EXACT BREATHER-TYPE SOLUTIONS AND RESONANCE-TYPE SOLUTIONS OF THE (2+1)-DIMENSIONAL POTENTIAL BURGERS SYSTEM

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1 EXACT BREATHER-TYPE SOLUTIONS AND RESONANCE-TYPE SOLUTIONS OF THE (+)-DIMENSIONAL POTENTIAL BURGERS SYSTEM YEQIONG SHI College of Science Guangxi University of Science Technology Liuzhou China Received February 0 07 Abstract. In this paper the (+)-dimensional potential Burgers system is investigated by means of the extended homoclinic (heteroclinic) test method. Some new types of exact solutions of the (+)-dimensional potential Burgers system are obtained such as the periodic breather-type solutions two-soliton fusion wave solutions Y-type soliton resonance solutions peakon solutions and so on. These results enrich the variety of the dynamics of the higher-dimensional nonlinear wave fields. Key words: (+)-dimensional potential Burgers equation extended homoclinic test method periodic breather-wave two-soliton solution.. INTRODUCTION Many phenomena in the nonlinear science can be modeled by the integrable nonlinear evolution equations. The investigation of integrable systems becomes nowadays a central theme in the study of nonlinear physical applications. Research on nonlinear waves has been active because of their rich properties. Consequently the construction of exact solutions of nonlinear equations plays an important role in the study of nonlinear wave phenomena. Nowadays with the rapid development of software technology solving nonlinear evolution equations via symbolic computation is taking an increasing role. Many solution methods have been proposed such as the tanh method [ ] the sine-cosine method [3 4] the Riccati equation expansion method [5 6] the generalized projective Riccati equation expansion method [7 8] the sine/sinh- Gordon method [9 0] the algebraic method [] the reduction mkdv method [] the extended Jacobi elliptic function method [3 4] and the tri-function method [5 7] the G / G-expansion method [8] exp-function method [9 0] Hirota bilinear method [ 3] F-expansion method [4 6] homoclinic test method [7 30] and so on. Romanian Journal of Physics 6 6 (07)

2 Article no. 6 Yeqiong Shi It is known that there are multiple wave solutions to nonlinear equations for instance multi-soliton solutions to many physically significant equations including the Korteweg de Vries (KdV) equation and the Toda lattice equation [3] and multiple periodic wave solutions to the Hirota bilinear equations [3 33]. In recent years the subject of new types of waves in nonlinear science has attracted particular attention such as lump waves rogue waves and parity-time-symmetric waves. Some methods have been developed to search for various new exact solutions of nonlinear wave equations see also a series of relevant recently published papers in this rapidly expanding research field [34 6]. We would in this paper like to study exact solitary solutions for the two-dimensional potential Burgers equation by means of the extended homoclinic test method [6]. Burgers equations are encountered in many fields [63 65] such as fluid mechanics traffic flows acoustic transmission and the structure of shock waves. The Burgers equation had been extensively investigated [66 73]. The variable separation solutions were obtained for the (+)-dimensional Burgers system () in Refs. [74] and [75] and some other methods are applied on this system such as the multiple linear variable separation method [76] the tanh function method [77] the Jacobi elliptic function method [78] the simplest equation-riccati equation and variable separation method [79]. In this work we consider a (+)-dimensional potential Burgers equation by the homoclinic test method and we show that breather-type waves and other new types of exact solutions may also appear in this system but through a different mechanism when compared with those reported so far in the literature.. THE EXACT SOLUTIONS FOR THE (+)-DIMENSIONAL POTENTIAL BURGERS Now we consider the (+)-dimensional Burgers equation u uu av u bu abu uvy 0. t y x x yy xx We transform () into the potential Burgers system vyt vyvyy avxvxy bvyyy abvxxy 0 () where a and b are arbitrary constants. By using the Cole-Hopf transformation method we have the solution of the Eq. () as 0 () v b[ln( f )] (3) where 0 is an arbitrary known seed solution of Eq. (). The Burgers equation () x

3 3 Exact solutions of the (+)-dimensional potential Burgers system Article no. 6 can be cast into a bilinear form ( ff f ) a ( ff f f ) ff a ff 0 y yy y 0x xy x y 0 yy y 0xy x ( f f )( f bf abf ) 0. y y t yy xx We find the corresponding solution for the potential function U U( x y t) ( U v / b) ; therefore the potential solution form of the Burgers equation () reads y U f f f f f b y xy x y 0. In the following Section we will solve the two-dimensional potential Burgers system () by using some new test functions. (4) (5).. THE PERIODIC SOLITARY WAVE SOLUTION AND TWO-SOLITIONS SOLUTIONS In this Section we take the seed solution 0 as follows: (6) 0 mx nt.... The periodic solitary wave solution of eq. () With regard to Eq. (4) using the homoclinic test technique we can seek for the solution in the form where f d cos d exp( ) d exp( ) (7) 3 px k y qt px ky qt and d d d 3 p p k k q q are real constants to be determined. Substituting (6) and (7) into (4) and equating the coefficients of sin( ) e e cos( ) e cos( ) sin( )e to zero we obtain the sets of algebraic equations of the parameters d d a b m n k k q q d 3 p p. Solving these equations with the help of Maple we get three types of periodic breather wave solutions as follows.

