Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

Hülya Durur; Asif Yokuş

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

Journal Details

Format: Journal
eISSN: 2444-8656
First Published: 01 Jan 2016
Publication timeframe: 2 times per year
Languages: English

Different studies can be seen on NPDEs in various disciplines such as physical events, applied sciences, natural events and engineering. Scientists have used many methods to find analytical solutions of NPDEs. Some of these methods are new sub equation method [1, 2], (1/G′)-expansion method [3, 4], Homotopy analysis and Homotopy-Pade methods [5], (G′/G)-expansion method [6, 7], Variational Iteration Algorithm-I [8], decomposition method [9,10,11], sumudu transform method [12], sub equation method [13, 14], collocation method [15], the auto-Bäcklund transformation method [16], the Clarkson-Kruskal (CK) direct method [17], first integral method [18], homogeneous balance method [19], SGEEM [20], residual power series method [21], Modified Kudryashov method [22], sine-Gordon expansion method (SGEM) [23, 24] and so on [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].

Many analytical methods have been added to the literature to find the exact solution of the NPDEs. We know that each method produces different types of solutions. For example, sub equation method produces dark solitons, Hirota bilinear form lump solitons. In this study, we aim to produce hyperbolic type wave solutions using the (1/G′)-expansion method. For this, we have dealt with the AKNS equation with strong nonlinearity. So let's take the AKNS equation [47], (1) $4 u_{xt} + u_{xxxz} + 8 u_{xz} u_{x} + 4 u_{z} u_{xx} = 0 .$ 4{u_{xt}} + {u_{xxxz}} + 8{u_{xz}}{u_x} + 4{u_z}{u_{xx}} = 0. Scientists related to the equation of AKNS equation have produced several studies in the literature. Some of these are obtained traveling wave reductions of AKNS equation applying classical Lie symmetries [48], new soliton like solutions of AKNS equation obtained using the extended auxiliary equation method [49], the truncated Painlevè analysis of AKNS equation obtained with their symmetries [50], complex combined dark-bright soliton solutions of AKNS equation attained using SGEM [51], general solutions of the AKNS system have been obtained via standard truncated Painlevè expansions [52].

Description of the Method

Consider form of NPDEs, (2) $† (u, \frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial z}, \frac{\partial^{2} u}{\partial x^{2}}, ...) = 0 .$ \dagger \left( {u,{{\partial u} \over {\partial t}},{{\partial u} \over {\partial x}},{{\partial u} \over {\partial z}},{{{\partial ^2}u} \over {\partial {x^2}}},...} \right) = 0. Let ξ = x + z − ct, u(x,z,t) = u(ξ) = u, where c is a constant and the speed of the wave. We can convert it into the following nODE (3) $θ (u, - {cu}^{'}, u^{″}, u^{‴}, ...) = 0 .$ \theta \left( {u, - cu',u'',u''',...} \right) = 0. The solution that makes the Eq. (3) correct is assumed to have the form (4) $u (ξ) = a_{0} + \sum_{i = 1}^{m} a_{i} {(\frac{1}{G^{'}})}^{i},$ u\left( \xi \right) = {a_0} + \sum\limits_{i = 1}^m {a_i}{\left( {{1 \over {G'}}} \right)^i}, where a_i, (i = 0,1,...,m) are constants, G = G(ξ) provides the following second order IODE (5) $G^{″} + λ G^{'} + μ = 0,$ G'' + \lambda G' + \mu = 0, where λ and μ are constants to be determined after, (6) $\frac{1}{G^{'} (ξ)} = \frac{1}{- \frac{μ}{λ} + B cosh (ξ λ) - B sinh (ξ λ)},$ {1 \over {G'\left( \xi \right)}} = {1 \over { - {\mu \over \lambda } + B\cosh \left( {\xi \lambda } \right) - B\sinh \left( {\xi \lambda } \right)}}, where B is integral constant. If the desired derivatives of the Eq. (4) are calculated and substituting in the Eq. (3), a polynomial with the argument (1/G′) is attained. An algebraic equation system is created by equalizing the coefficients of this polynomial to zero. The equation are solved using package program and put into place in the default Eq. (3) solution function. Lastly, the solutions of Eq. (2) are found.

