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Research article

Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations

  • Received: 22 June 2021 Accepted: 25 July 2021 Published: 30 July 2021
  • MSC : 35Q51, 37K40

  • In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.

    Citation: Wen-Xin Zhang, Yaqing Liu. Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations[J]. AIMS Mathematics, 2021, 6(10): 11046-11075. doi: 10.3934/math.2021641

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  • In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.



    Let ΓC be a closed Jordan curve and let Ω be the bounded component of CΓ. For any λ>0, the set

    Γλ:={zΩ:dist(z,Γ)=λ}.

    is called the constant distance boundary of Γ. Meanwhile, let

    Ωλ:={zΩ:dist(z,Γ)>λ}.

    Here dist(z,Γ):=inf{|zζ|:ζΓ}. In [2], Brown showed that for all but countable number of λ, every component of Γλ is a single point, or a simple arc, or a simple closed curve. It was also proved that Γλ is a 1-manifold for almost all λ in [3]. Blokh, Misiurewiczch and Oversteegen generalised Brown's result in [1], they provided that for all but countably many λ>0 each component of Γλ is either a point or a simple closed curve. If Γ is smooth or having positive reach, Γλ is called the offset of Γ in [6]. For λ within a positive reach, most nice properties are fulfilled by the Γλ. For instance, Γλ shares the class of differentiability of the Γ, since there is no ambiguity about the nearest point on Γ in such region. And points of Γλ project onto Γ along the normal to Γ through such point.

    In Figure 1, we display three examples to show the relationship between Ωλ and Γλ. In the left two graphs, Γ is the outside polygon. The interior "curve" of (A) is Γλ, which is not a real curve. The interior curve of (B) is Ωλ. In graph (C), the outer curve is Γ and the interior two curves make up Ωλ. So, in general Ωλ is not necessarily to be connected and its boundary Ωλ may not be equal with Γλ. However, it is not hard to see that ΩλΓλ. We would like to ask that what is the sufficient condition for Γλ to be a Jordan curve and what is the sufficient condition for Ωλ=Γλ? These questions are studied in [7]. If Γλ is a Jordan curve when λ is small enough, we find that, with a flatness condition, Γλ is approaching to Γ in the sense of Hausdorff distance as λ goes to 0. This means that Γλ is similar to Γ when λ is small enough. Thus we may expect Γλ inherits the geometric properties of Γ. This approximation property of Γλ could be applied in the theory of integration. In another paper we are preparing for, we show that ΓλfΓf with some geometric restriction on Γ.

    Figure 1.  In (A) and (B) λ=xy2. In (C) λ>zw2.

    Given two points x,yΓ, denotes Γ(x,y) by the subarc of Γ containing and connecting x and y which has a smaller diameter, or, to be either subarc when both have the same diameter. Let x,y be the infinite line through x and y, let

    ζΓ(x,y)=1|xy|sup{dist(z,x,y):zΓ(x,y)}.

    Following definition can be introduced.

    Definition 1.1. [7] A Jordan curve Γ is said to have the (ζ,r0)-chordal property for a certain ζ>0 and r0>0, if

    sup{ζΓ(x,y):x,yΓand|xy|r0}ζ.

    Also, denote

    ζΓ=limr0sup{ζΓ(x,y):x,yΓ and |xy|r}.

    These quantities are used to measure the local deviation of the subarcs from their chords. It is not hard to see that Γ is smooth if and only if ζΓ=0. Therefore all smooth curves have the (ζ,r0)-chordal property. Moreover, if a piecewise smooth Jordan curve only has corner points then it has the (ζ,r0)-chordal property. However if a piecewise smooth Jordan curve has a cusp point then it does not have the (ζ,r0)-chordal property.

    Theorem 1.1. [7] Let Γ be a Jordan curve in R2. If Γ has the (1/2,r0)-chordal property for some r0>0, then Γ has the level Jordan curve property i.e., there exists λ0>0 such that Γλ is a Jordan curve for each λ<λ0.

    This theorem provides us a method to verify whether Γλ is a Jordan curve. As we have seen in Figure 1, even through Γ is a simple Jordan curve, Γλ varies greatly. Based on this theorem, the authors of [7] also studied the quasi-circle property of Γλ. However, we are interested in the limit behaviour of Γλ as λ approaching to 0. The (ζ,r0)-chordal property of Jordan curves is an essential condition in the proof of the main theorem, we show that Γλ converges to Γ if Γ has the (1/2,r0)-chordal property.

