Studies in Applied Mathematics最新文献

{"title":"Semiclassical Dynamics and Coherent Soliton Ensembles in the Derivative Nonlinear Schrödinger Equation With Periodic Initial Conditions","authors":"Zachery Wolski,&nbsp;Zechuan Zhang,&nbsp;Gino Biondini,&nbsp;Gregor Kovačič","doi":"10.1111/sapm.70260","DOIUrl":"https://doi.org/10.1111/sapm.70260","url":null,"abstract":"<div>\u0000 \u0000 <p>The semiclassical limit of the derivative nonlinear Schrödinger equation with periodic initial conditions is studied analytically and numerically. The spectrum of the associated scattering problem for a certain class of initial conditions, referred to as periodic single-lobe potentials, is numerically computed, and it is shown that the spectrum becomes confined to the real and imaginary axes or the spectral parameter in the semiclassical limit. A formal Wentzel–Kramers–Brillouin expansion is computed for the scattering eigenfunctions, which allows one to obtain asymptotic expressions for the number, location, and size of the spectral bands and gaps. The results of these calculations suggest that, in the semiclassical limit, all excitations in the spectrum become effective solitons. Finally, the analytical predictions are compared with direct numerical simulations as well as with numerical calculations of the Lax spectrum, and the results are shown to be in excellent agreement.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148532917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Soliton and Breather Interactions in the Integrable Discrete Focusing Manakov System via Hirota's Method","authors":"Uyen Le,&nbsp;Alexander Chernyavsky,&nbsp;Barbara Prinari","doi":"10.1111/sapm.70259","DOIUrl":"https://doi.org/10.1111/sapm.70259","url":null,"abstract":"<div>\u0000 \u0000 <p>The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all “two-body” interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148532975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak Solutions for Image Inpainting Based on Anisotropic Degenerate Cahn–Hilliard Equation","authors":"Jihui Wu,&nbsp;Alain Miranville","doi":"10.1111/sapm.70264","DOIUrl":"https://doi.org/10.1111/sapm.70264","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper is concerned with a diffusion interface model for phase separation of the anisotropic Cahn–Hilliard image inpainting with a concentration-dependent degenerate mobility in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d=2,3$</annotation>\u0000 </semantics></math> dimensions. We present global existence of weak solutions to the anisotropic non-degenerate Cahn–Hilliard image inpainting with a smooth double-well potential. We prove that as the value of the fidelity parameter <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>λ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$lambda _{0}$</annotation>\u0000 </semantics></math> outside of the inpainting region gets large, the existing region imposes a certain type of effective boundary conditions on the inpainting region, and we present a proof for the steady-state problem. Furthermore, we obtain the global existence and regularity of weak solutions to the anisotropic degenerate Cahn–Hilliard image inpainting system with a logarithmic potential.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Issue Information-TOC","authors":"","doi":"10.1111/sapm.70266","DOIUrl":"https://doi.org/10.1111/sapm.70266","url":null,"abstract":"","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70266","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"From Polynomial Lie Algebras \u0000 \u0000 \u0000 gl\u0000 ∞\u0000 \u0000 (\u0000 n\u0000 )\u0000 \u0000 \u0000 $mathfrak {gl}_{infty }^{(n)}$\u0000 and \u0000 \u0000 \u0000 so\u0000 ∞\u0000 \u0000 (\u0000 n\u0000 )\u0000 \u0000 \u0000 $mathfrak {so}_{infty }^{(n)}$\u0000 to Matrix Lax Structures","authors":"Huizhan Chen,&nbsp;Weiqi Peng","doi":"10.1111/sapm.70257","DOIUrl":"https://doi.org/10.1111/sapm.70257","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper presents a unified and systematic approach to constructing the modified Kadomtsev–Petviashvili (mKP), BKP, and modified BKP (mBKP) hierarchies through the representation theory of polynomial Lie algebras. Based on the polynomial Lie algebras <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>gl</mi>\u0000 <mi>∞</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$mathfrak {gl}_{infty }^{(n)}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>so</mi>\u0000 <mi>∞</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$mathfrak {so}_{infty }^{(n)}$</annotation>\u0000 </semantics></math>, we naturally derive the complete structure of these integrable systems, including bilinear identities, tau functions, wave matrices, dressing operators, and Lax pairs, emerge naturally without external constraints. A key result is the automatic emergence of the BKP constraint <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>L</mi>\u0000 <mtext>BKP</mtext>\u0000 <mo>∗</mo>\u0000 </msubsup>\u0000 <mo>=</mo>\u0000 <mo>−</mo>\u0000 <mi>∂</mi>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mtext>BKP</mtext>\u0000 </msub>\u0000 <msup>\u0000 <mi>∂</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathcal {L}^{*}_{text{BKP}}=-partial mathcal {L}_{text{BKP}}partial ^{-1}$</annotation>\u0000 </semantics></math> from <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>so</mi>\u0000 <mi>∞</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$mathfrak {so}_{infty }^{(n)}$</annotation>\u0000 </semantics></math> symmetry, contrasting with traditional approaches where such constraints are artificially imposed. The polynomial algebra framework provides a unifying principle where matrix structures arise intrinsically through fermionic Fock space decompositions, offering a powerful methodology for constructing and analyzing integrable hierarchies in higher dimensions.