Solutions of The AKNS equation

Let's consider the Eq. (1). By applying the transmutation ξ = x + z − ct, allows us to convert Eq. (1) into an nonlinear ODE (7) $- 4 {cu}^{'} + u^{‴} + 6 {(u^{'})}^{2} = 0 .$ - 4cu' + u''' + 6{\left( {u'} \right)^2} = 0. In Eq. (7), m = 1 is found from the definition of balancing term and written according to Eq. (4) (8) $u (ξ) = a_{0} + a_{1} (\frac{1}{G^{'}}), a_{1} \neq 0 .$ u\left( \xi \right) = {a_0} + {a_1}\left( {{1 \over {G'}}} \right),\quad {a_1} \ne 0. If Eq. (8) is replacing in Eq. (7) and the coefficients are equal to zero, the following system is attained. (9) $\begin{array}{l} {(\frac{1}{G^{'} [ξ]})}^{1} : - 4 c λ a_{1} + λ^{3} a_{1} = 0, \\ {(\frac{1}{G^{'} [ξ]})}^{2} : - 4 c μ a_{1} + 7 λ^{2} μ a_{1} + 6 λ^{2} a_{1}^{2} = 0, \\ {(\frac{1}{G^{'} [ξ]})}^{3} : 12 λ μ^{2} a_{1} + 12 λ μ a_{1}^{2} = 0, \\ {(\frac{1}{G^{'} [ξ]})}^{4} : 6 μ^{3} a_{1} + 6 μ^{2} a_{1}^{2} = 0 . \end{array}$ \matrix{ {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^1}: - 4c\lambda {a_1} + {\lambda ^3}{a_1} = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^2}: - 4c\mu {a_1} + 7{\lambda ^2}\mu {a_1} + 6{\lambda ^2}a_1^2 = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^3}:12\lambda {\mu ^2}{a_1} + 12\lambda \mu a_1^2 = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^4}:6{\mu ^3}{a_1} + 6{\mu ^2}a_1^2 = 0.} \hfill \cr } When we solve Eq. (9) system with the help of computer ready package program, we can present the following situations.

Case1. (10) $a_{1} = - μ, c = \frac{λ^{2}}{4},$ {a_1} = - \mu ,\quad c = {{{\lambda ^2}} \over 4}, if the values in Eq. (10) are written in place of Eq. (8), the hyperbolic type solution below are obtain (11) $u_{1} (x, z, t) = - \frac{μ}{- \frac{μ}{λ} + B (cosh [λ (x + z - \frac{t λ^{2}}{4})] - sinh [λ (x + z - \frac{t λ^{2}}{4})])} + a_{0} .$ {u_1}\left( {x,z,t} \right) = - {\mu \over { - {\mu \over \lambda } + B\left( {\cosh \left[ {\lambda \left( {x + z - {{t{\lambda ^2}} \over 4}} \right)} \right] - \sinh \left[ {\lambda \left( {x + z - {{t{\lambda ^2}} \over 4}} \right)} \right]} \right)}} + {a_0}. As seen in figure 1, we can observe asymptotic behavior in the 0 < x < 5 range. Different interpretations and studies can be made for this situation.

3D, 2D and contour graphs respectively for B = 0.9, μ = 0.1, λ = 0.2, a₀ = 5, z = 1 values of Eq. (11).

