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Abstract

In this paper, the Khater II analytical technique is used to examine novel soliton structures for the fractional nonlinear third-order Schrödinger (3-FNLS) problem. The 3-FNLS equation explains the dynamical behavior of a system’s quantum aspects and ultra-short optical fiber pulses. Additionally, it determines the wave function of a quantum mechanical system in which atomic particles behave similarly to waves. For example, electrons, like light waves, exhibit diffraction patterns when passing through a double slit. As a result, it was fair to suppose that a wave equation could adequately describe atomic particle behavior. The correctness of the solutions is determined by comparing the analytical answers obtained with the numerical solutions and determining the absolute error. The trigonometric Quintic B-spline numerical (TQBS) technique is used based on the computed required criteria. Analytical and numerical solutions are represented in a variety of graphs. The strength and efficacy of the approaches used are evaluated.

Keywords

Quantum mechanical system Computational and numerical solutions Soliton waves Ultra-short optical fiber

AMS classification: 35J10; 35D30; 35R10; 65N15; 35Q51

Introduction

Numerous academics in diverse fields, including engineering and applied sciences, have recently concentrated their efforts on the physical interpretation of optical fibers.¹ Organic artificial materials used in the manufacturing of optical fibers include the well-known silica (drawing glass) and polymers with a thickness less than a human hair.² This procedure transforms the fiber into a translucent one with increased flexibility.³ This kind of fiber is often used to transfer light down a long rope of optical fiber, which is a critical step in fiber-optic communications.⁴ This fiber transfers data over a greater distance and at a higher bandwidth than electrical wires.⁵ Transmission procedures occur without a single signal being lost or with far less signal loss than metal lines do because fibers are resistant to electromagnetic interference.⁶ Illumination and imaging are regarded to be the second critical function of optical fiber.⁷ Where this occurs, the material is often wrapped in bundles to bring light into, or pictures out in small areas.⁸ Fiber scopes, fiber optic sensors, and fiber lasers are all examples of this approach.⁹

Optical fibers are composed of a kernel encased in a transparent substance with a lower refractive index.¹⁰ The phenomenon of perfect interior reflection, which enables the fiber to operate like a superconductor, permits light to be retained in the heart.¹¹ Multi-standard fibers are referred to as fibers that follow numerous pathways or circumferential modes.¹² In contrast, single-mode fibers (SMF) follow a single direction. On the other hand, multi-mode fibers have a larger core diameter and are often utilized for connections over short distances and for applications needing high power transfer.¹³ Single-mode fibers are being used for the majority of contact lines more than 1000 m (3300 ft) in width. It is crucial in fiber optic communication to be able to join optical fibers with little loss.¹⁴ It is more complex than electrical wiring or cable connections and involves meticulous fiber cleaning, perfect fiber core alignment, and coupling.¹⁵

Optical fiber is one of a number of various disciplines that have been described mathematically using fractional or integer order equations or systems.¹⁶ Mechanical engineering, electrochemistry, biology, quantum optics, plasma physics, chemistry, mathematical physics, and hydraulic engineering are just a few of these other subjects.¹⁷ That is why several scholars in a variety of fields of science are concentrating their efforts on obtaining accurate and numerical solutions to the specified nonlinear evolution equations.¹⁸ As a result, numerous computational, semi-analytical, and numerical techniques have been developed, including the extended simplest equation method, the extended tanh expansion method, the generalized Kudryashov method, the modified Khater method, the Sin–Cos expansion method, the Sech–Tanh expansion method, the Sine-Gordon expansion method, and the new auxiliary equation method.^19-23 However, there is no one-size-fits-all analytical technique for all nonlinear evolution problems.^24-26

In this context, we used the Khater II scheme to the 3-FNLS^27-31 to evaluate an unlimited number of solutions and then used the TQBS scheme^32,33 to determine the absolute value of error between the precise and numerical solutions. The three-FNLS equation is denoted by ref. 34

i (D_{t}^{γ} Q + Q_{x x x}) + | Q |^{2} (l_{1} Q + i l_{2} Q_{x}) + i l_{3} | Q |_{x}^{2} Q = 0,