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-Hump Solutions of Singularly Perturbed Korteweg-de Vries Equation","authors":"Shengfu Deng,&nbsp;Shu-Ming Sun","doi":"10.1111/sapm.70258","DOIUrl":"https://doi.org/10.1111/sapm.70258","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;The paper studies a singularly perturbed Korteweg-de Vries (KdV) equation &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;η&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$epsilon eta _{xxxx} + eta _{xx} - lambda eta + beta _1 eta ^2 = 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$x in mathbb {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$epsilon &gt; 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a small parameter and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$lambda, beta _1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are real constants. The equation arises in the study of surface waves in finite-depth water with small surface tension on the free surface. It is known that the equation admits homoclinic solutions with small oscillatory tails at infinity, called generalized solitary-wave solutions (or generalized one-hump solutions). Here, the multi-hump solutions are formally constructed. First, two-hump solutions are derived by drawing on ideas and methods from rigorous existence studies of two-hump solutions for various equations, including the singularly perturbed KdV equation. Three-hump solutions are then formally obtained via a matching procedure, although a rigorous mathematical justification for the existence of three-hump or other multi-hump solutions remains open. Finally, the ideas and me","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence of Solitary-Wave-Type Solutions for the Delayed NKdV-2 Equation","authors":"Ling Pan,&nbsp;Zhijun Qiao,&nbsp;Shihui Zhu","doi":"10.1111/sapm.70256","DOIUrl":"https://doi.org/10.1111/sapm.70256","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we provide a rigorous analysis of solitary-wave-type solutions for the second member in the negative-order Korteweg-de Vries equations, called the NKdV-2 equation. This system represents a significant nonlocal generalization of the classical KdV hierarchy. The existence of solitary wave solutions for the transformed NKdV-2 system is established for local, nonlocal, and peakon-type nonsmooth spatial nonlocal convolution kernels (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>τ</mi>\u0000 </mrow>\u0000 </mfrac>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mfrac>\u0000 <mi>t</mi>\u0000 <mi>τ</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$f_3(x)=frac{1}{2tau }e^{-frac{t}{tau }}$</annotation>\u0000 </semantics></math>). To the best of our knowledge, this is the first systematic persistence analysis for the NKdV-2 traveling-wave system under such local and nonlocal perturbations, with the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>f</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <annotation>$f_3$</annotation>\u0000 </semantics></math>-case providing the most distinctive new result. Consequently, the original NKdV-2 equation admits solitary-wave-type solutions corresponding to these perturbed homoclinic structures. These works extend the usual solitary wave analysis to a class of nonlocal integrable systems as well as providing new insights into their dynamical behavior under delay perturbations.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spreading Speed for a Vector-Borne Disease System on Non-Coincident Straight Infinite Cylinders","authors":"Arnaud Ducrot,&nbsp;Hongliang Li,&nbsp;David Manceau","doi":"10.1111/sapm.70255","DOIUrl":"https://doi.org/10.1111/sapm.70255","url":null,"abstract":"<p>Vector-borne diseases remain an increasing global public health concern. In this work, we investigate the spreading speed of vector-borne disease via a four-component reaction–diffusion system posed on non-coincident straight infinite cylinders, which stands for an unconventional spatial configuration. We first establish the existence and uniqueness of a global solution by using the integrated semigroup theory, then by using duality arguments and a local maximum principle, we show the boundedness of this solution. For the long time behavior of the solutions, we derive a criteria condition for whether the disease dies out or spreads, governed by the sign of a suitable elliptic principal eigenvalue. In the case where spreading occurs, we determine an associated spreading speed by constructing appropriate sub- and super-solutions.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70255","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Role of Inducible Defence in Ecological Models: Effects of Nonlocal Intraspecific Competition","authors":"Sangeeta Saha,&nbsp;Swadesh Pal,&nbsp;Roderick Melnik","doi":"10.1111/sapm.70253","DOIUrl":"https://doi.org/10.1111/sapm.70253","url":null,"abstract":"<p>Phenotypic plasticity significantly influences species interactions, especially via inducible defensive mechanisms in predator–prey dynamics. This work proposes and explores a predator–prey scenario whereby the prey species demonstrates inducible defence mechanisms against predators. The model demonstrates a diverse array of complex dynamical characteristics, notably highlighting the significant stabilizing influence of defence on population dynamics. We further investigate the system under conditions of spatio-temporal diffusion inside a confined region, demonstrating that the Turing instability domain diminishes as the defence level escalates, thereby decreasing the probability of spatial pattern emergence. To achieve more realistic ecological interactions, the model is augmented by including a nonlocal factor into the intraspecific competition of the prey population. The study shows that inducible defence suppresses pattern creation in the local model, but adding nonlocal interactions changes this behavior considerably by making the Turing domain wider as the range of interactions gets expanded. These results show how important inducible defence and spatial linkages are for keeping ecosystems stable and diverse in space. In general, the results give new insight into how behavioral adaptations and nonlocal impacts work together to affect species coexistence, pattern development, and the robustness of ecological systems.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Modified Camassa–Holm Equation on the Half Line: A Riemann–Hilbert Approach","authors":"Iryna Karpenko,&nbsp;Dmitry Shepelsky","doi":"10.1111/sapm.70254","DOIUrl":"https://doi.org/10.1111/sapm.70254","url":null,"abstract":"&lt;p&gt;We consider the initial-boundary value (IBV) problem for the modified Camassa–Holm (mCH) equation &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;/mfenced&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$tilde{m}_t+{left((tilde{u}^2-tilde{u}_x^2+2tilde{u})tilde{m}right)}_x = 0, qquad tilde{m}:=tilde{u}-tilde{u}_{xx}+1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on the half-line &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;se","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"156 6","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70254","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}