Case2. (12) $a_{1} = - μ, λ = 2 \sqrt{c},$ {a_1} = - \mu ,\quad \lambda = 2\sqrt c , if the values in Eq. (12) are written in place of Eq. (8), the hyperbolic type solution below are obtain (13) $u_{2} (x, z, t) = - \frac{μ}{- \frac{μ}{2 \sqrt{c}} + B (cosh [(- ct + x + z) 2 \sqrt{c}] - sinh [(- ct + x + z) 2 \sqrt{c}])} + a_{0} .$ {u_2}\left( {x,z,t} \right) = - {\mu \over { - {\mu \over {2\sqrt c }} + B\left( {\cosh \left[ {\left( { - ct + x + z} \right)2\sqrt c } \right] - \sinh \left[ {\left( { - ct + x + z} \right)2\sqrt c } \right]} \right)}} + {a_0}.

3D, 2D and contour graphs respectively for B = 0.7, μ = 2, z = 1, c = −1, a₀ = 3 values of Eq. (13).

Conclusions and Discussion

In this study, analytical solutions of the AKNS equation were obtained via the (1/G′)-expansion method. Although the nonlinear terms of this equation are strong, traveling wave solutions are obtained. The solutions obtained are of the hyperbolic type. By giving special values to the parameters in these solutions, 3D, 2D and contour graphics are presented. We consider these wave solutions are especially important for scientists studying shock wave structure and asymptotic behavior. In addition, it will be much more valuable when the physical meaning is loaded on the constants in these solutions. Algebraic equation obtained by classical expansion methods is simpler. However, the disadvantage of the expansion method we use is that it only provides hyperbolic type solution. The advantage of this method is that it is different from the solutions obtained in other expansion methods. While obtaining solutions in many NPDEs, we encountered less complex operations while solving this equation. However, this study also offers solutions with two cases. It is seen that the method applied in the study is easy, reliable, useful and can be used in future studies. Ready-made package programs are used to facilitate the complexity of the procedure in this package.

Figures & Tables

3D, 2D and contour graphs respectively for B = 0.9, μ = 0.1, λ = 0.2, a0 = 5, z = 1 values of Eq. (11).

3D, 2D and contour graphs respectively for B = 0.7, μ = 2, z = 1, c = −1, a0 = 3 values of Eq. (13).

References

[1] Kurt, A., Tasbozan, O., & Durur, H., (2019), Fundamental Journal of Mathematics and Applications, 2(2), 173–179. KurtA. TasbozanO. DururH. 2019 Fundamental Journal of Mathematics and Applications 2 2 173 179 10.33401/fujma.562819 Search in Google Scholar

[2] Tasbozan, O., Kurt, A., & Durur, H., (2019), International Journal of Engineering Mathematics and Physics. TasbozanO. KurtA. DururH. 2019 International Journal of Engineering Mathematics and Physics Search in Google Scholar

[3] Yokuş, A., & Durur, H., (2019), Journal of Balikesir University Institute of Science and Technology, 21(2), 590–599. YokuşA. DururH. 2019 Journal of Balikesir University Institute of Science and Technology 21 2 590 599 10.25092/baunfbed.631193 Search in Google Scholar

[4] Durur, H., & Yokuş, A., (2019), Afyon Kocatepe Universitesi Fen ve Muhendislik Bilimleri Dergisi, 19(3), 615–619. DururH. YokuşA. 2019 Afyon Kocatepe Universitesi Fen ve Muhendislik Bilimleri Dergisi 19 3 615 619 10.35414/akufemubid.559048 Search in Google Scholar

[5] Kheiri, H., Alipour, N., & Dehghani, R., (2011), Mathematical Sciences, 5(1), 33–50. KheiriH. AlipourN. DehghaniR. 2011 Mathematical Sciences 5 1 33 50 Search in Google Scholar

[6] Durur, H., (2020), Modern Physics Letters B, 34(03), 2050036. DururH. 2020 Modern Physics Letters B 34 03 2050036 10.1142/S0217984920500360 Search in Google Scholar