(1)

where l₁, l₂, l₃ are arbitrary constants and

Q

is a complicated function illustrating the physical behavior of light transmission in optical fibers when ultra-short pulses are generated. Equation (1) translates into the Hirota equation and the modified KdV in complex form for (l₁ = l₂). Using the following wave transformation

Q = S (T) e^{p_{1} x + p_{2} \frac{t^{γ}}{γ} + p_{3}}, T = q_{1} x - q_{2} t^{γ} / γ

equation (1) is transformed into the following ordinary differential system for the real and imaginary portions, respectively

{\begin{array}{l} 3 q_{1}^{2} p_{1} S ″ + (p_{2} - p_{1}^{3}) S + (l_{1} p_{1} - l_{1}) S^{3} = 0, \\ q_{1}^{3} S ‴ - (3 q_{1} p_{1}^{2} + q_{2}) S^{'} + q_{1} (l_{1} + 2 l_{3}) S^{2} S^{'} = 0. \end{array}

(2)

once integration of equation (2) with zero constant of the integration, gets

q_{1}^{3} S ″ - (3 q_{1} p_{1}^{2} + q_{2}) S + \frac{q_{1}}{3} (l_{1} + 2 l_{3}) S^{3} = 0.

(3)

comparing equation (3) and first equation in (2) shows their equivalence when

l_{3} = - - l_{1} / 2 p_{1}, p_{2} = - 3 p_{1} q_{2} - 8 p_{1}^{3} q_{1} / q_{1}

. Balancing the above-equation terms by using the homogeneous balance principles and the Khater II method’s auxiliary equation

ℱ^{'} (T) = - δ - ℱ {(T)}^{2}, ϕ^{'} (T) = - ℱ (T) ϕ (T), ϕ {(T)}^{2} = α (ℱ {(T)}^{2} / δ + 1)

, where δ is arbitrary constant and α = − 1, get n = 1. Thus, the general solution of the studied model is given by

G (T) = \sum_{i = 1}^{n} (a_{i} f {(T)}^{i} + b_{i} ϕ (T) f {(T)}^{i - 1}) + a_{0} = a_{1} f (T) + a_{0} + b_{1} ϕ (T),

(4)

where a₀, a₁, b₁, a₁ ≠ 0, b₁ ≠ 0.

This study paper’s remaining parts are organized as follows: The Khater II technique is used in the section Applications to find new solitary solutions to the 3-FNLS problem. Additionally, the TQBS numerical scheme is used to evaluate numerical solutions and determine the absolute error value between the computational and numerical solutions. The section Results and Discussion explains the findings and provides a physical description of the drawings that are shown. The section Conclusion summarizes the results of this investigation and the paper’s contributions.

Applications

Here, the analytical and approximate solutions of the conformable fractional NLS equation are investigated by employing the suggested schemes. This study’s goal is to construct novel solutions’ structures and check their accuracy by calculating the absolute value of error between analytical and approximate solutions (Figures 1–10).

Figure 1.

Graphical representations of equation (8) for the real, imaginary, and absolute values of the $Q_{I, 1} (x, t)$

Figure 2.

Graphical representations of equation (9) for the real, imaginary, and absolute values of the $Q_{I, 2} (x, t)$

Figure 3.

Graphical representations of equation (10) for the real, imaginary, and absolute values of the $Q_{II, 1} (x, t)$

Figure 4.

Graphical representations of equation (11) for the real, imaginary, and absolute values of the $Q_{II, 2} (x, t)$

Figure 5.

Graphical representations of equation (12) for the real, imaginary, and absolute values of the $Q_{III, 1} (x, t)$

Figure 6.

Graphical representations of equation (13) for the real, imaginary, and absolute values of the $Q_{III, 2} (x, t)$

Figure 7.

Computational, numerical, and absolute value of error according to the shown Table 1.

Figure 8.

Computational, numerical, and absolute value of error according to the shown Table 2.

Figure 9.

Computational, numerical, and absolute value of error according to the shown Table 3.

Figure 10.

Computational, numerical, and absolute value of error according to the shown Tables 1–3.

