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The utilization of a one-parameter gathering of microscopic changes decreases the quantity of autonomous factors by one, and therefore, the arrangement of fractional differential conditions, in two free factors lessens to an arrangement of standard differential conditions. The acquired differential conditions are fathomed in some unique cases. The outcomes are shown graphically for various parameters.
International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 12, December 2019, pp 264-283 Article ID: IJMET_10_12_030 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=12 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication THE TRADEMARK CAPACITY STRATEGY AND THE ARRANGEMENTS OF NONLINEAR HIROTA-SATSUMA CONDITIONS Medhat M Helal Civil Engineering Dept., College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, Saudi Arabia Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt A I Ismail Mechanical Engineering Dept., College of Engineering and Islamic Architecture, Umm AlQura University, Makkah, Saudi Arabia Faculty of Science, Mathematics Department, Tanta University, P.O Box 31527, Tanta, Egypt ABSTRACT The trademark work strategy has been utilized to decide and research certain classes of arrangement of an arrangement of third request non-straight of HirotaSatsuma conditions The outstanding Hirota-Satsuma coupled KdV condition are assessed and the purported summed up Hirota-Satsuma coupled KdV framework is likewise considered The utilization of a one-parameter gathering of microscopic changes decreases the quantity of autonomous factors by one, and therefore, the arrangement of fractional differential conditions, in two free factors lessens to an arrangement of standard differential conditions The acquired differential conditions are fathomed in some unique cases The outcomes are shown graphically for various parameters Key words: Trademark work technique; Hirota–Satsuma; KdV condition Cite this Article Medhat M Helal and A I Ismail, the Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions International Journal of Mechanical Engineering and Technology, 10(12), 2019, pp 264-283 http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=12 INTRODUCTION In this investigation, we consider two coupled KdV conditions were presented by Hirota– Satsuma [1] and an issue with three possibilities, inferred by Wu et al [2] Various research have been accomplished for the Hirota–Satsuma coupled KdV equations (1) – (2) Eqs (1)–(2) http://www.iaeme.com/IJMET/index.asp 264 editor@iaeme.com Medhat M Helal and A I Ismail are found as models emerging from the Drinfeld–Sokolov order [5], [7] and have been examined by different methodologies, for example, the bilinear strategy [3], [4], Lax pair [6], [8], [9], Bäcklund change [11], Darboux change [12], [13], [14], Painlevé property [9], [10] and interminably numerous balances and preservation laws [15] Soliton, intermittent and different sorts of arrangements were built by different strategies [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] The motivation behind this paper is to give a deliberate treatment of trademark work strategy for dissecting and ordering the arrangement of some nonlinear fractional differential conditions it is outstanding that the nonlinear halfway differential conditions are generally used to portray numerous significant wonders in material science, science, science, and so forth The Hirota–Satsuma