[7] Yokuş, A., & Kaya, D., (2015), Istanbul Ticaret Universitesi Fen Bilimleri Dergisi, 14(28). YokuşA. KayaD. 2015 Istanbul Ticaret Universitesi Fen Bilimleri Dergisi 14 28 Search in Google Scholar

[8] Ahmad, H., Rafiq, M., Cesarano, C., & Durur, H., (2020), Earthline Journal of Mathematical Sciences, 3(2), 229–247. AhmadH. RafiqM. CesaranoC. DururH. 2020 Earthline Journal of Mathematical Sciences 3 2 229 247 10.34198/ejms.3220.229247 Search in Google Scholar

[9] Kaya, D., & Yokuş, A., (2002), Mathematics and Computers in Simulation, 60(6), 507–512. KayaD. YokuşA. 2002 Mathematics and Computers in Simulation 60 6 507 512 10.1016/S0378-4754(01)00438-4 Search in Google Scholar

[10] Kaya, D., & Yokus, A., (2005), Applied Mathematics and Computation, 164(3), 857–864. KayaD. YokusA. 2005 Applied Mathematics and Computation 164 3 857 864 10.1016/j.amc.2004.06.012 Search in Google Scholar

[11] Yavuz, M., & Ozdemir, N., (2018), Konuralp Journal of Mathematics, 6(1), 102–109 YavuzM. OzdemirN. 2018 Konuralp Journal of Mathematics 6 1 102 109 Search in Google Scholar

[12] Yavuz, M., & Ozdemir, N., (2018), Mathematical Studies and Applications 2018 4–6 October 2018, 442. YavuzM. OzdemirN. 2018 Mathematical Studies and Applications 2018 4–6 October 2018 442 Search in Google Scholar

[13] Durur, H., Taşbozan, O., Kurt, A., & Şenol, M., (2019), Erzincan University Journal of the Institute of Science and Technology, 12(2), 807–815. DururH. TaşbozanO. KurtA. ŞenolM. 2019 Erzincan University Journal of the Institute of Science and Technology 12 2 807 815 Search in Google Scholar

[14] Durur, H., Kurt, A., & Taşbozan, O., (2020), Applied Mathematics and Nonlinear Sciences, 5(1), 455–460. DururH. KurtA. TaşbozanO. 2020 Applied Mathematics and Nonlinear Sciences 5 1 455 460 10.2478/amns.2020.1.00043 Search in Google Scholar

[15] Aziz, I., & Sarler, B., (2010), Mathematical and Computer Modelling, 52(9–10), 1577–1590. AzizI. SarlerB. 2010 Mathematical and Computer Modelling 52 9–10 1577 1590 10.1016/j.mcm.2010.06.023 Search in Google Scholar

[16] Kaya, D., Yokuş, A., & Demiroglu, U., (2020), In Numerical Solutions of Realistic Nonlinear Phenomena (pp. 53–65). Springer, Cham. KayaD. YokuşA. DemirogluU. 2020 In Numerical Solutions of Realistic Nonlinear Phenomena 53 65 Springer Cham 10.1007/978-3-030-37141-8_3 Search in Google Scholar

[17] Su-Ping, Q., & Li-Xin, T., (2007), Chinese Physics Letters, 24(10), 2720. Su-PingQ. Li-XinT. 2007 Chinese Physics Letters 24 10 2720 10.1088/0256-307X/24/10/002 Search in Google Scholar

[18] Darvishi, M., Arbabi, S., Najafi, M., & Wazwaz, A., (2016), Optik, 127(16), 6312–6321. DarvishiM. ArbabiS. NajafiM. WazwazA. 2016 Optik 127 16 6312 6321 10.1016/j.ijleo.2016.04.033 Search in Google Scholar

[19] Rady, A. A., Osman, E. S., & Khalfallah, M., (2010), Applied Mathematics and Computation, 217(4), 1385–1390. RadyA. A. OsmanE. S. KhalfallahM. 2010 Applied Mathematics and Computation 217 4 1385 1390 10.1016/j.amc.2009.05.027 Search in Google Scholar