Solitons wave solutions

Applying the Khater II method to the considered model for investigating the soliton wave solutions, gets the following values of the above-shown parameters: Set I

a_{0} \to 0, a_{1} \to \frac{\sqrt{6} q_{1}}{\sqrt{- l_{1} - 2 l_{3}}}, b_{1} \to 0, p_{1} \to \frac{i \sqrt{q_{2} - 2 δ q_{1}^{3}}}{\sqrt{3} \sqrt{q_{1}}} .

(5)

set II

a_{0} \to 0, a_{1} \to \frac{\sqrt{α} b_{1}}{\sqrt{δ}}, l_{1} \to \frac{- 4 α b_{1}^{2} l_{3} - 3 δ q_{1}^{2}}{2 α b_{1}^{2}}, p_{1} \to \frac{\sqrt{δ q_{1}^{3} - 2 q_{2}}}{\sqrt{6} \sqrt{q_{1}}} .

(6)

set III

a_{0} \to 0, a_{1} \to 0, l_{1} \to - \frac{2 (α b_{1}^{2} l_{3} + 3 δ q_{1}^{2})}{α b_{1}^{2}}, p_{1} \to \frac{\sqrt{- δ q_{1}^{3} - q_{2}}}{\sqrt{3} \sqrt{q_{1}}} .

(7)

thus, the traveling wave solutions of the conformable fractional nonlinear NLS equation are structured as follows

For δ ≠ 0, we obtain

Q_{I, 1} (x, t) = - \frac{\sqrt{6} \sqrt{δ} q_{1} e^{i (p_{2} t + p_{1} x + p_{3})}}{\sqrt{- l_{1} - 2 l_{3}}} t a n (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})),

(8)

Q_{I, 2} (x, t) = \frac{\sqrt{6} \sqrt{δ} q_{1} e^{i (p_{2} t + p_{1} x + p_{3})}}{\sqrt{- l_{1} - 2 l_{3}}} c o t (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})),

(9)

Q_{II, 1} (x, t) = b_{1} e^{i (p_{2} t + p_{1} x + p_{3})} (s e c (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})) - \sqrt{α} \tan (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ}))),

(10)

Q_{II, 2} (x, t) = b_{1} e^{i (p_{2} t + p_{1} x + p_{3})} (\sqrt{α} c o t (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})) + csc (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ}))),

(11)

Q_{III, 1} (x, t) = b_{1} e^{i (p_{2} t + p_{1} x + p_{3})} s e c (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})),

(12)

Q_{III, 2} (x, t) = b_{1} e^{i (p_{2} t + p_{1} x + p_{3})} c s c (\sqrt{δ} (q_{1} x - \frac{q_{2} t^{γ}}{γ})) .

(13)

Approximate solutions

Investigating the above-obtained solutions’ accuracy by calculating the absolute error between analytical and numerical solutions. Checking the following solutions equations (8), (10), (12) when δ = − 1, l₁ = − 7, l₃ = 2, p₁ = 2, p₃ = 4, p₂ = 6, q₁ = − 1, q₂ = 3 & b₁ = 0.5, δ = − 4, q₁ = 0.1, q₂ = − 0.2 & b₁ = 2, δ = − 9, q₁ = 6, q₂ = 4 to evaluate the initial and boundary conditions of the studied model, gets

Q (x, t) = {\begin{array}{l} \sqrt{2} tanh (3 t + x), \\ 0.5 (tanh (2 (0.2 t + 0.1 x)) + sech (2 (0.2 t + 0.1 x))), \\ 3 sech (4 t - 2 x) . \end{array}

(14)

using these conditions in the suggested numerical scheme’s framework demonstrates the matching between analytical and approximate solutions through the following Tables 1–3.

Table 1.

Computational, numerical, and absolute values of error with different values of x when r = 32.