4 Article no. 6 Yeqiong Shi 4 Case k a k 0 q amp q amp p p wherem n k p pare arbitrary real parameters that satisfy k 0 and p p 0. Substituting (8) into (7) using the potential (5) we get the periodic wave solution d Pk 4 d Pk B[cos( X )cosh( Y A ) sin( X )sinh( Y )] U (9) [ dcos( X ) cosh( Y )] where X px k y amp t Y px ky ampt 3 d ln d 3 d d (8). The typical structure of solutions (9) is shown in Fig. ; this is a breather-soliton wave. From (9) it can be seen clearly that the phase factor of periodic soliton is a function of time t in X-direction (here X px k y qt ). Thus it shows oscillation within the periodic solution in the X-direction that is the breathing effect (Fig. ). Case Fig. The periodic breather wave solution for the following parameters: b m n d d d k p p. 3 k k q ( p p ) qp a m q amp q (0) p p p ( k k ) p

5 5 Exact solutions of the (+)-dimensional potential Burgers system Article no. 6 where n d d d 3 k k are arbitrary constants that satisfy p p 0 k k 0. Substituting (0) into (7) using the potential (5) we have the X-periodic solitary wave solution d p 4 d B( p k k p )[sin( X )sinh( Y A ) cos X sinh( Y A )] U + [ dcos( X ) cosh( Y )] 8pkB cosh (Y A ) [ d cos( X ) cosh( Y )] () qp where X px ky t Y px ky ampt. p It can be seen from Fig. that the solution () is a periodic solitary wave solution with b n d d d3 k k p p q. Case 3 Fig. The periodic solitary wave solution for the following parameters b n d d d k k p p q. 3 q amp q ( k k ) b amp ab( p p ) m m n n d d d 0 d3 d3 k k k k p p p p. () Substituting (7) into (5) with () we obtain the periodic line solitary wave solution that spreads along the X-axis direction of propagation. It is a periodic solution and a Y-kink-type solitary wave solution:

6 Article no. 6 Yeqiong Shi 6 U d p k d d [( p k p k )cos( X ) ( p k p k )sin( X )](sinhy cosh Y ). 3 Y [ dcos( X ) d3e ] (3) Here X px k y amp t and Y p x k y [( k k ) b amp ab( p p )] t. Fig. 3 The singular periodic solitary wave solution (Eq. (3)) a b m n d d k k p p. with 3... The two soliton-type wave solutions With regard to Eq. (4) by means of the homoclinic test technique we can seek for the solution in the form f d sinh( ) d exp( ) d exp( ) (4) 3 where px k y qt px ky qt. Substituting (4) into (4) and equating the coefficients of cosh( ) e sinh( ) e cosh( ) e sinh( ) e to zero we obtain the set of algebraic equations for the parameters a b m n d d k k q q d 3 p p. Solving these equations with the help of Maple we get five sets of solutions as follows. Case k k ( k k ) m a p 0 q q 0 (5) p p

7 7 Exact solutions of the (+)-dimensional potential Burgers system Article no. 6 where m k k p are free parameters that satisfy p 0. Inserting (5) and (4) into (5) we have the singular two-solitary wave solution: U d k p 4 d p [ k sinh X cosh( Y A ) k cosh X sinh( Y )] (6) [ dsinh( X ) cosh( Y )] ( k where k) m X px ky t Y k y. p In Fig. 4 we show the shape of this solution (Eq. (6)) when taking b m n d d d k k p. 3 Case Fig. 4 The singular double solitary wave solution (Eq. (6)) with b m n d d d k k p. 3 d d k k p p q amp q q q. (7) 4 d3 Substituting (7) and (4) into (5) we get the linear travelling solitary wave solution as follows

8 Article no. 6 Yeqiong Shi 8 U pk[ cosh( X Y ln )] d3 [sinh( X) sinh( Y ln )] d d d3 where m k k p q p d d 3 are arbitrary constants that satisfy dd 3 0. (8) Case 3 Fig. 5 The travelling solitary wave (Eq. (8)) with a b m n d 4 d k 0.0 p q. k a ( p ) p 3 k mp k mp p p q q ( ) ( ) p p p p k 0. (9) Substituting (9) into (4) we have the double solitary wave solution from Eq. (5) U d p k 4d p k Bsinh X cosh( Y A ) 4 p d k sinh( Y ) cosh X [ dsinh( X ) cosh( Y )] (0) where m k p p d d 3 n are arbitrary constants that satisfy