[20] Sulaiman, T. A., & Bulut, H., (2019), Applied Mathematics and Nonlinear Sciences, 4(2), 513–522. SulaimanT. A. BulutH. 2019 Applied Mathematics and Nonlinear Sciences 4 2 513 522 10.2478/AMNS.2019.2.00048 Search in Google Scholar

[21] Durur, H., Şenol, M., Kurt, A., & Taşbozan, O., (2019), Erzincan University Journal of the Institute of Science and Technology, 12(2), 796–806. DururH. ŞenolM. KurtA. TaşbozanO. 2019 Erzincan University Journal of the Institute of Science and Technology 12 2 796 806 Search in Google Scholar

[22] Kumar, D., Seadawy, A. R., & Joardar, A. K., (2018), Chinese journal of physics, 56(1), 75–85. KumarD. SeadawyA. R. JoardarA. K. 2018 Chinese journal of physics 56 1 75 85 10.1016/j.cjph.2017.11.020 Search in Google Scholar

[23] Eskitaşcioglu, E. I., Aktaş, M. B., & Baskonus, H. M., (2019), Applied Mathematics and Nonlinear Sciences, 4(1), 105–112. EskitaşciogluE. I. AktaşM. B. BaskonusH. M. 2019 Applied Mathematics and Nonlinear Sciences 4 1 105 112 10.2478/AMNS.2019.1.00010 Search in Google Scholar

[24] Baskonus, H. M., Bulut, H., & Sulaiman, T. A., (2019), Applied Mathematics and Nonlinear Sciences, 4(1), 129–138. BaskonusH. M. BulutH. SulaimanT. A. 2019 Applied Mathematics and Nonlinear Sciences 4 1 129 138 10.2478/AMNS.2019.1.00013 Search in Google Scholar

[25] Dusunceli, F., (2019), Applied Mathematics and Nonlinear Sciences, 4(2), 365–370. DusunceliF. 2019 Applied Mathematics and Nonlinear Sciences 4 2 365 370 10.2478/AMNS.2019.2.00031 Search in Google Scholar

[26] Yokus, A., (2020), Boletim da Sociedade Paranaense de Matematica, (in Press). YokusA. 2020 Boletim da Sociedade Paranaense de Matematica (in Press). Search in Google Scholar

[27] Yokus, A., & Yavuz, M., (2018), Discrete & Continuous Dynamical Systems-S, 0. YokusA. YavuzM. 2018 Discrete & Continuous Dynamical Systems-S 0 Search in Google Scholar

[28] Yokuş, A., & Gulbahar, S., (2019), Applied Mathematics and Nonlinear Sciences, 4(1), 35–42. YokuşA. GulbaharS. 2019 Applied Mathematics and Nonlinear Sciences 4 1 35 42 10.2478/AMNS.2019.1.00004 Search in Google Scholar

[29] Rezazadeh, H., Kumar, D., Neirameh, A., Eslami, M., & Mirzazadeh, M., (2020), Pramana, 94(1), 39. RezazadehH. KumarD. NeiramehA. EslamiM. MirzazadehM. 2020 Pramana 94 1 39 10.1007/s12043-019-1881-5 Search in Google Scholar

[30] Gao, W., Silambarasan, R., Baskonus, H. M., Anand, R. V., & Rezazadeh, H., (2020), Physica A: Statistical Mechanics and its Applications, 545, 123772. GaoW. SilambarasanR. BaskonusH. M. AnandR. V. RezazadehH. 2020 Physica A: Statistical Mechanics and its Applications 545 123772 10.1016/j.physa.2019.123772 Search in Google Scholar

