Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3067999pt" id="M1" height="15.6075pt" version="1.1" viewBox="-0.0657574 -15.3007 17.6155 15.6075" width="17.6155pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-30" d="M597 536C514 627 443 676 315 676S72 616 72 475C72 255 443 260 443 129C443 65 375 30 319 30C204 30 143 77 70 158L35 130C115 36 196 -14 323 -14C443 -14 591 45 591 186C591 411 213 394 213 548C213 599 275 632 319 632C433 632 498 583 567 501L597 536ZM464 63C481 84 487 112 487 139C487 304 116 279 116 487C116 536 144 587 188 610L189 608C177 585 169 561 169 534C169 360 547 365 547 186C547 131 513 87 466 62L464 63Z"/></g><g transform="matrix(.012,0,0,-0.012,10.914,-7.578)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>

<div>Multibranch hyperbolic case I: distribution of <svg height="8.68572pt" id="M485" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt" 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="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g></svg> (<svg height="11.8174pt" id="M486" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 42.5437 11.8174" width="42.5437pt" 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="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.619,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.197,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,28.46,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,36.091,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg> and  <svg height="11.927pt" id="M487" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 63.4052 11.927" width="63.4052pt" 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="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.619,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.013,0,0,-0.013,17.197,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,28.46,0)"><path d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.0091,0,0,-0.0091,37.079,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path></g><g transform="matrix(.013,0,0,-0.013,45.658,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,56.92,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg>).</div>

Advances in Mathematical Physics

fig6

Figure 6

Figure 6: Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3067999pt" id="M1" height="15.6075pt" version="1.1" viewBox="-0.0657574 -15.3007 17.6155 15.6075" width="17.6155pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-30" d="M597 536C514 627 443 676 315 676S72 616 72 475C72 255 443 260 443 129C443 65 375 30 319 30C204 30 143 77 70 158L35 130C115 36 196 -14 323 -14C443 -14 591 45 591 186C591 411 213 394 213 548C213 599 275 632 319 632C433 632 498 583 567 501L597 536ZM464 63C481 84 487 112 487 139C487 304 116 279 116 487C116 536 144 587 188 610L189 608C177 585 169 561 169 534C169 360 547 365 547 186C547 131 513 87 466 62L464 63Z"/></g><g transform="matrix(.012,0,0,-0.012,10.914,-7.578)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg> 