conditions play a vital standard in applied arithmetic and material science and have numerous applications in material science and Building The trademark work technique [29, 30, 31] is an earth shattering, adaptable and essential to the improvement of efficient frameworks that lead to invariant course of action of the nonstraight issues Since the trademark work methodology didn't rely upon straight chairmen, superposition or various requirements of the immediate course of action methodologies, it is material to both immediate and nonlinear differential models The numerical framework in the present examination is the one-parameter bundle change The trademark work method, all things considered, is a class of progress which diminishes the amount of free factors in specific structures of deficient differential conditions by one The upsides of this procedure are relied upon to consider the course of action of partial differential conditions as a plan of arithmetical conditions, and in reducing the amount of free factors by one, it is possible to obtain another game plan of deficient differential conditions with/without continuing to get basic differential conditions, therefore, the trademark work strategy yields all out results with less effort Therefore, it is material to understand a progressively broad collection of nonlinear issues THE HIROTA-SATSUMA COUPLED KDV SYSTEMS In this paper we think about the issue for the accompanying framework u t a (6 uu x u x ,x ,x ) 2b x , (1) t xxx c ux d x , (2) wherever a, b, c, d stand actual numbers, and u, v are actual - esteemed functions of the double variables x and t once c , d , the scheme (1), (2) decreases to the subsequent ut a (6 uu x u x ,x ,x ) 2b x , (3) t xxx 3ux , (4) Which was planned by Hirota and Satsuma [1] to classical the communication of dual extended waves with different diffusion relations The original surroundings are traditional on the disposed line X : A x B C as u (X ,0) (X ) and (X ,0) (X ) (5) Anywhere A , B and C are numbers http://www.iaeme.com/IJMET/index.asp 265 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions THE GROUP METHODICAL PREPARATION To apply the trademark technique to the Hirota and Satsuma Eqs (1),(2) we consider the oneparameter gathering of little changes in (x , t , u , ) specified by x x 1 (t , x , u , ) O ( ) t t (t , x , u , ) O ( ) u u 1 (t , x , u , ) O ( ) (t , x , u , ) O ( ) u i u i 1i (t , x ,u , , u x , x , u t , t ) O ( ) i i i2 (t , x ,u , , u x , x , u t , t ) O ( ) u ij u ij 1ij (t , x ,u , , u x , x , , u xx , xx , ) O ( ) ij ij 2ij (t , x ,u , , u x , x , , u xx , xx , ) O ( ) u ijk u ijk 1ijk (t , x ,u , , u x , x , , u xx , xx , , u xxx , xxx ) O ( ) ijk ijk 1ijk (t , x ,u , , u x , x , , u xx , xx , , u xxx , xxx ) O ( ) (6) Wherever “ ” is the trivial group constraint, 1, 2 , 1, 2 , 1i , 1ij , 1ijk , i2 , ij2 , ijk are the infinitesimals of the group of conversions and i , j and k for x, t THE TRADEMARK WORK STRATEGY To produce the trademark functions "W " and "W " , let u (x , t ) u (x , t , u , , ) , (x , t ) (x , t , u , , ) , (7) Regarding the infinitesimals, Eqs (7) An be composed as u (x 1 , t ) u (x , t ) 1 (x , t , u , ) O ( ) , (x 1 , t ) (x , t ) (x , t , u , ) O ( ) , (8) Extending the left-hand sides of Eqs (8), we acquire http://www.iaeme.com/IJMET/index.asp 266 editor@iaeme.com Medhat M Helal and A I Ismail u u u (x 1 , t ) u (x , t ) 1 2 t x (x 1 , t 2 ) (x , t ) 1 O ( ) , x 2 O ( ) , t t (9) From Eqs (8) and (9), likening the coefficients of , we get 1 (t , x ,u , ) u u (t , x , u , ) 1 (t , x , u , ) , x t 1 (t , x ,u , ) (t , x , u , ) (t , x , u , ) , x t (10) Characterize the trademark capacities W1 and W2 as W1 (t , x ,u ,, u x , x , ut , t ) 1(t , x ,u ,) u x 2 (t , x ,u ,) ut 1(t , x ,u ,) , W (t , x ,u ,, u x , x , ut , t ) 1(t , x ,u ,) x 2 (t , x ,u ,) t 2 (t , x ,u ,) , (11) Where the infinitesimals of the group of transformations 1, 2 , 1 and 2 express in terms of the two the trademark work strategy functions W and W as: 1 1 W W W W , 2 , ut t ux x W W W W ux x ut W , t W u x x ut t (12) The three expressions of the above equations ( 1 , 2 1 and 2 ) comes from the direct derivative of Eq (11) The components 1i , i2 , 1ij , 2ij , 1ijk and 2ijk for i , j and k stand for x, t can be determined from the following expressions: 1i D i (1 ) u x D i (1 ) u t D i ( ) , i2 D i ( ) x D i (1 ) t D i ( ) , 1ji Di (1j ) u jx Di (1 ) u jt Di (2 ) , 2ji Di (2j ) jx Di (1 ) jt Di (2 ) , http://www.iaeme.com/IJMET/index.asp 267 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions 1kji Di (1kj ) u kjx Di (1 ) u kjt Di (2 ) , kj kji Di ( ) kjx Di (1 ) kjt Di ( ) , (13) where D is total derivative define by Di i u i u i u ix u x ix x u it ut it t , THE INVARIANCE EXAMINATION Under the infinitesimal group of conversion, the system of differential equations(1), (2) on the form Gi = 0, for i 1, , will be invariant if DGi = 0, where the operator D is written as D 1 2t 2 1 2 1x 1t 2x x t u u x ut x 1xx xx 1xt 2xt 1tt u xx xx u xt xt u tt t tt2 1xxx 1xxt 1xtt 1ttt u xxx u xxt u xtt u ttt tt xxx xxx 2xxt xxt 2xtt xtt ttt2 ttt (14) Apply the operator D from the Eq (14) on G i of Eqs (1), (2), gives the accompanying arrangement of straight incomplete differential conditions 1t 6a 1 u x 6a 1x u 2b x 2b x2 a 1xxx , 2t c 1 x c 2x u d x d 2x 2xxx (15) Utilizing the articulations given by Eqs (13), and comparing to zero the coefficients of subordinates of u and , we get the deciding conditions: 2b2u 1 a1 , a1xxx 3a1uxx 12au1x 6a1 2b2u 1t , 3a1xx 2b 2 6au 1 2b2 2b1u 4b1x acu 1 ad 1 , 6au 1x 2b2x 1t a1xxx , 6a1uu , 3a1 , a2 , 3a2xx , 3a2u , 3a1u 3a1x , a2uuu , 3a2xx 6au2 31 2b2u acu2 ad 2 , 3a2x , a2xxx 2t 31x , a2 2 , http://www.iaeme.com/IJMET/index.asp 268 6a2ux , 3a1xx 3a1ux , 3a1uxx 3a1uxx 18au1u , editor@iaeme.com Medhat M Helal and A I Ismail 3a1uux a1uuu , 3a1x 4b1 , 3a1x , a1uuu , 31u 6au2u 6au2u , a1 , 9a1ux 3a1uu , 3a1uu 6a1ux , 3a2uu , 3a1x 3a1u , 3a1xx 6a1ux 12au1 2b1u 8b1u acu1 ad 1 , 1 2b2x a1 , 3a1u , 6a2x , 3a2 , 3a2x , 3a2x , 6a1x 3a1u , 3a2uxx , 3a1u , a1 , 3a2uu , 3a2ux , 2b2 , 3a1u , 3a2u , 3a1 , 3a2u , 9a1u , 3a1 , 61ux 32u , cu 2u 6u 2u 18u1x d 2u 32uxx , acu1 3a2x ad 1 8b1 2b1u 3a1xx , cau1x a1xxx a1t ca1 ad 1x 2b2u 6b1x 3a2xx da2 , 32uu , 32 , 32ux , 31uu , 32u 31ux , cau2 a1 da2 2b2u 2b2 3a2xx , 62ux , 62ux , 91u , 32u , da1u 6a2ux 6b1u 24au1 6au1u acu1u 3a1uxx , 31uu , 32uux 18u1u , 2 , 2t d 2x cu2x 2xxx , 32u , cu 2x d 2x 2t 2xxx , 32x , 1uuu , 32 , 32uu , 62 , 1 , 31u , 32x , 31u , 31x 2 , d 2u 6u 2u 6u 2 32uxx cu 2u , 2uuu , 31xx 32x , 2 , 32u 61ux , 41 21u , 32ux , 31 , http://www.iaeme.com/IJMET/index.asp 269 32uu , 91x 32 , 31u , 32u , editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions 32 , 32uu , 32uux , 32xx , 31u , 62u , 32x , 32u , 32 , 61 , 2xxx , cu 2u , (16) Tackling the subsequent conditions, we get 1 k + k3 x , 2 k + k t , 1 k3 u , 2 k3 (17) Now, the trademark functions define by Eqs (11) Become W1 