by (37), then (a)
ifx<0, I(t,x) always increase with respect to time t;(b)
ifx>0, I(t,x) decrease when t<x/D and increase when t>x/D.There is no extremum forΓt(x) which is regarded either as a curve of t for given x or as a curve of x for fixed t.


### 3.4. Camassa-Holm Equation

By differential (12) with respect to x, we get(42)uxt,x=-ce-x-ctsgnx-ct,and then by direct computation, we have (43)uxxt,x=-ce-x-ct,uxtt,x=-c2e-x-ct.Let (44)g=1+c2e-2x-ct1/2;we have(45)Kt,x=cg3e-x-ct,(46)It,x=cge-x-ctsgn⁡x-ct.By differential (45) and (46) with respect to t, we get(47)dKt,xdt=c2g5sgn⁡x-cte-x-ct1-2c2e-2x-ct,(48)dIt,xdt=-c2g3e-x-ct.It is clear that the extremum point of the curveΓt which is defined by (12) develops to ct from 0, and there is no inflection point for the curve Γt.Theorem 9.
Suppose thatK(t,x) is given by (45), then (a)
ifx<-1/2ln⁡2c2, K(t,x) always decrease with respect to time t for any t>0;(b)
if-1/2ln⁡2c2<x<0, K(t,x) increase when t<x/c+ln⁡2c2/2c and decrease when t>x/c+ln⁡2c2/2c;(c)
if0<x<1/2ln⁡2c2, K(t,x) decrease when t<x/c, increase when x/c<t<x/c+ln⁡2c2/2c, and finally decrease again when t>x/c+ln⁡2c2/2c;(d)
ifx>1/2ln⁡2c2, Kt,x increase when t<x/c-ln⁡2c2/2c, decrease when x/c-ln⁡2c2/2c<t<x/c, increase again when x/c<t<x/c+ln⁡2c2/2c, and finally decrease again when t>x/c+ln⁡2c2/2c.From (48), we obtain that I(t,x) given by (46) always decrease with respect to time t for any t>0.For any givenx>0, the curve Γt always has extreme value at t∗=x/D, but it is not true for any given x<0.