[31] Durur, H., Tasbozan, O., & Kurt, A., (2020), Applied Mathematics and Nonlinear Sciences, 5(1), 447–454. DururH. TasbozanO. KurtA. 2020 Applied Mathematics and Nonlinear Sciences 5 1 447 454 10.2478/amns.2020.1.00042 Search in Google Scholar

[32] Rezazadeh, H., Osman, M. S., Eslami, M., Mirzazadeh, M., Zhou, Q., Badri, S. A., & Korkmaz, A., (2019), Nonlinear Engineering, 8(1), 224–230. RezazadehH. OsmanM. S. EslamiM. MirzazadehM. ZhouQ. BadriS. A. KorkmazA. 2019 Nonlinear Engineering 8 1 224 230 10.1515/nleng-2018-0033 Search in Google Scholar

[33] Osman, M. S., Rezazadeh, H., & Eslami, M., (2019), Nonlinear Engineering, 8(1), 559–567. OsmanM. S. RezazadehH. EslamiM. 2019 Nonlinear Engineering 8 1 559 567 10.1515/nleng-2018-0163 Search in Google Scholar

[34] Osman, M. S., (2019), Nonlinear Dynamics, 96(2), 1491–1496. OsmanM. S. 2019 Nonlinear Dynamics 96 2 1491 1496 10.1007/s11071-019-04866-1 Search in Google Scholar

[35] Osman, M. S., & Wazwaz, A. M., (2019), Mathematical Methods in the Applied Sciences, 42(18), 6277–6283. OsmanM. S. WazwazA. M. 2019 Mathematical Methods in the Applied Sciences 42 18 6277 6283 10.1002/mma.5721 Search in Google Scholar

[36] Javid, A., Raza, N., & Osman, M. S., (2019), Communications in Theoretical Physics, 71(4), 362. JavidA. RazaN. OsmanM. S. 2019 Communications in Theoretical Physics 71 4 362 10.1088/0253-6102/71/4/362 Search in Google Scholar

[37] Goyal, M., Baskonus, H. M., & Prakash, A., (2019), The European Physical Journal Plus, 134(10), 482. GoyalM. BaskonusH. M. PrakashA. 2019 The European Physical Journal Plus 134 10 482 10.1140/epjp/i2019-12854-0 Search in Google Scholar

[38] Prakash, A., & Verma, V., (2019), Pramana, 93(4), 66. PrakashA. VermaV. 2019 Pramana 93 4 66 10.1007/s12043-019-1819-y Search in Google Scholar

[39] Prakash, A., Goyal, M., Baskonus, H. M., & Gupta, S., (2020), AIMS Mathematics, 5(2), 979. PrakashA. GoyalM. BaskonusH. M. GuptaS. 2020 AIMS Mathematics 5 2 979 10.3934/math.2020068 Search in Google Scholar

[40] Kumar, D., Singh, J., Prakash, A., & Swroop, R., (2019), Progr Fract Differ Appl, 5(1), 65–77. KumarD. SinghJ. PrakashA. SwroopR. 2019 Progr Fract Differ Appl 5 1 65 77 10.18576/pfda/050107 Search in Google Scholar

[41] Prakash, A., & Kumar, M., (2019), India Section A: Physical Sciences, 89(3), 559–570. PrakashA. KumarM. 2019 India Section A: Physical Sciences 89 3 559 570 10.1007/s40010-018-0496-4 Search in Google Scholar

[42] Gao, W., Ismael, H. F., Husien, A. M., Bulut, H., & Baskonus, H. M., (2020), Applied Sciences, 10(1), 219. GaoW. IsmaelH. F. HusienA. M. BulutH. BaskonusH. M. 2020 Applied Sciences 10 1 219 10.3390/app10010219 Search in Google Scholar

[43] Garcia Guirao, J. L., Baskonus, H. M., Kumar, A., Rawat, M. S., & Yel, G., (2020), Symmetry, 12(1), 17. Garcia GuiraoJ. L. BaskonusH. M. KumarA. RawatM. S. YelG. 2020 Symmetry 12 1 17 10.3390/sym12010017 Search in Google Scholar