k1+ k x u x k + k t ut k u , W k1+ k x x k + k t t k , (18) We infer that the arrangement of Eqs (1), (2) will be invariant under a minute gathering of changes, if the trademark capacities, W and W are of the structure(18) In this manner, it should now be conceivable to decrease the quantity of autonomous factors by one The trademark arrangement of Eqs (1), (2) is given by: dx dt du d , 2 k + k t k1+ k x k3 u k3 3 (19) THE DROP TO ORDINARY DIFFERENTIAL EQUATIONS Our point is to utilize the trademark arrangement to speak to the issue as conventional differential conditions At that point we need to continue our investigation to finish the change in three cases Case 1: k3 ≠ Similarity transformation can be obtained by solving the trademark system (19) which gives the similarity for the independent variables that are: = k2 k3 t , k (3k k x )3 http://www.iaeme.com/IJMET/index.asp (20) 270 editor@iaeme.com Medhat M Helal and A I Ismail And the similarity for the dependent variables are: F1 ( ) (3k k x )2 u (x , t ) , (21) F2 ( ) (3k k x )2 (x , t ) , (22) Where F1 ( ) and F2 ( ) are arbitrary functions of By relieving the self-similar variables u, and w (in ) from the Eqs (20)-(22) in Eqs (1) And(2), we obtain a system of partial differential equations with one independent variables : 27ak 3 3F 162ak 3 F 186ak 3 F 6bk F F 3 3 2 18ak 3 F1F1 24ak 33F1 F1 4bk 3F22 12ak 3F12 , 27k 3 3F 162k 3 F k F 24k 2cF 3d F 3 3 (23) 2dk 3F22 1 186k 33 3ck 3 F1 F2 , (24) The initial conditions on the line 1 reduce to u (1 ) (1 ) and (1 ) (1 ) (25) Now we study the diagrams got by means of the fourth-order Runge-Kutta Method using big value for k The important of choice large value for k to deal with the singularity of the above equation, i.e we take k as The diagrams are as follows: http://www.iaeme.com/IJMET/index.asp 271 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Figure (1) Solution of Hirota and Satsuma equations (23) and (24) at k 10 , a b , c , d Case 2: k3 = and k k Similarity transformation in this case at k3 = can be attained by resolving the trademark system: d x dt du d , k1 k2 0 (26) The similarity for the independent variables is x k1 t, k2 http://www.iaeme.com/IJMET/index.asp (27) 272 editor@iaeme.com Medhat M Helal and A I Ismail And the similarity for the dependent variables are: u (x ,t ) F1 ( ) , (28) (x , t ) F2 ( ) , (29) We obtain a system of partial differential equations with one independent variables aF1 6aF1F1 k1 F1 2bF2 F2 , k2 (30) k F2 cF1F2 F2 dF2 F2 , k2 (31) The arrangements of the above conditions are plotted in Fig (2) With two diverse beginning qualities The figures showed that the vibrations decline with the expansion of the underlying qualities THE NEW GENERALIZED HIROTA–SATSUMA COUPLED KDV EQUATION As of late, by presenting an issue with three possibilities, Wu et al [2] determined another chain of command of nonlinear development conditions; two normal conditions in the progression are another summed up Hirota–Satsuma coupled KdV condition u t u xxx uu x 3(w ) x , (32) t xxx 3ux , (33) w t w xxx 3uw x , (34) http://www.iaeme.com/IJMET/index.asp 273 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Figure (2) Solution of Hirota and Satsuma equations (30) and (31) with two different initial conditions The infinitesimal transformation 1 k + k 4x , (35) 2 k + k 4t , 1 (36) k4 u , (37) 2 k , (38) The trademark system of Eqs (1), (2) is given by: 3 (3k 4k ) w , (39) dx dt du d dw (40) k + k t k k2+ k4 x k4 u (3k 4k ) w 3 We have to proceed our analysis to complete the transformation in three cases Case 1: k4 ≠ Similarity transformation can be obtained by solving the trademark system (40) which gives the similarity for the independent variables that are: http://www.iaeme.com/IJMET/index.asp 274 editor@iaeme.com Medhat M Helal and A I Ismail = 3k k x (k k t ) , (41) And the similarity for the dependent variables are: F1 ( ) (k k t ) u (x , t ) , F2 ( ) (k k t ) F3 ( ) (k k t ) k1 k4 (42) (x , t ) , k1 k4 (43) w (x , t ) , (44) Where F1 ( ) , F2 ( ) and F3 ( ) are arbitrary functions of By substituting the self-similar variables u, and w (in ) from the Eqs (41)-(44) In Eqs (32) - (34), we obtain a system of partial differential equations with one independent variables : F1 1 2 F1 4F1 18F1F1 18F3F2 18F2 F3 , 3k 42 F2 3k 42 3k F2 9F1F2 , F2 k4 (46) F3 3k 42 3k F3 9F1F3 4F3 , F3 k4 (47) (45) Fig (3) displays the solutions of Eqs (45)-(47), we take k but with two different values for k When k to be smaller, say k 0.5 , the vibrations of the variables F1 , F2 , F3 , F1 , F2 , F3 , F1 , F2 and F3 increase http://www.iaeme.com/IJMET/index.asp 275 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions http://www.iaeme.com/IJMET/index.asp 276 editor@iaeme.com Medhat M Helal and A I Ismail http://www.iaeme.com/IJMET/index.asp 277 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Figure (3) Solution of Hirota and Satsuma equations (45)-(47) with two different parameter k 0.5 and k at k Case 2: k2 = k4 = and k3 ≠ Similarity transformation on the singular case at k2 = k4 = can be derived by solving the trademark system: d x dt du d dw , k3 k k w (48) Which gives the similarity for the independent variables, that is: = x, (49) And the similarity for the dependent variables are: u F1 , e (50) k1 t k3 w e F2 , k1 t k3 (51) F3 , (52) We acquire an arrangement of halfway differential conditions with one free factor F1 3F1F1 3F2F3 3F2 F3 , (53) F2 F2 3F1F2 , (54) F3 F3 3F1F3 , (55) http://www.iaeme.com/IJMET/index.asp 278 editor@iaeme.com Medhat M Helal and A I Ismail Figure (4) Solution of Hirota and Satsuma equations (53)-(55) http://www.iaeme.com/IJMET/index.asp 279 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Case 3: k3 = k4 = and k2 ≠ For this special case, the characteristic system is given by: d x dt du d dw , k2 0 k k w (0.56) Gives the similarity variables; t, (0.57) The similarity forms are; u F1 , e (0.58) k1 x k2 w e F2 , k1 x k2 (0.59) F3 (0.60) Substituting Eqs (0.57)-(0.60) In Eqs Error! Reference source not found.Error! Reference source not found., we obtain: F1 , (0.61) k k F2 F2 F1F2 , k2 k2 (0.62) k k F3 F3 F1F3 k2 k2 In this case we obtain an exact solution as (0.63) F1 u , F2 0 e (0.64) k1 k1 3u t k2 k2 F3 w e , (0.65) k k 3u t k k (0.66) where u , 0 and w being arbitrary constants The exact solutions of the coupled KdV equations Error! Reference source not found.-Error! Reference source not found will be given be substituting Eqs (0.64)-(0.66) Into the similarity forms (0.58)-(0.60) as u (x , t ) u , ( x , t ) 0 e (0.67) k k u t x k2 k2 w (x , t ) w e , k k u t x k k (0.68) (0.69) For instance, if we take k / k i C , where i 1 and C a constant, then the above three equations become http://www.iaeme.com/IJMET/index.asp 280 editor@iaeme.com Medhat M Helal and A I Ismail u (x , t ) u , (0.70) ( x , t ) 0 e i C t x , (0.71) w (x , t ) w e i C t x , (0.72) where k u k2 u (x ,0) u , So we can give a new exact solution with initial