    The other parts of the paper will be organized as follows: In Section 2, we investigate some basic properties of constant distance boundary of Jordan curves. We prove that if Γ and Γλ are Jordan curves then there exist at least three points of Γ which have distance λ from Γλ. Also, we find out the relation between Γλ+μ and (Γμ)λ. This relation will be used in the proof of our main theorem. Section 3 is devoted to prove our main result, Theorem 3.1. The definition and some basic properties of Hausdorff distance, dH(,), are introduced firstly. We show that under the (1/2,r0)-chordal property of Γ, the upper and lower bounds of dH(Γ,Γλ) are obtained. Consequently, the main theorem can be obtained.

    In this section, we investigate several fundamental properties according to the (ζ,r0)-chordal property of Jordan curve Γ. In this paper, we always assume that λ>0 and that Γλ is non-empty. First we introduce a notation which will be used frequently through the paper. For each xΓ, define

    Γxλ:={yΓλ:|xy|=λ}.

    And for any yΓλ, define

    Γy:={xΓ:|xy|=λ}.

    In [5], the so called λ-parallel set of Γ is introduced. The definition is the following,

    Ωpλ:={zΩ:dist(z,Γ)<λ}.

    Recall that we already have the set

    Ωλ={zΩ:dist(z,Γ)>λ}.

    We have seen in Figure 1 that Ωλ is a proper subset of Γλ and Theorem 1.1 states that if Γ has the (1/2,r0)-chordal property for some r0>0 then Ωλ=Γλ whenever λ is small enough. However, the identical of Γλ and Ωpλ can be obtained directly without the (1/2,r0)-chordal property.

    Proposition 2.1. Γλ=Ωpλ.

    Proof. According to the continuity of the distance function, the relation of ΩpλΓλ is obvious.

    Let zΓλ. Then there exists xΓz. Consider an arbitrary point y on the segment (x,z). We know that dist(y,Γ)|xy|<λ. Thus yΩpλ. Since the point z is the limit of points along the segment [x,z], we know that z¯Ωpλ. Therefore we have ΓλΩpλ.

    In the above proof, [x,z] stands for the line segment connecting points x and z, while (x,z) is [x,z]{x,z}.

    Proposition 2.2. Let x,yΓλ be different points and let xΓx and yΓy. If the two segments [x,x] and [y,y] intersect at p, i.e., [x,x][y,y]={p} then x=y=p.

    Proof. If xy then {p}=(x,x)(y,y). We have

    |xp|+|px|=λ and |yp|+|py|=λ.

    Since

    |xp|+|py||xy|λ.

    It follows that

    |xp||yp|.

    Then

    |xy||xp|+|py|λ.

    Because of |xy|λ, we know that |xy|=|xp|+|py|=λ. This means that the points y, p and x are collinear, i.e., p(y,x). However p(y,y), this is impossible unless x=y. Therefore, we must have x=y=p.

    This proposition tells us that two such segments [x,x] and [y,y] could only intersect at the end points.

    Proposition 2.3. Let xΓ and yΓxλ. If z(x,y) such that |xz|=μ for some 0<μ<λ then zΓxμ.

    Proof. Since yΓxλ and |xz|=μ, we have |yz|=λμ and dist(z,Γ)|xz|=μ. Suppose that dist(z,Γ)<μ, there exists tΓ such that |zt|=dist(z,Γ)<μ. Then |yt||yz|+|zt|<λμ+μ=λ. It follows that dist(y,Γ)<λ. This contradicts to the fact that yΓxλΓλ. Thus dist(z,Γ)=μ and then zΓxμ.

    In the proofs of the above three propositions, the set Γ is not necessarily to be a Jordan curve. So these properties are correct for any compact subset of C. In the following context, we assume that Γ is a Jordan curve. The Lemma 4.2 of [7] states that if Γλ is a Jordan curve and if there exist distinct x,yΓzλ for some zΓ, then the subarc Γλ(x,y) is a circular arc of the circle centred at z and with radius λ, which denoted by γ(z,λ).

    Lemma 2.1. If Γ and Γλ are Jordan curves then there exist at least three points of Γ which all have distance λ from Γλ.

    Proof. Suppose that there is no point on Γ has distance λ from Γλ. It means that for any pΓ the distance dist(p,Γλ)λ. It is clear that dist(p,Γλ)<λ is incorrect. Thus dist(p,Γλ)>λ for all pΓ. It follows that for a fixed point qΓλ we know that |pq|>λ for all pΓ. Therefore dist(q,Γ)>λ. This contradicts the fact that qΓλ.