## 3.1. KdV Equation

In order to computeK(t,x) and I(t,x), firstly we need to calculate ux(x,t). And it follows from expression (6) that(13)uxt,x=-3D3/2sech2ξtanh⁡ξ,whereξ=D2x-Dt.Take a derivative (13) with respect to t, (14)uxt=3D32sech2ξ3sech2ξ-2.Take a derivative (13) with respect to x, (15)uxx=3D22sech2ξ2-3sech2ξ.Through above formulas, we obtain(16)It,x=-3D3/2gsech2ξtanh⁡ξ.By differential (16) with respect to t, (17)dIdt=32Dg3sech2ξ3sech2⁡ξ-2.And it is follows from the definition of K(t,x) that(18)Kt,x=32DDg3sech2ξ2-3sech2⁡ξ,(19)dKt,xdt=3D74g5Asech2⁡ξtanh⁡ξ,where A=A(ξ)=4-12sech2⁡ξ-72D3sech4ξ+216D3sech6ξ-135D3sech8ξ.From calculus theory, we know that the extremum point fromΓ0 at x=0 develops to Γt at x=Dt and the inflection point of Γ0 from (ln⁡(6±2)/2,2D) develops to (Dt+ln⁡((6±2)/2),2D). It is well known that the inflection point can change the sign of curvature K(t,x). For any fixed x∈R, the extremum will arrive at time t1(x)=x/D; it means that the curve over (-∞,0) will not reach extreme value and the curve over (0,+∞) must be an extremum at some time. The concave-convex of curve for given x will change twice at t2(x)=x/D+(1/D)ln⁡(6-2)/2 and t3(x)=x/D+(1/D)ln⁡((6+2)/2), respectively, where the interval of two times is invariant for every x≥ln⁡((6-2)/2). If x<ln⁡(6-2)/2, the change does not happen.Lemma 1.
Suppose0<D3<3/8 and A(x)=4-12x-72D3x2+216D3x3-135D3x4, then there exists a unique ζ∈(0,1), such that A(ζ)=0.Proof.
It is clear thatA(0)=4 and A(1)=-8+9D3<0, so we get ζ∈(0,1) which satisfies A(ζ)=0. (20)A′x=-12-144D3x+648D3x2-540D3x3,A′′x=4D3-36+324x-405x2.We know that A′(x) arrived maximum at x0=(135+8914)/405 and A′(x0)<32D3-12<0; it indicates that A′(x0)<0 and ζ is unique. Thus the proof of Lemma 1 is completed.Without loss of generality, we can assume thatζ=q/p, where p>q>0 and they are relatively prime. It means sech2ξ=ζ=q/p, then ξ1=ln⁡(p-p-q)/q and ξ2=ln⁡(p+p-q)/q solve the equation. By direct computation, we obtain that ξ1<0 and ξ2>0. The evolution procedure of the curvature at given x can be described as follows.Theorem 2.
Suppose0<D3<3/8, K(t,x) is given by (18), and then(a)
if0<x<(2/D)ξ1, the curvature K(t,x) decrease with respect to t;(b)
if(2/D)ξ1<x<0, the curvature K(t,x) increase when 0<t<(1/D)(x-(2/D)ξ1) and decrease if t>(1/D)(x-(2/D)ξ1) with respect to t;(c)
if0<x<(2/D)ξ2, the curvature K(t,x) decrease when 0<t<x/D; increase when x/D<t<(1/D)(x-(2/D)ξ1); and decrease again when t>(1/D)(x-(2/D)ξ1);(d)
ifx>(2/D)ξ2, the curvature K(t,x) increase when 0<t<(1/D)(x-(2/D)ξ2); decrease when (1/D)(x-(2/D)ξ2)<t<x/D and increase again if x/D<t<(1/D)(x-(2/D)ξ1); decrease again when t>(1/D)(x-(2/D)ξ1).For more perceptual intuition, we choose some different value forx to show above properties via graph; one can see Figure 1.Figure 1
The curvatureK(t,x) for different given x.
(a)
x
=
-
0.5
(b)
x
=
0.5On the other hand, we can obtain the description for evolution procedure of cumulative curvature at givenx. If x∈(-∞,2/Dln⁡(6-2)/2), the cumulative curvature I(t,x) is always decreased with respect to time t; if x∈(2/Dln⁡6-2/2,2/Dln⁡6+2/2), I(t,x) increase when t<1/D(x-2/Dln⁡6-2/2) and decrease when t>(1/D)(x-2/Dln⁡(6-2)/2). If x∈(2/Dln⁡(6+2)/2,+∞), I(t,x) decrease when t<1/D(x-2/Dln⁡(6+2)/2) and increase when 1/D(x-2/Dln⁡6+2/2)<t<1/D(x-2/Dln⁡(6-2)/2); meanwhile I(t,x) decrease again when t>(1/D)(x-2/Dln⁡(6-2)/2). For I(t,x), the changes of monotone indicate the orientation of curve bending of some segment of Γt changing at time t; it coincides with the existence of inflection point.Thus, for any givenx>0, the curve Γt(x) always has extreme value at t∗=x/D, but it is not true for any given x<0.