conditions in the form (x ,0) (x ), w (x ,0) (x ) , (0.73) By using Fourier and inverse Fourier transforms, the complete solution of Eqs (0.70)-(0.72) become u (x , t ) u , (0.74) iC t x iC d e , ( ) e 2 (x , t ) (0.75) iC t x iC w (x , t ) d e , (0.76) ( ) e 2 Case 4: k4 = and k3 ≠ Similarity transformation in this case at k4 = can be obtained by solving the characteristic system: d x dt du d dw , k2 k3 k k w (0.77) The similarity for the independent variables is x k2 t, k3 (0.78) And the similarity for the dependent variables are: u F1 , e (0.79) k1 t k3 w e F2 , k1 t k3 (0.80) F3 (0.81) We obtain a system of partial differential equations with one independent variables k F1 3F1F1 3F2F3 3F2 F3 F1 , k3 (0.82) k1 k F2 F2 3F1F2 , k3 k3 (0.83) F2 F3 k1 k F3 F3 3F1F3 , k3 k3 http://www.iaeme.com/IJMET/index.asp (0.84) 281 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions CONCLUSION In this paper, an answer philosophy dependent on the trademark work strategy has been applied to unravel third request non-straight of Hirota-Satsuma conditions By deciding the change bunch under which a given halfway differential conditions are invariant, we acquired the likeness factors that decreased the quantity of autonomous factors The subsequent frameworks of non-direct common differential Eqs Are fathomed systematically and numerically with its limit conditions We show the outcomes with certain investigations together with the impacts of various parameters The proposed examination performed here demonstrates the viability of this strategy in getting likeness factors that might be utilized to decrease the quantity of autonomous factors in incomplete respectful conditions, and the computations are likewise basic and direct The outcomes demonstrate that the trademark work strategy is a ground-breaking numerical device for tackling direct and nonlinear incomplete differential conditions, and in this manner, can be generally applied in building issues ACKNOWLEDGEMENTS The authors would like to express their gratitude to the Umm Al-Qura University for offering a research grant to support this work The authors would like to express their gratitude and thanks to the reviewers of the manuscript for their critical review and valuable comments that modified the paper greatly REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Hirota, R and Satsuma, J., Soliton solutions of a coupled KdV equation Phys Lett A 85, (1981), 407-409 Wu,Y T., Geng, X G., Hu, X B and Zhu, S M., A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformations, Phys Lett A 255 (1999), 259-264 Seshadri, R and Na, T.Y., Group invariance in engineering boundary value problems, Springer-Verlag, New York (1985) J Satsuma, R Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J Phys Soc Japan, 51 (1982), pp 3390-3397 V.G Drinfeld, V.V Sokolov, New evolutionary equations possessing an (L,A)-pair Partial Differential Equations, Proc S.L Sobolev seminar, No 2, Inst of Math., Siberian Branch of the USSR Acad Sci., Novosibirsk (1981) (in Russian) M Gürses, A Karasu, Integrable KdV systems: recursion operators of degree four, Phys Lett A, 251 (1999), pp 247-249 V Drinfel’d, V Sokolov, Lie algebras and equations of Korteweg–de Vries type, J Soviet Math., 30 (1985), pp 1975-2036 H.W Tam, W.X Ma, X.B Hu, D.L Wang, The Hirota–Satsuma coupled KdV equation and a coupled Ito system revisited, J Phys Soc Japan, 69 (2000), pp 45-52 R Dodd, A Fordy, on the integrability of a system of coupled KdV