    Suppose that there is only one point xΓ which has distance λ from Γλ. Therefore dist(x,Γλ)=λ and dist(p,Γλ)>λ for any pΓ when px. Thus for arbitrary qΓλ, we have |qp|>λ when px. It implies that |qx|=λ. Then ΓλΓxλ. So Γλ is a circular arc of the circle with center at x and with radius λ, i.e., Γλγ(x,λ). Because Γλ is a Jordan curve, we must have Γλ=γ(x,λ). Therefore Γ is the union of {x} and a certain subset of circle γ(x,2λ). In other words, Γ is separated by Γλ into two parts. This contradicts the fact that Γ is a Jordan curve.

    Suppose that there are only two points x,yΓ which have distance λ from Γλ. It means that dist(x,Γλ)=λ=dist(y,Γλ) and dist(p,Γλ)>λ for any pΓ when px,y. Similar to the one point case, we know that ΓλΓxλΓyλ. Since Γλ is a Jordan curve, there are three situations we should consider.

    (i) If |xy|=2λ then Γλ lies in the two tangential circles γ(x,λ) and γ(y,λ). Because Γλ is a Jordan curve, it could only contained in one circle. Then x and y are separated by this circle which contradicts that fact that Γ is a Jordan curve.

    (ii) If |xy|<2λ then Γλ is the curve looks like number eight which enclose x and y at the inside area. While Γ{x,y} is in the outside area otherwise Γ={x,y}. The both situations contradict the fact that Γ is a Jordan curve.

    (iii) If |xy|>2λ then Γλ lies in one of the disjoint two circles γ(x,λ) and γ(y,λ). Therefore x and y are not connected which contradicts that fact that Γ is a Jordan curve.

    By the above analysis we finished the proof.

    The constant distance boundary Γλ of Γ will be a Jordan curve under specific conditions (see Theorem 1.1). Thus we can consider the constant distance boundary of Γλ, denoted by (Γλ)μ if which is non-empty. Naturally we will investigate the relationship between (Γλ)μ and Γλ+μ.

    Lemma 2.2. Let λ0>0. Suppose that Γλ is a Jordan curve for each λ<λ0. Then for 0<μ<λ<λ0 we have

    Γλ(Γμ)λμ.

    Proof. Since 0<μ<λ<λ0, it follows from Proposition 2.1 that ΩpμΩpλ. For any yΓλ, there is xΓ such that |xy|=λ. This means that yΓxλ. Let z be a point of segment [x,y] such that |xz|=μ. By Proposition 2.3, we conclude that zΓxμ, i.e., zΓμ.

    Now we have dist(y,Γμ)|yz|=λμ. If the equality holds then y(Γμ)λμ. If dist(y,Γμ)<|yz| then there exists zΓμ such that dist(y,Γμ)=|yz|<|yz|=λμ. Because of zΓμ, there must exists xΓ such that |zx|=μ. Therefore dist(y,Γ)|yx||yz|+|zx|<λμ+μ=λ which contradicts to the fact of yΓλ. Therefore we must have dist(y,Γμ)=|yz|=λμ which means y(Γμ)λμ. It follows that Γλ(Γμ)λμ.

    In Lemma 2.2 even though we assume that the sets Γλ and Γμ are Jordan curves, but Γμ does not necessarily satisfy the (1/2,r0)-chordal property, thus the set (Γμ)λμ probably is not a Jordan curve (see Theorem 1.1).

    Corollary 2.1. Let Γ be a Jordan curve and has level Jordan curve property for some λ0>0. If Γμ has (1/2,r0)-chordal property for a μ<λ0, then Γλ=(Γμ)λμ when 0<μ<λ<λ0 and λμ<δ for some δ>0.

    Proof. By the assumption of level Jordan curve property of Γ, we know that Γλ and Γμ are Jordan curves if 0<μ<λ<λ0. Because of Lemma 2.2 we have

    Γλ(Γμ)λμ.

    By Theorem 1.1 and by the assumption of Γμ has (1/2,r0)-chordal property, we know that the curve Γμ has level Jordan curve property for some δ>0. Thus its constant distance boundary (Γμ)λμ is a Jordan curve when λμ<δ. Then both Γλ and (Γμ)λμ are Jordan curves, it implies that Γλ=(Γμ)λμ.

    In Corollary 2.1, the (1/2,r0)-chordal property of Γμ is crucial, because it is a necessary condition for the set (Γμ)λμ to be a Jordan curve, i.e., the curve Γμ has level Jordan curve property. So far, we only know that if Γμ has (1/2,r0)-chordal property then (Γμ)λμ is a Jordan curve when λμ<δ for some δ>0.