## 3.2. Modified KdV Equation

From (8), we obtain that(21)uxt,x=±D2sech2⁡ξ,where ξ=D/2(x+Dt), with D>0 being a constant.By differential (21) with respect to t, (22)uxtt,x=∓D2D2sech2⁡ξtanh⁡ξ.By differential (21) with respect to x, (23)uxxt,x=∓DD2sech2ξtanh⁡ξ.Denoting (24)g=1+D44sech4ξ1/2,direct computation yields(25)Kt,x=∓Dg3D2sech2ξtanh⁡ξ,(26)It,x=±D2gsech2⁡ξ.By differential (25) and (26) with respect to t, respectively, we obtain(27)dKt,xdt=∓D38g5Bξsech2⁡ξ,where B(ξ)=D2sech4⁡ξ+12sech2⁡ξ-8.(28)dIt,xdt=∓D2g3D2sech2⁡ξtanh⁡ξ.It is clear that the curveΓt 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 Γt develops to (-Dt,0) from (0,0) and there is no extremum.Lemma 3.
Suppose thatB(x)=D2x2+12x-8, then there is a unique x0∈(0,1) such that B(x0)=0.Proof.
By solving equation straightly,(29)Bx=D2x2+12x-8=0,we have (30)x0=36+8D2-6D2,which belongs to (0,1).
Thus, Lemma3 is proved.For convenience, we denotex0 by q/p, where p>q>0 are constants depending on D. Then the evolution procedure of the curvature at given x can be described as follows.Theorem 4.
The curvatureK(t,x) is given by (25), then (a)
if(31)x>2Dln⁡p+p-qq,thenK(t,x) always increase with respect to t for any t>0;(b)
if(32)2Dln⁡p-p-qq<x<2Dln⁡p+p-qq,thenK(t,x) decrease when t<1/D2/Dln⁡((p+p-q)/q)-x/D and increase when t>1/D2/Dln⁡((p+p-q)/q)-x/D;(c)
if(33)x<2Dln⁡p-p-qq,thenK(t,x) increase when t<1/D2/Dln⁡(p-p-q)/-q-x/D and decrease when 1/D2/Dln⁡(p-p-q)/-q-x/D<t<1/D2/Dln⁡((p+p-q)/q)-x/D and increase again when t>1/D2/Dln⁡((p+p-q)/q)-x/D.Theorem 5.
I
(
t
,
x
) is given by (26), then (a)
ifx>0, I(t,x) always decrease with respect to t;(b)
ifx<0, I(t,x) increase when t<-x/D and decrease when t>-x/D.There is no extremum forΓt(x) which is regarded either as a curve of t for given x or as a curve of x for fixed t.


## 3.3. Sine-Gordon Equation

By differential (10) with respect to x, we get(34)uxt,x=-1-D2x-Dt2+1-D2,and then by direct computation, we have (35)uxtt,x=-2D1-D2x-Dtx-Dt2+1-D22,uxxt,x=21-D2x-Dtx-Dt2+1-D22,uxxtt,x=2D1-D2x-Dt2+1-D2x-Dt2+1-D23.Let g=(1+(1-D2)/((x-Dt)2+1-D2)2)1/2>1; direct computation yields(36)Kt,x=21-D2x-Dtg3x-Dt2+1-D22,(37)It,x=-1-D2gx-Dt2+1-D2.By differential (36) and (37) with respect to t,(38)dKt,xdt=2D1-D2g5x-Dt2+1-D25Cx-Dt,where (39)Cx-Dt=x-Dt6+31-D2x-Dt4+31-D22x-Dt2-51-D2x-Dt2+1-D222-D2,dIt,xdt=-2D1-D2x-Dtg3x-Dt2+1-D22.Lemma 6.
Suppose thatD2≤3/4 and C(x)=(x)3+3(1-D2)(x)2+3(1-D2)2(x)-5(1-D2)(x)+(1-D2)2(2-D2), then there exist 0<x1<x2, such that C(x1)=C(x2)=0.Proof.
By differentialC(x) with respect to x, we can get (40)C′x=3x2+61-D2x-1-D23D2+2.Solving equation C′x=0, we obtain a stationary point x0=(151-D2-31-D2)/3>0. Throughout rigmarole computing, we obtain min⁡C(x)=C(x0), and (41)Cx0=-1591-D23/2+31-D22-51531-D25/2+31-D23<0.Noticing that C(0)>0 and C(x)>0 hold true for large enough x, according to calculus basic theory, we assert that there is 0<x1<x2 which satisfies, respectively, C(x1)=C(x2)=0. Then lemma is proved.From (38) and Lemma 3, we derive the following results.Theorem 7.
SupposeD2<3/4, x1, and x2 are given by Lemma 3, then (a)
ifx<-x2, K(t,x) defined by (36) always increase with respect to t;(b)
if-x2<x<-x1, K(t,x) decrease when t<(x+x2)/D and increase when t>(x+x2)/D;(c)
if-x1<x<x1, K(t,x) increase when t<(x+x1)/D, decrease when (x+x1)/D<t<(x+x2)/D, and increase again when t>(x+x2)/D;(d)
ifx1<x<x2, K(t,x) decrease when t<(x-x1)/D, increase when (x-x1)/D<t<(x+x1)/D, decrease again when (x+x1)/D<t<(x+x2)/D, and finally increase again when t>(x+x2)/D;(e)
ifx>x2, K(t,x) increase when t<(x-x2)/D, decrease when (x-x2)/D<t<(x-x1)/D, increase when (x-x1)/D<t<(x+x1)/D, decrease again when (x+x1)/D<t<(x+x2)/D, and finally increase again when t>(x+x2)/D.Theorem 8.
Assume thatI(t,x) is given by (37), then (a)
ifx<0, I(t,x) always increase with respect to time t;(b)
ifx>0, I(t,x) decrease when t<x/D and increase when t>x/D.There is no extremum forΓt(x) which is regarded either as a curve of t for given x or as a curve of x for fixed t.


## 3.4. Camassa-Holm Equation

By differential (12) with respect to x, we get(42)uxt,x=-ce-x-ctsgnx-ct,and then by direct computation, we have (43)uxxt,x=-ce-x-ct,uxtt,x=-c2e-x-ct.Let (44)g=1+c2e-2x-ct1/2;we have(45)Kt,x=cg3e-x-ct,(46)It,x=cge-x-ctsgn⁡x-ct.By differential (45) and (46) with respect to t, we get(47)dKt,xdt=c2g5sgn⁡x-cte-x-ct1-2c2e-2x-ct,(48)dIt,xdt=-c2g3e-x-ct.It is clear that the extremum point of the curveΓt which is defined by (12) develops to ct from 0, and there is no inflection point for the curve Γt.Theorem 9.
Suppose thatK(t,x) is given by (45), then (a)
ifx<-1/2ln⁡2c2, K(t,x) always decrease with respect to time t for any t>0;(b)
if-1/2ln⁡2c2<x<0, K(t,x) increase when t<x/c+ln⁡2c2/2c and decrease when t>x/c+ln⁡2c2/2c;(c)
if0<x<1/2ln⁡2c2, K(t,x) decrease when t<x/c, increase when x/c<t<x/c+ln⁡2c2/2c, and finally decrease again when t>x/c+ln⁡2c2/2c;(d)
ifx>1/2ln⁡2c2, Kt,x increase when t<x/c-ln⁡2c2/2c, decrease when x/c-ln⁡2c2/2c<t<x/c, increase again when x/c<t<x/c+ln⁡2c2/2c, and finally decrease again when t>x/c+ln⁡2c2/2c.From (48), we obtain that I(t,x) given by (46) always decrease with respect to time t for any t>0.For any givenx>0, the curve Γt always has extreme value at t∗=x/D, but it is not true for any given x<0.


## 4. Conclusion

From the above discussion, the behavior of the curve groupΓt as a function of t for any given x is different with that as a function of x for any given t. 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 (x,u(x,t)) possess different evolution procedures of curvature, all evolution results are the same. For some points, the curvature of Γt(x) uniformly monotonously tends to final state when t goes to infinity, but for other points, the curvature of Γtx tends to final state after several fluctuations. Besides this, this paper investigates fluctuation for every given x with respect to time t. We find that, for some x, u(x,t) will have its extreme value at some t∗, but for another case, it never can happen. It is an interesting phenomenon.



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*Source: 5460216-2017-10-29.xml*