Value of x	Computational	Numerical	$\| E r r o r \|$
0	0	−1.73 472 347 597 681E-18	1.73 472 347 597 681E-18
0.031 25	0.044 179 793 317 226 9	0.044 179 793 010 744 8	3.06 482 117 462 537E-10
0.062 5	0.088 273 438 196 663 7	0.088 273 437 431 072 9	7.65 590 868 478 938E-10
0.093 75	0.132 195 456 833 792	0.132 195 455 672 147	1.16 164 527 996 787E-09
0.125	0.175 861 701 627 197	0.175 861 700 086 312	1.54 088 480 863 734E-09
0.156 25	0.219 189 992 933 662	0.219 189 991 053 136	1.88 052 587 390 963E-09
0.187 5	0.262 100 724 868 095	0.262 100 722 690 284	2.17 781 059 852 484E-09
0.218 75	0.304 517 429 891 506	0.304 517 427 465 103	2.42 640 380 010 428E-09
0.25	0.346 367 294 049 603	0.346 367 291 426 698	2.62 290 461 572 334E-09
0.281 25	0.387 581 616 033 836	0.387 581 613 268 638	2.76 519 818 154 242E-09
0.312 5	0.428 096 204 683 757	0.428 096 201 830 851	2.8 529 058 004 878E-09
0.343 75	0.467 851 711 078 912	0.467 851 708 191 721	2.88 719 115 282 277E-09
0.375	0.506 793 892 924 419	0.506 793 890 053 757	2.87 066 281 856 596E-09
0.406 25	0.544 873 810 463 186	0.544 873 807 655 996	2.807 189 480 869E-09
0.437 5	0.582 047 954 600 852	0.582 047 951 899 162	2.70 169 009 386 478E-09
0.468 75	0.618 278 309 264 827	0.618 278 306 704 927	2.55 989 918 152 011E-09
0.5	0.653 532 351 202 406	0.653 532 348 814 284	2.38 812 181 141 412E-09
0.531 25	0.687 782 991 430 057	0.687 782 989 237 068	2.19 298 923 465 061E-09
0.562 5	0.721 008 463 360 719	0.721 008 461 379 492	1.98 122 651 617 894E-09
0.593 75	0.753 192 163 251 809	0.753 192 161 492 368	1.75 944 159 241 936E-09
0.625	0.784 322 449 034 849	0.784 322 447 500 91	1.53 393 897 583 953E-09
0.656 25	0.814 392 403 816 965	0.814 392 402 506 399	1.31 056 587 804 323E-09
0.687 5	0.843 399 570 399 408	0.843 399 569 304 817	1.09 459 052 932 692E-09
0.718 75	0.871 345 663 057 613	0.871 345 662 166 995	8.90 617 024 573 714E-10
0.75	0.898 236 262 593 165	0.898 236 261 890 635	7.02 529 812 102 171E-10
0.781 25	0.924 080 500 323 984	0.924 080 499 790 513	5.33 471 045 116 585E-10
0.812 5	0.948 890 736 249 156	0.948 890 735 863 314	3.85 842 024 996 919E-10
0.843 75	0.972 682 236 132 641	0.972 682 235 871 305	2.6 133 606 390 033E-10
0.875	0.995 472 851 717 594	0.995 472 851 556 635	1.60 958 468 775 618E-10
0.906 25	1.017 282 707 730 2	1.017 282 707 645 01	8.51 805 292 967 356E-11
0.937 5	1.038 133 898 776 29	1.038 133 898 742 63	3.36 539 685 008 574E-11
0.968 75	1.058 050 198 690 37	1.058 050 198 683 76	6.61 048 993 322 311E-12
1	1.077 056 784 376 73	1.077 056 784 376 73	0

Table 2.

Computational, numerical, and absolute values of error with different values of x when r = 32.

