On the Nonlinear Perturbation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.8905pt" id="M1" height="15.1511pt" version="1.1" viewBox="-0.0657574 -12.2606 55.2666 15.1511" width="55.2666pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-76" d="M771 650H519L511 619L537 615C578 610 579 600 539 564C444 476 379 428 336 396C312 378 281 357 252 347L283 514C300 605 304 610 388 619L395 650H134L127 619C212 610 216 605 199 514L130 140C112 41 103 38 28 31L18 0H281L288 31C201 40 198 41 216 140L246 311C274 328 289 324 315 284C393 167 458 86 528 0H686L695 31C635 37 612 45 570 97C521 156 420 285 361 367L596 549C663 601 684 609 764 620L771 650Z"/></g><g transform="matrix(.017,0,0,-0.017,13.079,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,19.222,0)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.017,0,0,-0.017,28.008,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.017,0,0,-0.017,34.987,0)"><path id="g50-110" d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z"/></g><g transform="matrix(.017,0,0,-0.017,48.8,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave

<div>The special solution for alpha = 0.55 and <svg height="6.20645pt" id="M94" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -6.00005 10.6039 6.20645" width="10.6039pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z" id="g50-110"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="805" vert-adv-y="805"></glyph.data></g></svg> = 50.</div>

Advances in Mathematical Physics

fig2

Figure 2

Figure 2: On the Nonlinear Perturbation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.8905pt" id="M1" height="15.1511pt" version="1.1" viewBox="-0.0657574 -12.2606 55.2666 15.1511" width="55.2666pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-76" d="M771 650H519L511 619L537 615C578 610 579 600 539 564C444 476 379 428 336 396C312 378 281 357 252 347L283 514C300 605 304 610 388 619L395 650H134L127 619C212 610 216 605 199 514L130 140C112 41 103 38 28 31L18 0H281L288 31C201 40 198 41 216 140L246 311C274 328 289 324 315 284C393 167 458 86 528 0H686L695 31C635 37 612 45 570 97C521 156 420 285 361 367L596 549C663 601 684 609 764 620L771 650Z"/></g><g transform="matrix(.017,0,0,-0.017,13.079,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,19.222,0)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.017,0,0,-0.017,28.008,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.017,0,0,-0.017,34.987,0)"><path id="g50-110" d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z"/></g><g transform="matrix(.017,0,0,-0.017,48.8,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave 

Advances in Mathematical Physics

On the Nonlinear Perturbation Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave

Figure 2