    In this section, we study the limit behaviour of Γλ as λ tends to 0. All the limits are considered in the sense of Hausdorff distance. For the convenience of readers, we briefly introduce the concept and some elementary properties of Hausdorff distance, which can be found in [4].

    Definition 3.1. Let X and Y be two non-empty subsets of C. The Hausdorff distance of X and Y, denoted by dH(X,Y), is defined by

    dH(X,Y):=max{supxXinfyY|xy|,supyYinfxX|xy|}.

    Denote by

    d(X,Y):=supxXdist(x,Y)andd(Y,X):=supyYdist(y,X)

    the distance from X to Y and Y to X respectively. We could rewrite

    dH(X,Y)=max{d(X,Y),d(Y,X)}. (3.1)

    Note that d(X,Y)d(Y,X) usually happens.

    For non-empty subsets X and Y of C, we know that

    d(X,Y)=0xX,dist(x,Y)=0xX,x¯YX¯Y.

    Here ¯Y is the closure of Y in C. We summarize these equivalence relations in the following proposition.

    Proposition 3.1. Let X and Y be two non-empty subsets of C. Then d(X,Y)=0 if and only if X¯Y. Furthermore, dH(X,Y)=0 if and only if ¯X=¯Y.

    The triangle inequality is true not only for dH but also for d. That is for any compact subsets A, B and C of C we have

    d(A,B)d(A,C)+d(C,B). (3.2)

    We left the proof of (3.2) for interested readers as an exercise.

    Denote by Π the set of compact subsets of C. Federer shows in [4] that (Π,dH) is a complete metric space. According to our consideration, Γ is a Jordan curve, so it is compact. By the definition of constant distance boundary, Γλ is compact as well. Thus we have Γ,ΓλΠ. Observe that Γ0=Γ, so we want to know whether the limit of Γλ, in (Π,dH), is Γ or not as λ approaching to zero. The first proposition we obtained is the following.

    Proposition 3.2. If there exists LΠ such that limλ0Γλ=L then LΓ.

    Proof. If limλ0Γλ=L then limλ0dH(Γλ,L)=0. It follows that limλ0d(L,Γλ)=0. We know that d(Γλ,Γ)=λ, since

    d(Γλ,Γ)=supxΓλdist(x,Γ)=λ.

    It follows from (3.2) that d(L,Γ)d(L,Γλ)+d(Γλ,Γ). Letting λ tends to 0 implies that d(L,Γ)=0, thus L¯Γ. By the compactness of Γ, we have that L¯Γ=Γ.

    Proposition 3.2 states that if the limit of Γλ exists in Π then it must be a subset of Γ. But we still cannot confirm whether this limit is a proper subset of Γ or equal to Γ. While if Γ has the (1/2,r0)-chordal property, we obtain the following result.

    Lemma 3.1. Suppose that Γ has (1/2,r0)-chordal property and that λr0/2. Then λdH(Γλ,Γ)(25+1)λ.

    Proof. Because that Γ has the (1/2,r0)-chordal property, we may assume that Γλ in consideration is a Jordan curve. Recall that d(Γλ,Γ)=λ. By (3.1), we already have

    dH(Γλ,Γ)=max{d(Γλ,Γ),d(Γ,Γλ)}λ.

    Because of d(Γ,Γλ)λ we assume that d(Γ,Γλ)>λ, otherwise dH(Γλ,Γ)=λ.

    Now suppose that there exists a wΓ such that dist(w,Γλ)>λ. By Lemma 2.1 there are at least three points of Γ which have distance λ from Γλ. So we can choose a subarc Γ(x,y) of Γ such that wΓ(x,y) and d(p,Γλ)λ for all pΓ(x,y), especially, the equality holds only when p{x,y}. The reason is that if there is a zΓ(x,y){x,y} such that d(z,Γλ)=λ then one of the two subarcs Γ(x,z) or Γ(z,y) contains w. Thus only need to replace Γ(x,y) by this subarc. By the compactness of Γλ, there exist xΓxλ and yΓyλ.

    (i) Consider the case when x=y=q. It is not hard to know that |xy||xq|+|yq|=2λ and thus |xy|r0. By the (1/2,r0)-chordal property of Γ, we obtain that dist(p,x,y)1/2|xy|λ for every pΓ(x,y). The straight line x,y separates the complex plane into two parts, which denoted by CR and CL.

    Firstly, we assume that ΓCR and ΓCL are non-empty. Let p0CRΓ(x,y) such that

    dist(p0,x,y)=max{dist(p,x,y):pCRΓ(x,y)}.