equations, Phys Lett A, 89 (1982), pp 168-170 J Weiss, the Sine-Gordon equations: complete and partial integrability, J Math Phys., 25 (1984), pp 2226-2235 J Weiss, Modified equations, rational solutions, and the Painlevé property for the Kadomtsev–Petviashvili and Hirota–Satsuma equations, J Math Phys., 26 (1985), pp 2174-2180 D Levi, A hierarchy of coupled Korteweg–de Vries equations, Phys Lett A, 95 (1983), pp 7-10 http://www.iaeme.com/IJMET/index.asp 282 editor@iaeme.com Medhat M Helal and A I Ismail [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] S Leble, N Ustinov, Darboux transforms, deep reductions and solitons, J Phys A: Math Gen., 26 (1993), p 5007 H.C Hu, Q Liu, New darboux transformation for Hirota–Satsuma coupled KdV system, Chaos Solitons Fractals, 17 (2003), pp 921-928 H Hu, Y Liu, New positon, negaton and complexiton solutions for the Hirota–Satsuma coupled KdV system, Phys Lett A, 372 (2008), pp 5795-5798 W Oevel, on the integrability of the Hirota–Satsuma system, Phys Lett A, 94 (1983), pp 404-407 E Fan, Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled mKdV equation, Phys Lett A, 282 (2001), pp 18-22 E Fan, B.Y Hon, Double periodic solutions with jacobi elliptic functions for two generalized Hirota–Satsuma coupled KdV systems, Phys Lett A, 292 (2002), pp 335-337 E Zayed, H.A Zedan, K.~A Gepreel, On the solitary wave solutions for nonlinear Hirota– Satsuma coupled KdV of equations, Chaos Solitons Fractals, 22 (2004), pp 285-303 Y Chen, Z Yan, Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system, Appl Math Comput., 177 (2006), pp 85-91 Y Chen, Z Yan, B Li, H Zhang, New explicit exact solutions for a generalized Hirota– Satsuma coupled KdV system and a coupled mKdV equation, Chin Phys., 12 (2003), p Y Yu, Q Wang, H Zhang, The extended jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equations, Chaos Solitons Fractals, 26 (2005), pp 1415-1421 D Ganji, M Rafei, Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method, Phys Lett A, 356 (2006), pp 131-137 X Geng, H Ren, G He, Darboux transformation for a generalized Hirota–Satsuma coupled Korteweg–de Vries equation, Phys Rev E, 79 (2009), p 056602 J.P Wu, X.G Geng, X.L Zhang, N-soliton solution of a generalized Hirota–Satsuma coupled KdV equation and its reduction, Chin Phys Lett., 26 (2009), p 020202 R Hirota, the Direct Method in Soliton Theory, Cambridge University Press (2004) M Iwao, R Hirota, Soliton solutions of a coupled modified KdV equations, J Phys Soc Japan, 66 (1997), pp 577-588 R Hirota, X.B Hu, X.Y Tang, A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and lax pairs, J Math Anal Appl., 288 (2003), pp 326-348 W.F Ames, Similarity for nonlinear diffusion equation, I & EcFund (1965) 72–76 W.F Ames, M.C Nucci, Analysis of fluid equations by group methods, J Eng Math 20 (1985) 181–187 G.W Bluman, J.D Cole, Similarity Methods of Differential Equations, Springer, New York, 1974 http://www.iaeme.com/IJMET/index.asp 283 editor@iaeme.com ... editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Figure (2) Solution of Hirota and Satsuma equations (30) and (31) with two different initial conditions. .. 271 editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions Figure (1) Solution of Hirota and Satsuma equations (23) and (24) at k 10 ,... editor@iaeme.com The Trademark Capacity Strategy and the Arrangements of Nonlinear Hirota-Satsuma Conditions CONCLUSION In this paper, an answer philosophy dependent on the trademark work strategy has