[44] Cattani, C., & Rushchitskii, Y. Y., (2003), International applied mechanics, 39(10), 1115–1145. CattaniC. RushchitskiiY. Y. 2003 International applied mechanics 39 10 1115 1145 10.1023/B:INAM.0000010366.48158.48 Search in Google Scholar

[45] Cattani, C., (2003), International Journal of Fluid Mechanics Research, 30(5). CattaniC. 2003 International Journal of Fluid Mechanics Research 30 5 10.1615/InterJFluidMechRes.v30.i5.10 Search in Google Scholar

[46] Cattani, C., Chen, S., & Aldashev, G., (2012), Mathematical Problems in Engineering, 2012. CattaniC. ChenS. AldashevG. 2012 Mathematical Problems in Engineering 2012 10.1155/2012/868413 Search in Google Scholar

[47] Attia, R. A., Lu, D., & Khater, M. M., (2018), Phys. J, 1(3), 234–254. AttiaR. A. LuD. KhaterM. M. 2018 Phys. J 1 3 234 254 Search in Google Scholar

[48] Bruzon, M. S., Gandarias, M. L., Muriel, C., Ramirez, J., & Romero, F. R., (2003), Theoretical and mathematical physics, 137(1), 1378–1389. BruzonM. S. GandariasM. L. MurielC. RamirezJ. RomeroF. R. 2003 Theoretical and mathematical physics 137 1 1378 1389 10.1023/A:1026092304047 Search in Google Scholar

[49] Helal, M. A., Seadawy, A. R., & Zekry, M. H., (2013), Applied Mathematical Sciences, 7(65–68), 3355–3365. HelalM. A. SeadawyA. R. ZekryM. H. 2013 Applied Mathematical Sciences 7 65–68 3355 3365 10.12988/ams.2013.34239 Search in Google Scholar

[50] Ping, L., Bao-Qing, Z., Jian-Rong, Y., & Bo, R., (2015), Chinese Physics B, 24(1), 010202. PingL. Bao-QingZ. Jian-RongY. BoR. 2015 Chinese Physics B 24 1 010202 Search in Google Scholar

[51] Gao, W., Yel, G., Baskonus, H. M., & Cattani, C., (2019, June), In Book of Abstracts (p. 86). GaoW. YelG. BaskonusH. M. CattaniC. 2019 June In Book of Abstracts 86 Search in Google Scholar

[52] Chun-Long, Z., & Jie-Fang, Z., (2002), Chinese Physics Letters, 19(10), 1399. Chun-LongZ. Jie-FangZ. 2002 Chinese Physics Letters 19 10 1399 10.1088/0256-307X/19/10/301 Search in Google Scholar

Applied Mathematics and Nonlinear Sciences

Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

Hülya Durur and
Hülya Durur and
Asif Yokuş
Asif Yokuş

Published Online: 31 Dec 2020

Volume & Issue: Volume 6 (2021) - Issue 2 (July 2021)

Page range: 381 - 386

Received: 08 Apr 2020

Accepted: 20 Jun 2020

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

PDF Preview

Article

Fig. 1

Fig. 2

Figures & Tables

Fig. 1

Fig. 2

References

Recent Articles

Applied Mathematics and Nonlinear Sciences

Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

Hülya Durur andHülya Durur andAsif YokuşAsif Yokuş

Published Online: 31 Dec 2020

Volume & Issue: Volume 6 (2021) - Issue 2 (July 2021)

Page range: 381 - 386

Received: 08 Apr 2020

Accepted: 20 Jun 2020

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

PDF Preview

Article

Fig. 1

Fig. 2

Figures & Tables

Fig. 1

Fig. 2

References

Recent Articles

Hülya Durur and
Hülya Durur and
Asif Yokuş
Asif Yokuş