9 9 Exact solutions of the (+)-dimensional potential Burgers system Article no. 6 p X px ky p 0 k mp t ( p ) p k mp Y p x t. ( ) p p Case 4 k k a ( p ) p q ( p p ) qp m q n n. () p ( k ) p k Substituting () into (4) using the potential (5) we have the two solitary wave solution U d p k d k B( p p )sinh( X Y A ) 4 p k B cosh(y ) () [ dsinh( X ) cosh( Y )] qp where X px ky t Y px k y qt m k k p p d n are p arbitrary constants that satisfy ( k k )( p p ) 0. U Case 5 k k mp k mp a p p q q p p p p p p m m n n d d d d d3 d3 k k k p p 0. (3) Substituting (3) and (4) into Eq. (5) we get the exact solution (8 p k d p k ) 4 p k d sinh X cosh( Y A ) 4d p k B cosh X sinh( Y A ) (4) [ dsinh( X ) cosh( Y )] k where mp k X px ky t mp Y p x ky t p p p p p p 0... THE INTERACTION RESONANCE-TYPE SOLITON SOLUTIONS In this Section we take the seed solution 0 as follows 0 0. (5)

10 Article no. 6 Yeqiong Shi 0 By means of the homoclinic test technique we can take the test function to be where f d sin( )sinh( ) d e d e (6) 3 px k y qt px ky qt. Substituting (6) into (4) with (5) and equating the coefficients of cos ( ) cosh( ) cos( ) e sinh( ) e cos( ) cosh( ) e cos( ) sinh( ) e cos( ) sinh( )sin( ) cosh( ) sin( ) e sinh( )sin( ) e cosh( ) sin( ) e sinh( ) sin( ) e cosh ( ) sinh( ) cosh( ) to zero we obtain the sets of algebraic equations for the parameters a b m n d d k k q q d 3 p p. Solving these equations with the help of Maple software we can get k k p a d d d d d d k k k p p p p q 0 q 0 p (7) 3 3 p Substituting (7) and (6) using the potential (5) we obtain the singular breather-like solution with double-ring shape see Fig. 6 for the set of parameters: k p k p b d d d3. Fig. 6 The singular double-ring wave with the parameters b d d d k k p p. 3

11 Exact solutions of the (+)-dimensional potential Burgers system Article no The two-soliton fusion interaction wave solution We set the test function as f d cosh ( p x k y q t) d e d e (8) where p x k y q t pxk yqt pxk yqt 3 p x k y q t are real constants to be determined. Proceeding as before using the potential (5) we obtain k a d d d 0 d3 d3 k k k k p p p p q 0 p q b( k p k p ) (9) p where d d 3 p p k k are free parameters. Substituting (8) into (5) with (9) we get the two-soliton fusion-type interaction wave solution see Fig. 7 for the set of parameters b d d3 k k p p. Fig. 7 The two-soliton fusion interaction wave for b d d k k p p. 3

12 Article no. 6 Yeqiong Shi Let the test function be... The double resonance-type solutions f d sin ( ) d sinh ( ) d exp( ) d exp( ) (30) 0 3 where p x k y q t px ky qt and d 0 d d d 3 p p k k q q are real constants to be determined. Proceeding as before using the potential (5) we can get three sets of solutions: Case k n( k p k p ) p q d 0 d d k k k k 3 3 a p p p p p q 0 d0 d0 d d. (3) Case k k p a d d d d d d d d k k k p p p p p p q q 0. (3) 0 Case 3 q abp bk d d d d d d d 0 k 0 k k p 0 p p q 0. (33) Substituting (3) into (30) using the potential (5) we obtain the double-kink resonance wave solution see Fig. 8 for the following set of solution parameters: b d d d k k p p. 3 Substituting (3) into (30) using the potential (5) we can get the double peakon wave solution see Fig. 9 for the following set of solution parameters: b d d d k k p p. 3 Inserting (3) into (33) using the potential (5) we can get the Y-type resonance soliton wave solution see Fig. 0 for the following set of solution parameters: b d d d3 k k p p.

13 3 Exact solutions of the (+)-dimensional potential Burgers system Article no. 6 Fig. 8 The double-kink resonance wave. Fig. 9 The double resonance peakon wave.

14 Article no. 6 Yeqiong Shi 4 Fig. 0 The Y-type interaction resonance soliton with the parameters: b d d d k k p p CONCLUSION The (+)-dimensional potential Burgers system has been investigated by means of the extended homoclinic test method with different test functions. With the help of Maple and Matlab routines some new types of exact solutions of the (+)-dimensional potential Burgers system have been obtained such as periodic breather-type wave solutions two-soliton resonance two-soliton fusion wave solutions Y-type resonance wave solutions etc. These exact solutions may be of significant importance for the study of a variety of physical phenomena described by the potential Burgers system. Acknowledgments. This work is supported by the NSF of Guangxi under Grant No. 04GXNSF-AA803 Guangxi DER under Grant No. 03YB78. REFERENCES. W. Malfleit Am. J. Phys (99).. E. Fan Phys. Lett. A 77 (000). 3. C. Yan Phys. Lett. A 4 77 (996). 4. Z. Yan H. Zhang Phys. Lett. A 5 5 (999). 5. Z. Yan Phys. Lett. A (00).

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