Nonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulations

<div>Elevation <svg height="12.7178pt" id="M154" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 47.0682 12.7178" width="47.0682pt" 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="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z" id="g113-229"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.097,3.132)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 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616H322C398 616 419 607 436 579C449 559 455 539 464 499L493 503Z" id="g190-71"></path></g><g transform="matrix(.0091,0,0,-0.0091,22.681,3.132)"><path d="M372 352C427 368 507 407 507 502C507 548 486 592 451 616C420 638 376 650 301 650H44V622C124 616 128 610 128 527V123C128 40 122 34 36 28V0H257C336 0 402 12 453 40C512 72 548 124 548 191C548 291 471 335 372 350V352ZM213 362V558C213 591 216 602 226 608C235 614 258 618 282 618C377 618 415 558 415 490C415 406 369 362 260 362H213ZM213 328H258C380 328 450 283 450 181C450 76 377 35 296 34C231 34 213 53 213 125V328Z" id="g190-67"></path></g><g transform="matrix(.013,0,0,-0.013,28.584,0)"><path d="M162 -163V703H101V-163H162Z" id="g113-9"></path></g></svg> (blue), and MTA (red) of the BBM wave packet for <span class="nowrap"><svg height="15.8677pt" id="M156" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -12.5766 132.11 15.8677" width="132.11pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g 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International Journal of Differential Equations

fig8

Figure 8

Figure 8: Nonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulations 