Existence of exotic waves for the nonlinear dispersive mKdV equation

Applied Mathematics and Computation 229 (2014) 499–504

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Existence of exotic waves for the nonlinear dispersive mKdV equation Jiuli Yin ⇑, Shanyu Ding, Lixin Tian, Xinghua Fan, Xiaoyan Deng Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China

a r t i c l e

i n f o

Keywords: Nonlinear dispersive mKdV equation Painleve property Exotic traveling wave Composite wave

a b s t r a c t The nonlinear dispersive mKdV equation ut þ ðu3 Þx þ ðu2 Þxxx ¼ 0 is proved to be Painleve integrable. Detailed classification of its travelling waves under certain parameter conditions are obtained by the improved qualitative analysis method. Abundant type of solutions are shown to exist including exotic traveling waves, peaked waves, compacted waves, looped and cusped waves, very particular composite waves having two singular points. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Many new nonlinear wave structure properties due to the saturation of nonlinearity were reported. We know that, under the effect of nonlinear convection, the well-known Korteweg-de Vries (KdV) equation

ut þ uux þ uxxx ¼ 0; has smooth solitary waves [1]. This means that linear dispersion might be responsible for smooth solitary wave. With the co-effect of nonlinear convection and nonlinear dispersion, the nonlinear dispersive KdV equation

ut þ ðu2 Þx þ ðu2 Þxxx ¼ 0; presents abundant nonlinear singular phenomena such as compacton, Loopon [2–10]. Another change is that, the integrable KdV equation becomes non-integrable due to the possessing of nonlinear dispersion [9]. Our research interest in this paper is that, under the nonlinear dispersion, whether the well-known integrable mKdV equation [11,12]

ut þ ðu3 Þx þ uxxx ¼ 0; preserves integrability and admits more abundant singular solutions. After introducing a nonlinear dispersion term, the nonlinear dispersive mKdV equation is formed as

ut þ ðu3 Þx þ ðu2 Þxxx ¼ 0:

ð1:1Þ

It is noted that we will improve the work in [13] to study the structures of travelling wave solutions to Eq. (1.1). The method in [13] can deal with the existence of solutions to a type of nonlinear PDEs. After integration, these equation has a term like Pð/Þ ¼ /2 ðb/2 þ c/ þ dÞ. Solutions would be determined by analyzing the zeros of Pð/Þ. While in our work, the polynomial form is Pð/Þ ¼ /2 ða/3 þ b/2 þ c/ þ dÞ and the situation of zeros is more complicated. So the previous method in [13] does not work well. We solve the problem by checking the sign of some function values at selected points. By the improved method, we obtain very detailed classification of travelling waves in Eq. (1.1). ⇑ Corresponding author. E-mail address: [email protected] (J. Yin). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.043

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This paper is organized as follows. In Section 2, the nonlinear dispersive mKdV equation is proved to be integrable. In Section 3, the theorem of the classification of travelling waves is given. In Section 4, the proof of the theorem is presented. Section 5 is the conclusion. 2. Painleve property Painleve property is a method to study the integrability properties of nonlinear evolution equations. A nonlinear partial differential equation is called Painleve integrable when it has Painleve property which means its solutions are single valued about an arbitrary singular manifold. Similar proof about the integrability of a nonlinear wave equation can be seen in [15]. For the sake of completeness, we will give a simple proof. By the Kruskal method [14], we assume the solution of Eq. (1.1) is

u¼

1 X uj /jþa :

ð2:1Þ

j¼0

Substituting (2.1) into (1.1) leads to conditions on a and recursion relation for the functions uj. If a is a negative integer and the recursion relation is consistent, then we say system (1.1) is integrable. To determine the leading order behavior, we substitute u  uj /a into Eq. (1.1). Hence we have a = 2. The recursion relation of the expansion coefficients uj is yielded as

ðj þ 1Þðj  6Þðj  10Þuj ¼ F j ðux ; ut ; :::; u0 ; u1 ; :::; uj1 Þ;

ð2:2Þ

where Fj is a function of u0, u1, ..., uj1 and the derivatives of /. It is easy to find that the resonances occur at j = 1, 6, 10. The resonance at j = 1 represents the arbitrariness of the singular manifold /(x, t) = 0. Hence we only prove the existence of arbitrary functions at the other two cases j = 6, 10 for the integrability of Eq. (1.1). According to the Kruskal’s method, we take / = x + w(t), where w(t) is an arbitrary function of t. After a lengthy computation, we obtain

u0 ¼ 20; u1 ¼ u2 ¼ u3 ¼ 0; 1 u4 ¼  wt ; 60 u5 ¼ u6 ¼ u7 ¼ 0; 1 w2 ; u8 ¼  216000 t 1 u9 ¼ w : 72000 tt

ð2:3Þ

Substituting (2.3) into the recursion relations (2.2), one can find that (2.2) are satisfied identically. Hence equation (1.1) is Painleve integrable. 3. Classification of travelling waves For the transformation u(x, t) = /(x  ct), equation (1.1) becomes

c/x þ 3/2 /x þ 2//xxx þ 6/x /xx ¼ 0;

ð3:1Þ

where c is the wave speed. By integrating with respect to x, Eq. (3.1) becomes

ð/2 Þxx ¼ /3 þ c/ þ a;

ð3:2Þ

where a is an integral constant. Definition 3.1. A function / 2 H1loc ðRÞ is a travelling wave solution of equation (1.1) if / satisfies (3.2) in distribution sense for some a 2 R. Similar to Definition 1 and Lemmas 4 and 5 in [13], we will present the following definition of weak traveling wave solutions. Definition 3.2. Any bounded function / belongs to H1loc ðRÞ and is a travelling wave solution of equation (1.1) with speed c if satisfying the following two statements: (A) There are disjoint open intervals Ei, i P 1, and a closed set C such that R n C S x2 1 i¼1 Ei and /ðxÞ ¼ 0 for x 2 C. (B) There is an a 2 R such that (i) For each a 2 R, there exists b 2 R such that

/2x ¼ Fð/Þ; x 2 Ei ;

S1

i¼1 Ei ,

/ 2 C 1 ðEi Þ for i P 1, /ðxÞ–0 for

ð3:3aÞ

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where

/2x ¼ Fð/Þ ¼

   25 /2 /3  5c /  52a þ b 3 2/2

ð3:3bÞ

:

(ii) If C satisfies l(C) > 0, we have a = 0. We state our main result as follows, Theorem 3.1. All travelling wave solutions /(x  ct) of Eq. (1.1) are smooth except at points where u = 0. Any travelling wave solution (see Fig. 1) falls into one of the following categories: pffiffiffiffi 3 (1) For a 6  23pcffiffi3 , there are looped wave solutions and cusped wave solutions with decay for some positive b. pffiffiffiffi pffiffiffipffiffiffiffiffi 3 4 (2) For  23pcffiffi3 < a <  27 5 c3 , there are looped wave solutions, rotated looped wave solutions, smooth periodic wave solutions and cusped solutions. ffiffiffiffiffi pffiffiffipwave 4 (3) For a ¼  27 5 c3 , there are looped wave solutions, rotated looped wave solutions, smooth periodic wave solutions, smooth wave solutions with decay and cusped periodic wave solutions. ffi pffiffiffipffiffiffiffi 4 (4) For  27 5 c3 < a < 0, there are smooth periodic wave solutions, looped wave solutions, smooth wave solutions with decay and cusped periodic wave solutions. (5) For a = 0 there are smooth periodic wave solutions, compacted wave solutions and looped wave solutions and cusped wave solutions.

f=

f=

(a) Solitary wave solutions

(b) Peaked wave solutions

f=

f=

(c) Cusped waves solutions

(d) Looped wave solutions

f=

(e) Compacted wave solutions

(f) Composite wave solutions

Fig. 1. Travelling wave solutions of Eq. (1.1).

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pffiffiffipffiffiffiffiffi 4 (6) For 0 < a < 27 5 c3 , there are smooth periodic wave solutions, peaked wave solutions pointing downwards and looped periodic wave solutions and cusped periodic wave solutions. pffiffiffipffiffiffiffiffi 4 (7) For a ¼ 27 5 c3 , there are smooth periodic wave solutions, peaked solitary wave solutions and cusped wave solutions. pffiffiffiffi pffiffiffipffiffiffiffiffi 3 4 (8) For 27 5 c3 < a < 23pcffiffi3 , there are smooth periodic wave solutions and smooth wave solutions with decay, peaked periodic wave psolutions, and cusped wave solutions. ffiffiffi3ffi (9) a P 23pcffiffi3 , there are smooth periodic wave solutions, peaked periodic wave solutions and cusped wave solutions. (10) (Composite waves) Composite waves will appear when some travelling waves are joined at points where / = 0. For example, cusped wave solutions and compacted wave solutions can form the composite wave with two singular points (see (f) of Fig. 1). 4. Proof of Theorem 3.1 Assume / is a function such that (A) and (B). From (A), we can observe that the solution / can consist of a countable number of smooth wave segments separated by a closed set C. In view of (B), each wave segment satisfying the following equation

/2x ¼ Fð/Þ; Fð/Þ ¼

 25 /2



x 2 E;

 /3  5c /  52a þ b 3 2/2

ð4:1Þ

;

for some interval E and constants c0, b. Furthermore, under the condition that we obtain all travelling wave solutions of equation (4.1) for different intervals E, and different values of c0, b, these obtained solutions defined on intervals whose union is R n C for some closed set C of measure zero can be combined. It is noted that the solution which is defined on R will satisfy (A) and (B) if and only if all wave segments satisfy equation (4.1) with the same a. Furthermore, if l(C) > 0 for a = 0, this process will yield all solutions satisfying (A) and (B). This will finish the proof. To prove Theorem 3.1, we will present the following facts. The qualitative behavior of solutions of /2x ¼ Fð/Þ near points where F has a zero or a pole is as follows. (1) If F(/) has a simple zero at / = /0, the solution / to Eq. (4.1) satisfies /ðxÞ  m þ ðx  x0 Þ2 , where /(x0) = /0. It is easy to find that periodic smooth solutions exist if F(/) has two simple zeros. (2) If F(/) has a double zero at / = /0, the solution / of Eq. (4.1) satisfies /ðxÞ  m þ expðjxjÞ as x ? 1. It is easy to find that smooth solitary wave solutions exist if F(/) has a simple zero and a double zero.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