    Similarly, let p1CLΓ(x,y) such that

    dist(p1,x,y)=max{dist(p,x,y):pCLΓ(x,y)}.

    Construct straight lines p0 and p1 pass through p0 and p1 respectively and parallel to x,y. Thus the arc Γ(x,y) is bounded in the strip region between p0 and p1 which has width at most 2λ. It is needed to explain that Γ(x,y) may only at one side of the line x,y. Thus p0 or p1 may does not exist. However, Γ(x,y) can not be a straight line otherwise x and y must be different. Therefore, at least, one of p0 or p1 must exists. Then the mentioned strip region now is between p1 and x,y if p0 does not exist, while the strip region is between p0 and x,y if p1 does not exist. In these cases, the width of the strip region is at most λ.

    Construct straight lines x and y pass through x and y respectively and perpendicular to x,y. Choose x and x on xΓ(x,y) such that

    |xx|=max{|st|:s,txΓ(x,y)}.

    Because Γ(x,y) is bounded in the strip region with width at most 2λ, we must have |xx|2λr0. Thus the (1/2,r0)-chordal property implies that the arc Γ(x,x) is bounded in a strip region which has width at most 2λ. By the similar argument for y, we obtain that Γ(x,y) is bounded in a rectangular with width 2λ and length 4λ. We denote this rectangular by Δ. Thus |wx|diamΔ=25λ. Here diamΔ is the diameter of Δ. So |wq||wx|+|xq|25λ+λ=(25+1)λ. This implies that dist(w,Γλ)(25+1)λ.

    (ii) Consider the case when xy. For every qΓλ(x,y) there exists pΓq.

    If pΓ(x,y) the selection condition of Γ(x,y) implies that p{x,y}. We can see that if p=x then replace x by q, also denoted by x, and if p=y then replace y by q, also denoted by y. Choose another point qΓλ(x,y) and continuous the above process, finally we must have that x=y. Then repeat the proof of case (i), we also have dist(w,Γλ)(25+1)λ.

    Suppose that pΓ(y,x). Here Γ(y,x)=ΓΓ(x,y). Because the segment [p,q] does not intersect with Γλ(y,x). Then Γλ(y,x) is enclosed by the union of arcs L:=Γ(x,y)[y,y]Γλ(x,y)[x,x]. It implies that for every qΓλ(y,x) there must exists pΓq such that pΓ(x,y). If this is not the case then pΓ(y,x), and then [p,q] intersects L which is impossible. Repeat the analysis of the case when pΓ(x,y) for pΓ(x,y), it follows that dist(w,Γλ)(25+1)λ.

    In the above analysis, we have considered all the possible situations. As a conclusion, we obtain that d(Γ,Γλ)(25+1)λ. Therefore we have the inequalities λdH(Γλ,Γ)(25+1)λ.

    In Lemma 3.1, the condition which Γ has (1/2,r0)-chordal property is crucial, because of the (1/2,r0)-chordal property the curve Γ has level Jordan property and then the upper bound of dH(Γλ,Γ) can be decided. However this condition is rigorous, we should consider the questions for curves without this restriction in the future work.

    Theorem 3.1. If Jordan curve Γ has (1/2,r0)-chordal property then limλ0Γλ=Γ in (Π,dH).

    Proof. By Lemma 3.1, we immediately obtain that dH(Γλ,Γ)(25+1)λ when 2λr0. It implies that limλ0Γλ=Γ.

    This theorem provides us a sufficient condition for Γ such that its constant distance boundaries converging to itself. Now let λ take discrete values {1n}n=1, we have the following corollary.

    Corollary 3.1. Let Γ be a Jordan curve and has level Jordan curve property for some λ0>0. If Γ1n has (1/2,r0)-chordal property when 1/n<λ0. Then the limit limnΓ1n exists.

    Proof. The Corollary 2.1 implies that Γ1n=(Γ1m)1n1m when 0<1m<1n<λ0 and 1n1m<δ for some δ>0. By Lemma 3.1, it follows that

    dH(Γ1m,(Γ1m)1n1m)(25+1)(1n1m),

    when 1n1m<r0/2. Therefore when 1n1m<min{δ,r0/2}, we obtain that

    dH(Γ1m,Γ1n)=dH(Γ1m,(Γ1m)1n1m)(25+1)(1n1m).

    Therefore {Γ1n} is a Cauchy sequence in (Π,dH), and then the limit limnΓ1n exists.

    The first author is supported by NSFC (12101453).

    The authors declared that they have no conflicts of interest to this work.



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