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>Evolution of <svg height="8.68572pt" id="M428" 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> from (<a href="https://static.hindawi.com/articles/amp/volume-2018/5370349/figures/#EEq48">87</a>) with <svg height="11.439pt" id="M429" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 52.7077 11.439" width="52.7077pt" 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="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z" id="g113-115"></path></g><g transform="matrix(.013,0,0,-0.013,9.12,0)"><path d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z" id="g117-173"></path></g><g transform="matrix(.013,0,0,-0.013,19.616,0)"><path d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z" id="g113-92"></path></g><g transform="matrix(.013,0,0,-0.013,24.101,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,30.341,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path></g><g transform="matrix(.013,0,0,-0.013,35.484,0)"><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><g transform="matrix(.013,0,0,-0.013,41.724,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,48.045,0)"><path d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z" id="g113-94"></path></g></svg> and <span class="nowrap"><svg height="11.439pt" id="M430" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 56.5179 11.439" width="56.5179pt" 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><g transform="matrix(.013,0,0,-0.013,11.527,0)"><path d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z" id="g117-173"></path></g><g transform="matrix(.013,0,0,-0.013,22.023,0)"><path d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z" id="g113-92"></path></g><g transform="matrix(.013,0,0,-0.013,26.508,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,34.139,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z" id="g113-53"></path></g><g transform="matrix(.013,0,0,-0.013,40.379,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path></g><g transform="matrix(.013,0,0,-0.013,45.522,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z" id="g113-53"></path></g><g transform="matrix(.013,0,0,-0.013,51.82,0)"><path d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z" id="g113-94"></path></g></svg>.</span></div>

Advances in Mathematical Physics

fig4

Figure 4

Figure 4: 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> 