  Fig. 2. The graph of the polynomial Pð/Þ ¼  25 /2 /3  5c /  52a . 3

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jx  x0 j1=3 ; x # x0 ; where /(x0) = c and g is some constant. It is jx  x0 j1=3 ; x " x0 ; easy to find that looped wave solutions exist if F(/) has a pole lying between the maximum value and the minimum value of the wave. Cusped wave solutions exist if this pole is just the maximum value or the minimum value of the wave. (4) Peaked waves appear when the solution / according to Eq. (4.1) suddenly changes direction at / = c where F(0) – 0 and u(x0) = 0, that is, limx"x0 /x ðxÞ ¼ limx#x0 /x ðxÞ –  1. (5) Compactons appear when the solution / according to Eq. (4.1) suddenly stops and remains constant at / = 0 where F(0) = 0. (6) Double kinked waves appear when the right hand side of Eq. (4.1) has two double zeros which are not opposite numbers, and u = 0 is not between the two zeros. When the pole u = 0 falls in the interval of the two zeros, butterfly-like waves occur. One reason for the occurrence of these two new solutions is that the solutions to Eq. (1.1) must be symmetry because this equation is invariable under the transformation x ? x.

(3) If F(/) has a double pole at / = /0, we obtain ux ðxÞ 

If we apply these remarks to

/2x ¼ Fð/Þ ¼

   25 /2 /3  5c /  52a þ b 3 2/2

;

we can classify the solutions of (4.1). Consider the polynomial

  2 5c 5a ; Pð/Þ ¼  /2 /3  /  5 3 2 with a double root at / = c. The categories of Theorem 3.1 correspond to different behaviors of this polynomial. Once a is fixed, a change in b will shift the graph vertically up or down. Hence we can easily determine which b’s that yield bounded traveling waves. There are nine qualitatively different cases (see Fig. 2): pffiffiffiffi 3 (1) For a 6  23pcffiffi3 , there are looped wave solutions and cusped wave solutions with decay for some positive b. pffiffiffiffi pffiffiffipffiffiffiffiffi 3 4 (2) For  23pcffiffi3 < a <  27 5 c3 , there are looped wave solutions, rotated looped wave solutions, smooth periodic wave solutions, and cusped wave solutions for some positive b. pffiffiffipffiffiffiffiffi 4 (3) For a ¼  27 5 c3 , there are looped wave solutions, rotated looped wave solutions, smooth periodic wave solutions, smooth wave solutions with decay and cusped periodic wave solutions for some positive b. pffiffiffipffiffiffiffiffi 4 5 c3 < a < 0, there are smooth periodic wave solutions for some negative b or b = 0, and looped wave solu(4) For  27 tions, smooth wave solutions with decay and cusped periodic wave solutions for some positive b.

(5) For a = 0 there are smooth periodic wave solutions for some negative b, compacted wave solutions for b = 0, and looped wave solutions and cusped wave solutions for some positive b. pffiffiffipffiffiffiffiffi 4 5 c3 , there are smooth periodic wave solutions for some negative b, peaked wave solutions pointing (6) For 0 < a < 27 downwards for b = 0, and looped periodic wave solutions and cusped periodic wave solutions for some positive b. pffiffiffipffiffiffiffiffi 4 5 c3 , there are smooth periodic wave solutions for some negative b, peaked solitary wave solutions for (7) For a ¼ 27 b = 0, and cusped wave solutions for some positive b. pffiffiffiffi pffiffiffipffiffiffiffiffi 3 4 (8) For 27 5 c3 < a < 23pcffiffi3 , there are smooth periodic wave solution and smooth wave solutions with decay for some negative b, peaked periodic wave solutions for b = 0, and cusped wave solution for some positive b. pffiffiffiffi 3 (9) a P 23pcffiffi3 , there are smooth periodic wave solutions for some negative b, peaked periodic wave solutions for b = 0 and cusped wave solutions for some positive b. what is more interesting, by Theorem 3.1, composite waves will appear when some travelling waves corresponding to the same value of a are joined at points where / = 0. For example, cusped wave solutions and compacted wave solutions can form the composite wave with two singular points (see Fig. 1(f)). 5. Conclusions By a detailed qualitative analysis method, we have obtained the distribution of travelling waves under different parameter conditions. Specially, some singular solutions have been got including compacted wave, looped wave solutions, cusped wave solutions, peaked wave solutions, and some composite solutions. Our method can be widely applied to other nonlinear wave equations.

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Acknowledgement This work is supported by the National Nature Science Foundation of China (No. 11101191). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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