Value of x	Computational	Numerical	$\| E r r o r \|$
0	3	3	0
0.031 25	2.994 150 146 617 61	2.907 049 078 869 01	0.087 101 067 748 600 2
0.062 5	2.976 714 124 425 38	2.771 190 435 822 95	0.205 523 688 602 427
0.093 75	2.948 027 212 409 39	2.656 364 739 437 76	0.291 662 472 971 63
0.125	2.908 630 887 420 64	2.544 533 792 435 94	0.364 097 094 984 699
0.156 25	2.859 248 393 880 66	2.439 754 078 703 29	0.419 494 315 177 372
0.187 5	2.800 753 429 958 68	2.340 249 313 337 19	0.460 504 116 621 49
0.218 75	2.734 134 379 187 01	2.245 886 556 346 23	0.488 247 822 840 776
0.25	2.660 456 651 910 22	2.156 160 157 641 18	0.504 296 494 269 047
0.281 25	2.580 825 561 894 87	2.070 723 055 859 36	0.510 102 506 035 508
0.312 5	2.496 351 804 479 87	1.989 236 457 796 47	0.507 115 346 683 402
0.343 75	2.408 121 101 798 13	1.911 403 520 692 14	0.496 717 581 105 985
0.375	2.317 169 021 571 79	1.836 954 698 069 16	0.480 214 323 502 63
0.406 25	2.224 461 433 240 06	1.765 646 878 421 11	0.458 814 554 818 942
0.437 5	2.130 880 594 195 65	1.697 259 516 956 4	0.433 621 077 239 242
0.468 75	2.037 216 492 411 14	1.631 592 151 540 65	0.405 624 340 870 492
0.5	1.944 162 820 991 66	1.568 462 032 598 11	0.375 700 788 393 548
0.531 25	1.852 316 819 534 48	1.507 702 131 452 99	0.344 614 688 081 49
0.562 5	1.762 182 169 905 33	1.449 159 393 287 63	0.313 022 776 617 705
0.593 75	1.674 174 158 100 91	1.392 693 218 999 94	0.281 480 939 100 97
0.625	1.588 626 386 418 12	1.338 174 135 325 61	0.250 452 251 092 507
0.656 25	1.505 798 420 641 29	1.285 482 633 797 44	0.220 315 786 843 846
0.687 5	1.425 883 868 768 94	1.234 508 131 628 14	0.191 375 737 140 794
0.718 75	1.349 018 498 753 87	1.185 148 122 805 29	0.163 870 375 948 588
0.75	1.275 288 104 826 84	1.137 307 176 426 57	0.137 980 928 400 267
0.781 25	1.204 735 920 724	1.090 896 986 323 18	0.113 838 934 400 823
0.812 5	1.137 369 451 722 62	1.045 832 917 331 58	0.091 536 534 391 041 3
0.843 75	1.073 166 655 845 95	1.002 043 982 008 22	0.071 122 673 837 734 2
0.875	1.012 081 449 128 72	0.959 433 040 539 974	0.052 648 408 588 740 6
0.906 25	0.954 048 542 257 522	0.918 022 939 299 28	0.036 025 602 958 242
0.937 5	0.898 987 638 265 157	0.877 409 361 547 204	0.021 578 276 717 953 3
0.968 75	0.846 807 035 284 597	0.838 799 785 663 173	0.008 007 249 621 423 74
1	0.797 406 686 502 239	0.797 406 686 502 239	0

Table 3.

Computational, numerical, and absolute values of error with different values of x when r = 32.

Value of x	Computational	Numerical	$(Error)$
0	0.5	0.5	0
0.031 25	0.503 115 193 844 475	0.501 937 305 889 842	0.001 177 887 954 633 03
0.062 5	0.506 210 614 542 48	0.503 257 774 697 122	0.002 952 839 845 358 43
0.093 75	0.509 286 023 769 424	0.504 758 545 778 733	0.004 527 477 990 690 53
0.125	0.512 341 187 163 975	0.506 233 282 790 14	0.006 107 904 373 835 13
0.156 25	0.515 375 874 399 654	0.507 744 877 966 136	0.007 630 996 433 517 72
0.187 5	0.518 389 859 254 452	0.509 284 219 977 527	0.009 105 639 276 924 87
0.218 75	0.521 382 919 678 45	0.510 861 728 140 095	0.010 521 191 538 355 2
0.25	0.524 354 837 859 385	0.512 482 845 660 283	0.011 871 992 199 101 2
0.281 25	0.527 305 400 286 126	0.514 154 650 904 803	0.013 150 749 381 322 4
0.312 5	0.530 234 397 810 024	0.515 884 127 303 917	0.014 350 270 506 106 7
0.343 75	0.533 141 625 704 09	0.517 678 673 029 703	0.015 462 952 674 386 5
0.375	0.536 026 883 719 975	0.519 546 013 571 158	0.016 480 870 148 816 3
0.406 25	0.538 889 976 142 713	0.521 494 278 349 788	0.017 395 697 792 924 7
0.437 5	0.541 730 711 843 2	0.523 532 037 788 795	0.018 198 674 054 405 1
0.468 75	0.544 548 904 328 377	0.525 668 356 009 301	0.018 880 548 319 075 4
0.5	0.547 344 371 789 091	0.527 912 845 019 664	0.019 431 526 769 426 9
0.531 25	0.550 116 937 145 612	0.530 275 724 958 807	0.019 841 212 186 805 3
0.562 5	0.552 866 428 090 777	0.532 767 889 997 535	0.020 098 538 093 241 6
0.593 75	0.555 592 677 130 748	0.535 400 981 167 391	0.020 191 695 963 356 1
0.625	0.558 295 521 623 361	0.538 187 466 323 522	0.020 108 055 299 838 9
0.656 25	0.560 974 803 814 055	0.541 140 730 780 888	0.019 834 073 033 166 9
0.687 5	0.563 630 370 869 359	0.544 275 170 785 648	0.019 355 200 083 710 2
0.718 75	0.566 262 074 907 929	0.547 606 325 171 496	0.018 655 749 736 433 2
0.75	0.568 869 773 029 139	0.551 150 920 528 514	0.017 718 852 500 625 7
0.781 25	0.571 453 327 339 189	0.554 927 303 141 453	0.016 526 024 197 736
0.812 5	0.574 012 604 974 758	0.558 954 502 662 785	0.015 058 102 311 973 4
0.843 75	0.576 547 478 124 174	0.563 256 483 612 032	0.013 290 994 512 142 2
0.875	0.579 057 824 046 117	0.567 847 150 439 095	0.011 210 673 607 022 3
0.906 25	0.581 543 525 085 849	0.572 787 237 420 961	0.008 756 287 664 887 67
0.937 5	0.584 004 468 688 968	0.577 973 277 044 526	0.006 031 191 644 442 21
0.968 75	0.586 440 547 412 716	0.583 925 785 320 522	0.002 514 762 092 194 26
1	0.588 851 658 934 815	0.588 851 658 934 815	0

Results and discussion

This part explores the paper’s findings and innovation by exhibiting the benefits of the analytical and numerical techniques utilized, the acquired results, and their comparison to previously published solutions, ultimately proving the correctness of the found solutions. Finally, we will analyze the physical meaning of the numbers presented. Mostafa M. A. Khater discovered the Khater II technique for the first time. He created this extended technique in order to get innovative structures for explicit wave solutions to a class of nonlinear evolution equations. The efficacy and potency of this technique have been established.³⁵ The TQBS scheme has been employed to find the numerical solutions of the investigated model based on the obtained solutions (8), (10), (12). The analytical solutions have been explained through some different graphs (1, 2, 3, 4, 5, 6). While the matching between both solutions is explained through some distinct plots (7, 8, 9). Comparing our solutions to show the solutions’ accuracy leads to verify our obtained solutions and superiority of equation (8) over other constructed solutions.

Conclusion

The 3-FNLS model was effectively studied in this research work using the Khater II approach. Numerous separate precise solutions for moving and isolated waves have been discovered. These solutions have been illustrated using a variety of drawings that demonstrate additional and unexpected aspects of the fractional models under consideration. Our acquired answers have been discussed in terms of their correctness and uniqueness. Additionally, the potency and usefulness of the approaches utilized are discussed and validated.

Footnotes

Acknowledgment

We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

Authors’ contribution

Dexu Zhao and Mostafa Khater have revised the conceptualization, data curation, and methodology. Dianchen Lu and Mostafa Khater have revised Data curation, Investigation, and Software. Samir Salama and Piyaphong Yongphe have revised the physical meaning of the obtained solutions and raised the given graphs resolutions. All authors have read and agreed to the published version of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

Availability of data and material

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The used code of this study is available from the corresponding author upon reasonable request.

ORCID iD

Mostafa MA Khater

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Novel and accurate solitary wave solutions of the conformable fractional nonlinear Schrödinger equation

Abstract

Keywords

Introduction

Applications

Solitons wave solutions

Approximate solutions

Results and discussion

Conclusion

Footnotes

Acknowledgment

Authors’ contribution

Declaration of conflicting interests

Funding

Availability of data and material

Code availability

ORCID iD

References